Regularizing Fractional Brownian Motion with a View ...patrickc/webdiss.pdfRegularizing Fractional...

Diss. ETH No. 14051

Regularizing Fractional BrownianMotion with a View towards Stock

Price Modelling

A dissertation submitted to theSWISS FEDERAL INSTITUTE OF TECHNOLOGY

ZURICH

for the degree ofDoctor of Mathematics

presented byPATRICK CHERIDITO

Dipl. Math. ETHborn January 28, 1969citizen of Zurich ZH

accepted on the recommendation ofProf. Dr. F. Delbaen, examiner

Prof. Dr. P. Embrechts, co-examinerProf. Dr. A.N. Shiryaev, co-examiner

Prof. Dr. M. Yor, co-examiner

2001

Dank

Diese Doktorarbeit entstand unter der Leitung von Prof. Freddy Delbaen.Ich danke ihm fur alles, was er mir beigebracht hat, fur sein Interesse andieser Arbeit und fur die vielen Kontakte, die er mir ermoglicht hat. Prof.Paul Embrechts, Prof. Albert Shiryaev und Prof. Marc Yor danke ich furdie Ubernahme des Korreferats und die vielen Anmerkungen, die zu einerVerbesserung der Doktorarbeit gefuhrt haben.

Sehr wichtig fur mich war auch der Gedankenaustauch mit weiteren Mitar-beitern der ETH. Markus Melenk war immer bereit,uber mathematische Prob-leme zu diskutieren, und Prof. Eugene Trubowitz hat mir gezeigt, wie mathe-matische Physiker Integralgleichungen losen. Chantal Buteau, Giovanni Gen-tile, Ivan Jecic, Sven Jossen, Franz Muller, Dominik Schotzau und MichaelStuder verdanke ich viel Freude wahrend und neben der Arbeit.

Die Credit Suisse hat einen Teil dieser Arbeit gesponsert.

iii

Abstract

There have been several attempts to remedy some of the shortcomings of theSamuelson model for stock price movements using fractional Brownian mo-tion.

In the first part of this thesis we construct arbitrage strategies for two dif-ferent models based on fractional Brownian motion and show how arbitragecan be ruled out by putting restrictions on the trading strategies. Since thesemodels with the restricted trading strategies are incomplete, it is not clear howto price derivatives within them.

Alternatively, arbitrage can be excluded from fractional Brownian motionmodels by regularizing the local path behaviour of fractional Brownian mo-tion. We introduce two different ways of regularizing fractional Brownianmotion and discuss the pricing of a European call option in regularized frac-tional Samuelson models.

v

Contents

1 Preliminaries 1

1.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Fractional Brownian motion . . . . . . . . . . . . . . . . . . 1

1.3 Weak semimartingales . . . . . . . . . . . . . . . . . . . . . 12

1.4 The market . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2 Arbitrage in fractional Brownian motion models 23

2.1 Introduction . . . .. . . . . . . . . . . . . . . . . . . . . . . 23

2.2 The trading strategies . . . . . . . . . . . . . . . . . . . . . . 25

2.3 Construction of arbitrage . . . . . . . . . . . . . . . . . . . . 31

2.4 Exclusion of arbitrage . . . . . . . . . . . . . . . . . . . . . . 39

3 Regularized fractional Brownian motion and option pricing 49

3.1 Introduction . . . .. . . . . . . . . . . . . . . . . . . . . . . 49

3.2 Regularizing fractional Brownian motion . . . . . . . . . . . 50

3.2.1 General idea . . . . . . . . . . . . . . . . . . . . . . 50

3.2.2 Rϕ and its semimartingale decomposition . . . . . . . 56

3.2.3 Equivalence of(

1ϕ(0) Rϕ

)t∈[0,T ] to Brownian motion . 58

vii

viii Contents

3.3 Option pricing with regularized fractional Brownian motion . 63

3.3.1 Naive option pricing in regularized fractional Samuel-son models . . . . . . . . . . . . . . . . . . . . . . . 64

3.3.2 Discussion . . . . . . . . . . . . . . . . . . . . . . . 66

4 Mixed fractional Brownian motion 73

4.1 Introduction . . . .. . . . . . . . . . . . . . . . . . . . . . . 73

4.2 Proof of Theorem 4.2 forH ∈(0, 1

2

). . . . . . . . . . . . . 75

4.3 Proof of Theorem 4.2 forH ∈ (12,

34] . . . . . . . . . . . . . . 77

4.4 Proof of Theorem 4.2 forH ∈ (34, 1] . . . . . . . . . . . . . . 82

4.5 Option pricing with mixed fractional Brownian motion . . . . 86

4.6 Representations of Gaussian processes that are equivalent toBrownian motion . . . . . . . . . . . . . . . . . . . . . . . . 88

4.6.1 The representations of Shepp and Hitsuda . . . . . . . 88

4.6.2 The Girsanov-Hitsuda representation and relations be-tween different representations . . . . . . . . . . . . . 96

4.7 The Shepp-represention of mixed fractional Brownian motion 99

4.8 The Hitsuda-representation of mixed fractional Brownian mo-tion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

Bibliography 119

Chapter 1

Preliminaries

1.1 Notation

Throughout this thesis(�,A, P) will be a probability space.Let I ⊂ IR be an interval and(Xt )t∈I a stochastic process. We callX

continuous, right-continuous or cadlag (continua droit, limitesa gauche) if allpaths have the corresponding property. If almost all paths have the property,we call X a.s. continuous, a.s. right-continuous or a.s. cadlag. We sayX isstochastically right-continuous if for allt ∈ I \ {supI }, lims↘t Xs = Xt inprobability.

By lFX we denote the filtration generated byX, i.e. lFX = (F X

t

)t∈I , where

F Xt := σ (Xs : s ∈ I , s ≤ t) , t ∈ I .

Let T ∈ (0,∞). We say that a filtrationlF = (Ft )t∈[0,T] satisfies theusual assumptions if it is right-continuous,FT is complete andF0 containsall null sets ofFT . If lF = (Ft )t∈[0,T ] is an arbitrary filtration, we denote bylF = (

Ft)t∈[0,T ] the smallest filtration that containslF and satisfies the usual

assumptions.

1.2 Fractional Brownian motion

Definition 1.1 A fractional Brownian motion with Hurst parameterH ∈ (0, 1], is a continuous, centred Gaussian process

(BH

t

)t∈IR with

Cov(

BHt , BH

s

)= 1

2

(|t |2H + |s|2H − |t − s|2H

), t, s ∈ IR . (1.2.1)

1

2 Chapter 1. Preliminaries

These processes were first studied by Kolmogorov (1940) within a Hilbertspace framework. ForH = 1, fractional Brownian motion can be constructedas follows:

B1t = tξ , t ∈ IR , (1.2.2)

whereξ is a standard normal random variable. ForH = 12, fractional Brow-

nian motion is a two-sided Brownian motion. It can be constructed by takingtwo independent one-sided Brownian motions

(W1

t

)t≥0,

(W2

t

)t≥0 and setting

B12t =

{W1

t if t ≥ 0W2−t if t < 0

.

For H ∈ (0, 1), Mandelbrot and Van Ness (1968) gave the following construc-tion of fractional Brownian motion:

BHt = cH

∫IR

[ϕH (t − s)− ϕH (−s)] dWs , t ∈ IR , (1.2.3)

where(Ws)s∈IR is a two-sided Brownian motion,

ϕH (x) = 1{x≥0}xH− 12 , x ∈ IR , (1.2.4)

andcH is a normalizing constant. IfH = 12, it is clear that for allt ∈ IR,∫

IR[ϕH (t − s)− ϕH (−s)] dWs = Wt .

For H ∈(0, 1

2

)∪(

12, 1

), the integrals

∫IR

[ϕH (t − s)− ϕH (−s)] dWs , t ∈ IR

can be understood asL2-limits or almost sure-limits.In order to define

∫IR [ϕH (t − s)− ϕH (−s)] dWs for every t ∈ IR in the L2-

sense, we define for step-functions

f =n−1∑k=0

ak1(tk,tk+1] ,

wherea0, . . . , an−1 ∈ IR and−∞ < t0 < · · · < tn < ∞,

L2-∫

IRf (s)dWs :=

n−1∑k=0

ak(Wtk+1 − Wtk

).

1.2. Fractional Brownian motion 3

1

2

–5 –4 –3 –2 –1 1 2 3

s

1

2

–5 –4 –3 –2 –1 1 2 3

s

Figure 1.1: Left: The functionsϕH (t − s) andϕH (−s) for H = 34 and t = 3.

Right: The functionsϕH (t − s) andϕH (−s) for H = 14 and t = 3.

Since the step-functions are dense inL2(IR) and

E

[(L2-∫

IRf (s)dWs

)2]

=∫

IRf 2(s)ds

for all step-functionsf , L2-∫

IR can be extended continuously to a linear, norm-preserving mapping fromL2(IR) to L2(�).

YHt = L2-

∫IR

[ϕH (t − s)− ϕH (−s)] dWs

is then for allt ∈ IR, an L2-limit of linear combinations of random variablesfrom {Wt : t ∈ IR}. Hence,

(YH

t

)t∈IR is a centred Gaussian process. It is easy

to see that it has stationary increments. Furthermore, we obtain fort ≥ 0,

Var(YH

t

)=

∫ 0

−∞

[(t − s)H− 1

2 − (−s)H− 12

]2ds+

∫ t

0(t − s)2H−1ds

= t2H(∫ ∞

0

[(1 + x)H− 1

2 − xH− 12

]2dx + 1

2H

).

It follows that for allt, s ∈ IR,

Cov(YH

t ,YHs

)= 1

2

[Var

(YH

t

)+ Var

(YH

s

)− Var

(YH

t − YHs

)]= 1

2

[Var

(YH

t

)+ Var

(YH

s

)− Var

(YH

t−s

)]= 1

2

[Var

(YH|t|

)+ Var

(YH|s|

)− Var

(YH|t−s|

)]


=(∫ ∞

0

[(1 + x)H− 1

2 − xH− 12

]2dx + 1

2H

)1

2

(|t |2 + |s|2 − |t − s|2

).

Hence, for

cH =(∫ ∞

0

[(1 + x)H− 1

2 − xH− 12

]2 + 1

2H

)− 12

,

(cH YH

t

)t∈IR is a centred Gaussian process with covariance (1.2.1) (In contrast

to this, Mandelbrot and Van Ness (1968) chose to setcH = 0(H + 12)

−1).We could now apply the Kolmogorov-Centsov Theorem (compare Theorem2.2.8 of Karatzas and Shreve (1988)) to obtain a continuous modification of(cH YH

t

)t∈IR. But we can also prove that

(cH YH

t

)t∈IR has a continuous mod-

ification by showing that almost surely,∫

IR [ϕH (t − s)− ϕH (−s)] dWs canbe understood as an improper Riemann-Stieltjes integral for allt ∈ IR. Toprove this we need the following lemma, which follows from Theorem 2.21of Wheeden and Zygmund (1977) and Remark 2 on page 23 of the same book.

Lemma 1.2 Let [a, b] be a finite interval, f ∈ C[a, b] and φ ∈ C1[a, b].Then the Riemann-Stieltjes integralRS-

∫ ba φ(s)d f (s) exists and equals

−R-∫ b

af (s)φ′(s)ds+ f (b)φ(b)− f (a)φ(a) ,

whereR-∫ b

a f (s)φ′(s)ds is the Riemann integral.

Proposition 1.3 Let H ∈ (0, 12) ∪ (1

2, 1), and let (Wt)t∈IR be a two-sidedBrownian motion. Then there exists a measurable� ⊂ � with P[�] = 1such that for eachω ∈ � the improper Riemann-Stieltjes integral

iRS-∫

IR[ϕH (t − s)− ϕH (−s)] dWs(ω)

exists for all t∈ IR and is continuous in t . If we set

ZHt (ω) :=

{iRS-

∫IR [ϕH (t − s)− ϕH (−s)] dWs(ω) if ω ∈ �

0 if ω ∈ �c ,

(ZH

t

)t∈IR is a continuous modification of

(YH

t

)t∈IR. Hence,

(cH ZH

t

)t∈IR is a

fractional Brownian motion.


Proof. It follows from the law of the iterated logarithm (see e.g. Theorem2.9.24 of Karatzas and Shreve (1988)) that there exists a measurable set�0 ⊂� with P[�0] = 1 such that for allω ∈ �0,

limt→∞

W−t(ω)√t log t

= 0 (1.2.5)

Furthermore, it follows from Theorem 2.9.25 of Karatzas and Shreve (1988)that for all n ∈ IN, there exists a measurable set�n ⊂ � with P[�n] = 1such that for allω ∈ �n and allt ∈ [−n, n],

lims→t

Wt(ω)− Ws(ω)√|t − s| log(

1|t−s|

) = 0 (1.2.6)

We set� = ⋂∞n=0�n. It is clear thatP[�] = 1. We assumet > 0. Fort ≤ 0,

the proof is analogous. Let us first treat the caseH ∈(

12, 1

). It follows from

Lemma 1.2 that for eachω ∈ � and allx ∈ (0, t),

RS-∫ x

0ϕH (t − s)dWs(ω)

= (H − 1

2)R-∫ x

0Ws(ω)(t − s)H− 3

2 ds+ (t − x)H− 12 Wx(ω)

Since limx↗t (t − x)H− 12 Wx(ω) = 0 and the improper Riemann integral

iR-∫ t

0Ws(ω)(t − s)H− 3

2 ds = limx↗t

R-∫ x

0Ws(ω)(t − s)H− 3

2 ds

exists, the improper Riemann-Stieltjes integral

iRS-∫ t

0ϕH (t − s)dWs(ω) = lim

x↗tRS-∫ x


exists too and equals

(H − 1

2)iR-

∫ t

0Ws(ω)(t − s)H− 3

2 ds. (1.2.7)

To show that (1.2.7) is continuous int we set fort > 0,

f tω(s) = 1[0,t](s)Ws(ω)(t − s)H− 3

2 , s ∈ IR ,


and observe that for allT > 0, the family(

f tω

)t∈(0,T) is uniformly integrable

with respect to Lebesgue measure. Therefore, thet-continuity of (1.2.7) fol-lows from a generalized version of Lebesgue’s Dominated Convergence The-orem (see e.g. Theorem II.6.4.b of Shiryaev (1984)). Lemma 1.2 implies thatfor all x > 0,

RS-∫ − 1

x

−x[ϕH (t − s)− ϕH (−s)] dWs(ω)

= (H − 1

2)R-∫ − 1

x

−xWs(ω)

[(t − s)H− 3

2 − (−s)H− 32

]ds

+[(t + 1

x)H− 1

2 − (1

x)H− 1

2

]W− 1

x(ω)−

[(t + x)H− 1

2 − xH− 12

]W−x(ω)

It follows from (1.2.6) that

limx→∞

[(t + 1

x)H− 1

2 − (1

x)H− 1

2

]W− 1

x(ω) = 0 .

Moreover, for allx > 0,∣∣∣(t + x)H− 12 − xH− 1

2

∣∣∣ ≤ t (H − 1

2)xH− 3

2 .

This together with (1.2.5) implies that

limx→∞

[(t + x)H− 1

2 − xH− 12

]W−x(ω) = 0 .

Furthermore, it follows from (1.2.5) that the improper Riemann integral

iR-∫ 0

−∞Ws(ω)

[(t − s)H− 3

2 − (−s)H− 12

]ds

= limx→∞ R-

∫ − 1x

−xWs(ω)

[(t − s)H− 3

2 − (−s)H− 32

]ds

exists. Hence, the improper Riemann-Stieltjes integral

iRS-∫ 0

−∞[ϕH (t − s)− ϕH (−s)] dWs(ω)

= limx→∞ RS-

∫ − 1x




(H − 1

2)iR-∫ 0

−∞Ws(ω)

[(t − s)H− 3

2 − (−s)H− 32

]ds,

which is continuous int by Lebegue’s Dominated Convergence Theorem.

iRS-∫ t

−∞[ϕH (t − s)− ϕH (−s)] dWs(ω)

can now be defined as

iRS-∫ 0

−∞[ϕH (t − s)− ϕH (−s)] dWs(ω)+ iRS-

∫ t

0ϕH (t − s)dWs(ω) .

It is continuous int because

iRS-∫ t

0ϕH (t − s)dWs(ω) and iRS-

∫ 0

−∞[ϕH (t − s)− ϕH (−s)] dWs(ω)

are.Also for H ∈

(0, 1

2

), we define

iRS-∫ t

−∞[ϕH (t − s)− ϕH (−s)] dWs(ω)

as

iRS-∫ 0

−∞[ϕH (t − s)− ϕH (−s)] dWs(ω)+ iRS-

∫ t

0ϕH (t − s)dWs(ω) .

It can be deduced from Lemma 1.2 that for eachω ∈ � and allx ∈ (0, t),

RS-∫ x

0ϕH (t − s)dWs(ω) = RS-

∫ x

0(t − s)H− 1

2 d(Ws(ω)− Wt (ω))

= (H − 1

2)R-∫ x

0(Ws(ω)− Wt(ω))(t − s)H− 3

2 ds

+(t − x)H− 12 (Wx(ω)− Wt (ω))+ t H− 1

2 Wt (ω) .


(t − x)H− 12 (Wx(ω)− Wt (ω))

(x↗t)−→ 0 ,


and that the improper Riemann integral

iR-∫ t

0(Ws(ω)− Wt (ω))(t − s)H− 3

2 ds

= limx↗t

R-∫ x

0(Ws(ω)− Wt(ω))(t − s)H− 3

2 ds

exists. Therefore the improper Riemann-Stieltjes integral

iRS-∫ t

0ϕH (t − s)dWs(ω) = lim

x↗tRS-∫ x



(H − 1

2)iR-

∫ t

0(Ws(ω)− Wt (ω))(t − s)H− 3

2 ds+ t H− 12 Wt(ω) ,

Thatt H− 12 Wt (ω) is continuous int is clear. Thet-continuity of

iR-∫ t

0(Ws(ω)− Wt (ω))(t − s)H− 3

2 ds

can as before be derived from Theorem II.6.4.b of Shiryaev (1984). As in thecaseH ∈ (1

2, 1), we have for allx > 0,

RS-∫ − 1

x


= (H − 1

2)R-∫ − 1

x

−xWs(ω)

[(t − s)H− 3

2 − (−s)H− 32

]ds

+[(t + 1

x)H− 1

2 − (1

x)H− 1

2

]W− 1

x(ω)−

[(t + x)H− 1

2 − xH− 12

]W−x(ω)

As before,

limx→∞

[(t + 1

x)H− 1

2 − (1

x)H− 1

2

]W− 1

x(ω) = 0 ,

limx→∞

[(t + x)H− 1

2 − xH− 12

]W−x(ω) = 0

and the improper Riemann integral

iR-∫ 0

−∞Ws(ω)

[(t − s)H− 3

2 − (−s)H− 32

]ds


= limx→∞ R-

∫ − 1x

−xWs(ω)

[(t − s)H− 3

2 − (−s)H− 32

]ds

exists. Hence, the improper Riemann-Stieltjes integral

iRS-∫ 0

−∞[ϕH (t − s)− ϕH (−s)] dWs(ω)

= limx→∞ RS-

∫ − 1x



(H − 1

2)iR-∫ 0

−∞Ws(ω)

[(t − s)H− 3

2 − (−s)H− 32

]ds,

which is continuous int by Lebegue’s Dominated Convergence Theorem. Toshow that

(ZH

t

)t∈IR is a modification of

(YH

t

)t∈IR we set for allt ∈ IR,

f H,t(s) = ϕH (t − s)− ϕH (−s) , s ∈ IR ,

and for alln ∈ IN ands ∈ IR,

f H,tn (s) =

n2∑k=−n2

f H,t

(k + 1

2

n

)1( k

n ,k+1

n ](s) .

Since limn→∞ f H,tn = f H,t in L2, YH

t = L2-limn→∞∫

IR f H,tn (s)dWs. At

the same time

ZHt (ω) = lim

n→∞

∫IR

f H,tn (s)dWs(ω)

for all ω ∈ �. Hence, for allt ∈ IR, ZHt is measurable andZH

t = YHt almost

surely. 2

It can be deduced from (1.2.1) that fractional Brownian motions divideinto three different families.B

12 has independent increments. ForH ∈ (1

2, 1],the covariance between two increments over non-overlapping time-intervals ispositive, forH ∈ (0, 1

2) it is negative.From the representations (1.2.2) and (1.2.3) it can be seen that fractional

Brownian motion has stationary increments. Furthermore, it can easily bechecked thatBH is stochastically self-similar with self-similarity parameterH , i.e. for alla > 0,(

aH BHta

)t∈IR

has the same distribution as(

BHt

)t∈IR

.


Figure 1.2: Simulation of a typical path of fractional Brownian motion forH= 0.1, H=0.5 and H=0.8

Let (Xt )t≥0 be a stochastic process with stationary increments. We say thatthe increments ofX exhibit long-range dependence if for allh > 0,

∞∑n=1

∣∣Cov(Xh − X0, Xnh − X(n−1)h

)∣∣ = ∞ .

It can be derived from (1.2.1) that forH ∈ (0, 12) ∪ (1

2, 1] and fixedh > 0,

limt→∞

Cov(BH

h , BHt+h − BH

t

)t2(H−1)

= H (2H − 1)h2 .

This implies that the increments of(BH

t

)t≥0 exhibit long-range dependence if

and only if H ∈ (12, 1).

In the following lemma we collect some facts about fractional Brownianmotion that we will need throughout the thesis. They are already well-known.

Lemma 1.4 Let BH be a fractional Brownian motion for some H∈ (0, 1],and T, p,q > 0. Then:


a) For all γ < H there exist a constantδ and an almost every-where positive random variableξ such that

P

[ω : sup t,u∈[0,T];

0<t−u<ξ(ω)

∣∣BHt (ω)−BH

u (ω)∣∣

(t−u)γ ≤ δ

]= 1

b) npH−1∑n−1j =0

∣∣∣∣BH( j +1)

n T− BH

jn T

∣∣∣∣p(n→∞)−→ E

[∣∣BHT

∣∣p] in L1

c) npH−1−q ∑n−1j =0

∣∣∣∣BH( j +1)

n T− BH

jn T

∣∣∣∣p(n→∞)−→ 0 in L1

d) npH−1+q ∑n−1j =0

∣∣∣∣BH( j +1)

n T− BH

jn T

∣∣∣∣p(n→∞)−→ ∞ in probability,

i.e. for all L > 0, there exists an n0 such that for all n≥ n0 ,

P

[npH−1+q ∑n−1

j =0

∣∣∣∣BH( j +1)

n T− BH

jn T

∣∣∣∣p

< L

]< 1

L

Proof. a) follows from the Kolmogorov-Centsov Theorem (see e.g. Theorem2.2.8 of Karatzas and Shreve (1988)).

To prove b) we recall that the sequence(

BH( j +1)T − BH

jT

)∞j =0

is stationary.

Since it is Gaussian and

Cov(

BHT − BH

0 , BH( j +1)T − BH

jT

)( j →∞)−→ 0 ,

it is also mixing. Hence, the Ergodic theorem (see e.g. Theorem V.3.3 ofShiryaev (1984)) implies

1

n

n−1∑j =0

∣∣∣BH( j +1)T − BH

jT )

∣∣∣p (n→∞)−→ E[∣∣∣BH

T

∣∣∣p] in L1 . (1.2.8)

On the other hand, it follows from the self-similarity ofBH that for alln,

npH−1n−1∑j =0

∣∣∣∣BH( j +1)

n T− BH

jn T

∣∣∣∣p

has the same distribution as

1

n

n−1∑j =0

∣∣∣BH( j +1)T − BH

jT )

∣∣∣p .This together with (1.2.8) implies b).c) follows immediately from b).


To prove d) we chooseL > 0. It follows from b) that there exists ann1 ∈ INsuch that

P

∣∣∣∣∣∣E[∣∣∣BH

T

∣∣∣p]− npH−1n−1∑j =0

∣∣∣∣BH( j +1)

n T− BH

jn T

∣∣∣∣p∣∣∣∣∣∣ >

1

2E[∣∣∣BH

T

∣∣∣p] < 1

L

for all n ≥ n1. This implies that for alln ≥ n1,

P

npH−1

n−1∑j =0

∣∣∣∣BH( j +1)

n T− BH

jn T

∣∣∣∣p

<1

2E[∣∣∣BH

T

∣∣∣p] < 1

L

or, equivalently,

P

npH−1+q

n−1∑j =0

∣∣∣∣BH( j +1)

n T− BH

jn T

∣∣∣∣p

< nq 1

2E[∣∣∣BH

T

∣∣∣p] < 1

L.

This shows that there exists ann0 ∈ IN such that

P

npH−1+q

n−1∑j =0

∣∣∣∣BH( j +1)

n T− BH

jn T

∣∣∣∣p

< L

< 1

Lfor all n ≥ n0 ,

and d) is proved. 2

1.3 Weak semimartingales

The classical notion of a semimartingale stands at the end of a chain of gener-alizations of Brownian motion, each of which extended the class of stochasticprocesses that can play the role of the integrator in stochastic integration inthe Ito-sense (see Ito (1944) for Ito’s construction of the stochastic integral).It reached its final form in Doleans-Dade and Meyer (1970). In their paper astochastic process(Xt ) that is adapted to a filtrationlF = (Ft ) satisfying theusual assumptions is called anlF-semimartingale if it admits a decompositionof the form

Xt = X0 + Mt + At , (1.3.1)

where X0 is anF0-measurable random variable,M0 = A0 = 0, M is ana.s. right-continuous local martingale with respect tolF and A an a.s. right-continuous,lF-adapted finite variation process. Later it was found that if for

1.3. Weak semimartingales 13

T ∈ (0,∞), a filtration lF = (F )t∈[0,T] satisfies the usual assumptions, ana.s. right-continuous,lF-adapted stochastic process(Xt )t∈[0,T] is of the form(1.3.1) if and only ifX fulfils the following condition:

IX (β (lF)) is bounded inL0 , (1.3.2)

where

β (lF) =

n−1∑j =0

gj 1(t j ,t j +1] : n ∈ IN, 0 ≤ t0 < · · · < tn ≤ T,

∀ j , gj is Ft j -measurable and∣∣gj∣∣ ≤ 1 a.s.

}(1.3.3)

and

IX(ϑ) =n−1∑j =0

gj(Xt j +1 − Xt j

)for ϑ =

n−1∑j =0

gj 1(t j ,t j +1] ∈ β (lF) .

This result is usually referred to as the Bichteler-Dellacherie theorem (seee.g. Section VIII.4 of Dellacherie and Meyer (1980) for a proof). For ourpurposes it is more convenient to work with condition (1.3.2) than with thedecomposition property (1.3.1). If one does not require the process to be a.s.right-continuous and the filtration to satisfy the usual assumptions, one obtainsa weaker form of the semimartingale property than the classical one.

Definition 1.5 A stochastic process(Xt )t∈[0,T ] is a weak semimartingale withrespect to a filtrationlF = (Ft )t∈[0,T] if X is lF-adapted and satisfies (1.3.2).

Let (Xt )t∈[0,T] be a stochastic process. IflF1 = (F 1

t

)t∈[0,T ] and lF2 =(

F 2t

)t∈[0,T ] are two filtrations withF 1

t ⊂ F 2t for all t ∈ [0, T], thenβ

(lF1) ⊂

β(lF2). Hence,L0-boundedness ofIX

(β(lF2)) implies L0-boundedness of

IX(β(lF1)). This shows that ifX is not a weak semimartingale with respect

to the filtration generated byX, then it is not a weak semimartingale withrespect to any other filtration. Therefore it is natural to introduce the followingdefinition.

Definition 1.6 Let(Xt )t∈[0,T] be a stochastic process. We call X a weak semi-martingale if it is a weak semimartingale with respect tolFX. We call X a

semimartingale if it is a semimartingale with respect tolFX

.


Example 1.7 It is easy to see that the deterministic process

Xt ={

0 for t ∈ [0, 1]1 for t ∈ (1, 2] ,

is a weak semimartingale. But it is not a semimartingale because it is not a.s.right-continuous.

However, the following proposition shows that every a.s right-continuouslF-weak semimartingale is also anlF-semimartingale.

Proposition 1.8 Let lF = (Ft )t∈[0,T] be a filtration. Then every stochasticallyright-continuouslF-weak semimartingale is also anlF-weak semimartingale.In particular, if X is a.s. right-continuous, it is anlF-semimartingale.

Proof. DefinelF0 = (F 0

t

)t∈[0,T ] as follows: LetF 0

T be the completion ofFT ,

N the null sets ofF 0T and set

F 0t = σ (Ft ∪ N ) , t ∈ [0, T ] .

Let t ∈ [0, T ] andg ∈ L0(F 0

t

)such that|g| ≤ 1 almost surely. We set

A = {g > E[g|Ft ]} and B = {g < E[g|Ft ]} .Since

F 0t = {G ⊂ � : ∃F ∈ Ft such thatG4F ∈ N } ,

there existA, B ∈ Ft with A4 A, B4B ∈ N . The equalities∫A

g − E[g|Ft ] d P =∫

Ag − E[g|Ft ] d P = 0

and ∫B

g − E[g|Ft ] d P =∫

Bg − E[g|Ft ] d P = 0

imply P [ A] = P [B] = 0. Hence,

g = E[g|Ft ] almost surely. (1.3.4)

Let (Xt )t∈[0,T] be anlF-weak semimartingale. It follows from (1.3.4) that foreveryϑ ∈ β (lF0) there exists aϑ ∈ β (lF)with IX(ϑ) = IX (ϑ) almost surely.Therefore,

IX (β (lF)) = IX

(β(lF0))

in L0 .


This shows thatX is also anlF0-weak semimartingale.Now let

γ =n−1∑j =0

gj 1(t j ,t j +1] ∈ β (lF) .For all t ∈ [0, T ],

Ft =⋂s>t

F 0s∧T .

Therefore,

γ ε =n−1∑j =0

gj 1(t j +ε,t j +1] is in β(lF0)

(1.3.5)

for all ε with 0 < ε < min j(t j +1 − t j

). If (Xt )t∈[0,T ] is stochastically right-

continuous, then

limε↘0

IX(γ ε) = IX (γ ) in probability.

This, together with (1.3.5) and the fact thatIX(β(lF0)) is bounded inL0,

implies thatIX(β(lF))

is also bounded inL0, and thereforeX is anlF-weaksemimartingale. 2

It follows from Lemma 1.4 d) that forH ∈(0, 1

2

), BH has infinite

quadratic variation. The next proposition shows that this implies thatBH

cannot be a weak semimartingale ifH ∈(0, 1

2

).

Proposition 1.9 Let(Xt)t∈[0,T ] be an a.s. cadlag process and denote byτ theset of all finite partitions

0 = t0 < t1 < · · · < tn = T , n ∈ IN ,

of [0, T ]. If

n−1∑j =0

(Xt j +1 − Xt j

)2 : (t0, t1, . . . , tn) ∈ τ

is unbounded in L0, then X is not a weak semimartingale.


Proof. To simplify calculations we defineYt = Xt − X0, t ∈ [0, T ]. Then(Yt )t∈[0,T ] is anlFX-adapted, a.s. cadlag process withY0 = 0. It is clear thatIY = IX and

n−1∑j =0

(Yt j +1 − Yt j

)2 =n−1∑j =0

(Xt j +1 − Xt j

)2for all partitions

(t0, t1, . . . , tn) ∈ τ .To prove the lemma we must show thatIY

(β(lFX)) is unbounded inL0. The

key ingredient in our derivation of this from theL0-unboundedness of

n−1∑j =0

(Yt j +1 − Yt j

)2 : (t0, t1, . . . , tn) ∈ τ

is the equality

n−1∑j =0

(Yt j +1 − Yt j

)2 = Y2T − 2

n−1∑j =1

Yt j

(Yt j +1 − Yt j

), (1.3.6)

which holds for all partitions

(t0, t1, . . . , tn) ∈ τ .

That

n−1∑j =0

(Yt j +1 − Yt j

)2 : (t0, t1, . . . , tn) ∈ τ

is unbounded inL0 means that

c := limL→∞ sup

τP

n−1∑

j =0

(Yt j +1 − Yt j

)2> L

> 0 . (1.3.7)

We will deduce from this that

limL→∞ sup

ϑ∈β(lFX

) P [|IX (ϑ)| > L] ≥ c

4, (1.3.8)


which impliesL0-unboundedness ofIY(β(lFX)). To do this we chooseL >

0. SinceY is a.s. cadlag, supt∈[0,T ] |Yt | < ∞ almost surely. Therefore thereexists anN > 0 such that

P

[sup

t∈[0,T]|Yt | > N

]<

c

4. (1.3.9)

(1.3.7) implies that there exists a partition

(t0, t1, . . . , tn) ∈ τ

with

P

n−1∑

j =0

(Yt j +1 − Yt j

)2> 2L N + N2

> c

2. (1.3.10)

It follows from (1.3.9) and (1.3.10) that

P

{

supt∈[0,T]

|Yt | > N

}∪

n−1∑j =0

(Yt j +1 − Yt j

)2 ≤ 2L N + N2

≤ P

[sup

t∈[0,T]|Yt | > N

]+ P

n−1∑

j =0

(Yt j +1 − Yt j

)2 ≤ 2L N + N2

< 1 − c

4.

Hence,

P

{ sup

t∈[0,T ]|Yt | ≤ N

}∩

n−1∑j =0

(Yt j +1 − Yt j

)2> 2L N + N2

> c

4.

(1.3.11)It is clear that

ϑ =n−1∑j =1

−1{∣∣∣Yt j

∣∣∣≤N}Yt j

N1(t j ,t j +1]

is in β(lFX) and it can be seen from (1.3.6) that on the event

{sup

t∈[0,T ]|Yt | ≤ N

}∩

n−1∑j =0

(Yt j +1 − Yt j

)2> 2L N + N2

,


we have

IY (ϑ) = 1

2N

n−1∑

j =0

(Yt j +1 − Yt j

)2 − Y2T

>1

2N

(2L N + N2 − N2

)= L .

Together with (1.3.11), this implies that

P [ IY (ϑ) > L] >c

4.

Since L was chosen arbitrarily, this shows (1.3.8), and the proposition isproved. 2

Corollary 1.10(BH

t

)t∈[0,T ] is not a weak semimartingale if H∈

(0, 1

2

).

Proof. It follows from Lemma 1.4 d) that

n−1∑j =0

(BH( j +1)

n T− BH

jn T

)2(n→∞)−→ ∞ in probability.

This implies that

n−1∑j =0

(BH( j +1)

n T− BH

jn T

)2

: n ∈ IN

is unbounded inL0. Since BH is continuous, the corollary follows fromProposition 1.9. 2

For H ∈(

12, 1

), a direct proof of the fact that

(BH

t

)t∈[0,T] is not a weak

semimartingale seems to be difficult. But Proposition 1.8 permits us to usealready existing results on classical semimartingales.

Proposition 1.11 Let (Xt )t∈[0,T] be an a.s. right-continuous process suchthat

P[(Xt )t∈[0,T] is of finite variation

]< 1 (1.3.12)

and, for allε > 0, there exists a partition

0 = t0 < t1 < · · · < tn = T , n ∈ IN ,


withmax

0≤ j ≤n−1

(t j +1 − t j

)< ε (1.3.13)

and

P

n−1∑

j =0

(Xt j +1 − Xt j

)2> ε

< ε . (1.3.14)

Then X is not a weak semimartingale.

Proof. SupposeX is a weak semimartingale. By Proposition 1.8,X is also an

lFX

-semimartingale. Hence,X is of the form

Xt = X0 + Mt + At ,

where X0 is an F0-measurable random variable,M0 = A0 = 0, M is ana.s. right-continuous local martingale with respect tolF and A an a.s. right-continuous,lF-adapted finite variation process. It follows from (1.3.13),(1.3.14) and Theorem II.22 of Protter (1990) that

[X, X]t = X0 a.s., t ∈ [0, T ] .Hence,

[M,M ]t = 0 a.s., t ∈ [0, T ] .Therefore, Theorem II.27 of Protter (1990) impliesMt = 0 a.s., t ∈ [0, T ].Hence,X is a finite variation process. This contradicts (1.3.12). ThereforeXcannot be a weak semimartingale. 2

Corollary 1.12(BH

t

)t∈[0,T ] is not a weak semimartingale if H∈

(12, 1

).

Proof. It follows from Lemma 1.4 d) that

n−1∑j =0

∣∣∣∣BH( j +1)

n T− BH

jn T

∣∣∣∣ (n→∞)−→ ∞ in probability.

Therefore, there exists a sequence(nk)∞k=0 of natural numbers such that

nk−1∑j =0

∣∣∣∣BH( j +1)

nkT

− BHj

nkT

∣∣∣∣ (k→∞)−→ ∞ almost surely.


Hence,

P

[(BH

t

)t∈[0,T ] is of finite variation

]= 0 .

On the other hand, Lemma 1.4 c) shows that

n−1∑j =0

(BH( j +1)

n T− BH

T jn T

)2(n→∞)−→ 0 in L1 .

Hence,(BH

t

)t∈[0,T] satisfies the assumptions of Proposition 1.11. Therefore

it is not a weak semimartingale. 2

1.4 The market

Throughout this thesis we will consider a market that consists of a moneymarket account and a stock that pays no dividends. All economic activitytakes place in a time interval[0, T ] for someT ∈ (0,∞). Borrowing andshort-selling are allowed, the borrowing rate is equal to the lending rate, andit is possible to buy and sell any fraction of stock shares. Moreover, thereexist no transaction costs and stock shares can be bought and sold at the sameprice. We assume that money in the money market account evolves according

to a stochastic process(

S0t

)t∈[0,T ] and the stock price follows a stochastic

process(

St

)t∈[0,T ]. Since we want to useS0 as a numeraire, we require it

to be positive. ByS we denote the discounted stock priceS/S0. To makeclear how derivative prices depend on the explicit modelling of(S0, S), wewill analyse the price of a European call option on the stock. Such an optionis specified by its maturityT and the strike priceK . It has a random pay-offat timeT which is given by (

ST − K)+

.

The first continuous-time stochastic model for a financial asset appearedin the thesis of Bachelier (1900). He proposed modelling the price of a stockas follows:

St = S0 + µt + σ Bt ,

whereS0, µ andσ are constants andB is a Brownian motion. The drawbacksof this model are thatSt can become negative and the relative returns are lowerfor higher stock prices.

1.4. The market 21

Samuelson (1965) introduced the more realistic model

St = S0 exp

({µ− σ 2

2

}t + σ Bt

), (1.4.1)

where S0, µ and σ are constants andB is a Brownian motion. Black andScholes (1973) noticed that ifS is as in (1.4.1) and there is a constantr suchthat S0

t = exp(r t ), then the pay-off of a European call option onS can bereplicated by continuous trading inS0 and S, and they derived an explicitformula for the price of such an option. However, the Samuelson model alsohas deficiencies and up to now there have been many efforts to build bettermodels. Cutland et al. (1995) discuss the empirical evidence that suggeststhat long-range dependence should be accounted for when modelling stockprice movements and present a fractional version of the Samuelson model.

For constantsS0 > 0, ν, σ > 0 andr , we call

S0t = 1, St = S0 + νt + σ BH

t , t ∈ [0, T ] , (1.4.2)

the fractional Bachelier model and

S0t = exp(r t ), St = S0 exp

({r + ν} t + σ BH

t

), t ∈ [0, T ] , (1.4.3)

the fractional Samuelson model or, alternatively, the fractional Black-Scholesmodel.

Chapter 2

Arbitrage in fractionalBrownian motion models

2.1 Introduction

In Section 1.3 we showed that forH ∈(0, 1

2

)∪(

12, 1

),(BH

t

)t∈[0,T] is not

a weak semimartingale. In particular, it is not alFBH

-semimartingale, nei-ther is S = S/S0 in the models (1.4.2) and (1.4.3). Therefore, it followsimmediately from Theorem 7.2 of Delbaen and Schachermayer (1994) that(1.4.2) and (1.4.3) admit a “free lunch with vanishing risk” consisting of sim-

ple predictable integrands adapted tolFBH

. Rogers (1997), Shiryaev (1998)and Salopek (1998) even give arbitrage strategies for fractional Brownian mo-tion models.

Rogers (1997) constructs arbitrage for the fractional Bachelier model(1.4.2). His strategy consists of a combination of buy and hold strategies

and works for all Hurst parametersH ∈(0, 1

2

)∪(

12, 1

). However, as self-

similarity of S is essential for its construction, Rogers’ arbitrage only existsin the caseν = 0, i.e. St = S0 + σ BH

t . Moreover, Rogers modelsSt fort ∈ (−∞, 0] and to generate a profit on the time interval[−1, 0), his arbitrageneeds to know the whole history ofS from time−∞ until the present.

In Shiryaev (1998) only the caseH ∈(

12, 1

)is treated. An integral with

respect toBH is defined and it is indicated how it can be shown that for regular

23

24 Chapter 2. Arbitrage in fBm models

enough functionsF , the modified Ito formula

d F(t, BHt ) = ∂1F(t, BH

t )dt + ∂2F(t, BHt )d BH

t (2.1.1)

holds. Using this for the fractional Bachelier model (1.4.2) withH ∈(

12, 1

),

one can choose ac > 0 and set

ϑ0t = −c

(νt + σ BH

t

)2 − 2cS0

(νt + σ BH

t

), ϑ1

t = 2c(νt + σ BH

t

)to obtain

ϑ0t S0

t + ϑ1t St = ϑ0

0 S00 + ϑ1

0 S0 +∫ t

0ϑ1

udSu = c(νt + σ BH

t

)2.

Hence, if continuous adjustment of the portfolio is allowed,(ϑ0, ϑ1) is a self-financing arbitrage strategy for the fractional Bachelier model.

For the fractional Samuelson model (1.4.3) withH ∈(

12, 1

), one can set

for all c > 0,

ϑ0t = cS0

(1 − exp

(2νt + 2σ BH

t

)), ϑ1

t = 2c(exp

(νt + σ BH

t

)− 1

).


ϑ0t S0

t + ϑ1t St = ϑ0

0 S00 + ϑ1

0 S0 +∫ t

0ϑ0

udS0u +

∫ t

0ϑ1

udSu

= cS0 exp(r t )(exp

(νt + σ BH

t

)− 1

)2,

which shows that(ϑ0, ϑ1) is a self-financing arbitrage strategy for the frac-tional Samuelson model.

More generally, it is shown in Salopek (1998) that if a stochastic pro-cess(Xt )t≥0 is almost surely continuous and of boundedp-variation for somep < 2 (this is the case for the processesS0 andS in (1.4.2) and (1.4.3) when

H ∈(

12, 1

)), then for a real functionf on IR that is locally Lipschitz,t ≥ 0

and a sequence of partitions 0= tn0 < tn

1 < · · · < tnJ(n) = t, n ∈ IN with

limn→∞ max

j

∣∣∣tnj +1 − tn

j

∣∣∣ = 0 ,

the finite sumsJ(n)−1∑

j =0

f(

Xtnj

)(Xtn

j +1− Xtn

j

)

2.2. The trading strategies 25

almost surely converge to a limit∫ t

0 f (Xu)d Xu and

∫ t

0f (Xu)d Xu

a.s.= F(Xt )− F(X0) ,

whereF(x) = ∫ x0 f (u)du, x ∈ IR. This is used in Salopek (1998) to construct

a self-financing arbitrage strategy for two financial assetsX andY that are bothalmost surely continuous, of boundedp-variation for somep < 2 and suchthat Xt 6= Yt almost surely for allt .

In this chapter we construct arbitrage strategies for a class of fractional

Brownian motion models that contains (1.4.2) and (1.4.3) for allH ∈(0, 1

2

)∪(

12, 1

), and we show how arbitrage can be excluded from these models by

putting restrictions on the class of trading strategies.In Section 2 we define the notions of ’free lunch with vanishing risk’,

’arbitrage’ and ’strong arbitrage’. Then we introduce different classes of trad-ing strategies. In Section 3 we construct arbitrage strategies. As in the caseof Rogers (1997) our arbitrage strategies consist of combinations of buy andhold strategies. Therefore we need no integration theory for fractional Brow-nian motion. Moreover, to generate a profit on the time interval[0, T ], ourstrategies need only know the history ofS = S/S0 on [0, T ]. However, toperform these strategies it must be allowed to buy and sell within arbitrarilysmall time intervals. In Section 4 we show that arbitrage can be ruled outfrom models of the form (1.4.2) and (1.4.3) by introducing a minimal amountof timeh > 0 that must lie between two consecutive transactions.

2.2 The trading strategies

In this section the time interval is an arbitrary closed interval[a, b]. Moneycan be invested in a money market account where money grows according to

a positive stochastic process(

S0t

)t∈[a,b] and a stock whose price follows a

stochastic process(

St

)t∈[a,b]. A trading strategy is a pairϑ = (

ϑ0, ϑ1)

of

stochastic processes(ϑ0

t

)t∈[a,b] and

(ϑ1

t

)t∈[a,b]. ϑ

0t S0

t describes the money in

the money market account at timet andϑ1t the number of stock shares held at

time t . Hence, the evolution of the portfolio value of a strategyϑ is given by

Vϑt = ϑ0

t S0t + ϑ1

t St , t ∈ [a, b] .


We set

Vϑt = Vϑ

t

S0t

= ϑ0t + ϑ1

t St , t ∈ [a, b] .

Definition 2.1 Letξ be a[0,∞]-valued random variable with P[ξ > 0] > 0.

a) A sequence of trading strategies{ϑ(n)}∞n=1 is a ξ -FLVR (ξ -free lunchwith vanishing risk) if

limn→∞

(Vϑ(n)

b − Vϑ(n)a

)= ξ in probability

and

limn→∞

∥∥∥∥(Vϑ(n)b − Vϑ(n)

a

)−∥∥∥∥∞= 0 .

{ϑ(n)}∞n=1 is a FLVR if it is aξ ′-FLVR for some[0,∞]-valued randomvariableξ ′ with P[ξ ′ > 0] > 0.

b) A trading strategyϑ is a ξ -arbitrage if

Vϑb − Vϑ

a = ξ almost surely.

ϑ is an arbitrage if it is aξ ′-arbitrage for some[0,∞]-valued randomvariableξ ′ with P[ξ ′ > 0] > 0.

c) A trading strategyϑ is a strong arbitrage if there exists a constant c> 0such that

Vϑb − Vϑ

a ≥ c almost surely.

It is clear that we must put certain restrictions on a trading strategy to give itan economic meaning. First of all, trading strategies should only be based onavailable information. To describe the evolution of information we introducea family ofσ -algebraslF = (Ft )t∈[a,b]. We assume that at any timet ∈ [a, b],S0

t andSt can be observed and no information is lost over time. In other words,lF is a filtration and

F S0,St := σ

((S0

u

)u∈[0,t] ,

(Su

)u∈[0,t]

)⊂ Ft for all t ∈ [a, b] .

Note that

F St := σ

((Su)u∈[0,t]

) ⊂ F S0,St for all t ∈ [a, b] .

Furthermore, we requireS0 andS to be progressively measurable with respectto lF. This is in particular the case whenS0 andS are right-continuous, and it


ensures that for alllF-stopping timesτ , the stopped processes(

S0τ∧t

)t∈a,b

and(Sτ∧t

)t∈a,b

are also progressively measurable with respect tolF. To construct

arbitrage in fractional Brownian models of the form (1.4.2) or (1.4.3) it isenough to consider combinations of buy and hold strategies. We start ourdiscussion of different classes of combinations of buy and hold strategies byrecalling the definition of the classS(lF) of simple predictable integrands andintroducing the classaS(lF) of almost simple predictable integrands.

Definition 2.2

a) S(lF) := {g01{a} +∑n−1j =1 gj 1(τ j ,τ j +1] : n ≥ 2, a = τ1 ≤ · · · ≤ τn

= b; all τ j ’s are lF-stopping times; g0 is a real,Fa-measurable random variable; and the other gj ’s arereal, Fτ j -measurable random variables}

b) aS(lF) := {g01{a} +∑∞j =1 gj 1(τ j ,τ j +1] : a = τ1 ≤ τ2 ≤ · · · ≤ b;

all τ j ’s are lF-stopping times; g0 is a real,Fa−measurable random variable; the other gj ’s are real,Fτ j -measurable random variables;P[ ∃ j such thatτ j = b

] = 1}c) For ϑ1 = g01{a} +∑∞

j =1 gj 1(τ j ,τ j +1] ∈ aS(lF) we define(ϑ1 · S

)t:= ∑∞

j =1 gj (Sτ j +1∧t − Sτ j ∧t ), t ∈ [a, b] .(Note that this is almost surely a sum of finitely many terms,

and the process((ϑ1 · S

)t

)t∈[a,b]

is progressively measurable

because(

St

)t∈[a,b] is.)

Remark 2.3 For ϑ1 = g01{a} + ∑∞j =1 gj 1(τ j ,τ j +1] ∈ aS(lF) we can define

the setsAn = {τn < b} ∩ {τn+1 = b} , n ∈ IN. Then P[⋃∞

n=1 An] = 1, the

function N : � → IN defined by

N(ω) :={

n, ω ∈ An

0, ω 6∈ ⋃∞n=1 An

is Fb-measurable and

ϑ1 = g01{a} +∞∑j =1

gj 1(τ j ,τ j +1] = g01{a} +N∑

j =1

gj 1(τ j ,τ j +1] almost surely.

If an investor buys and sells stock shares according toϑ1, he will almost surelycarry out only finitely many transactions. But he does not know from the


beginning how many. Note that if we take an arbitraryFb-measurable functionN : � → IN, an increasing sequence oflF-stopping timesa = τ1 ≤ τ2 ≤· · · ≤ b, a real,Fa-measurable functiong0 and real,Fτ j -measurable functionsgj , j ∈ IN,then

g01{a} +N∑

j =1

gj 1(τ j ,τ j +1] = g01{a} +∞∑j =1

1{ j ≤N}gj 1(τ j ,τ j +1]

need not be inaS(lF).

Definition 2.4

2S(lF) :={ϑ : ϑ0, ϑ1 ∈ S(lF)

}, 2aS(lF) :=

{ϑ : ϑ0, ϑ1 ∈ aS(lF)

}.

Definition 2.5 Letϑ = (ϑ0, ϑ1

) ∈ 2aS(lF). There existlF-stopping times

a = τ1 ≤ τ2 ≤ · · · ≤ b

such thatϑ0 andϑ1 can be written in the form

ϑ0 = f01{a} +∞∑j =1

f j 1(τ j ,τ j +1], ϑ1 = g01{a} +∞∑j =1

gj 1(τ j ,τ j +1] . (2.2.1)

We setτ0 = a − 1 and call ϑ self-financing for(S0, S) if for all j ≥ 1,k = 1, . . . , j and l ≥ 0,

1{τ j −k<τ j −k+1=τ j +l<τ j +l+1}{(

f j +l − f j −k)

S0τ j

+ (gj +l − gj −k

)Sτ j

}a.s.= 0 .

(2.2.2)(Note that the property (2.2.2) is independent of the representation (2.2.1) ofϑ .)

2Ssf(lF) :=

{ϑ ∈ 2S(lF) : ϑ is self-financing

}.

2aSsf (lF) :=

{ϑ ∈ 2aS(lF) : ϑ is self-financing

}.

Proposition 2.6 Letϑ = (ϑ0, ϑ1

) ∈ 2aS(lF). Then the following are equiv-alent:

(i) ϑ is self-financing for(S0, S)

(ii) Vϑt

a.s.= Vϑa +

(ϑ0 · S0

)t+(ϑ1 · S

)t

for all t ∈ [a, b](iii) ϑ is self-financing for(1, S)

(iv) Vϑt

a.s.= Vϑa + (

ϑ1 · S)t for all t ∈ [a, b]


Proof. Let a = τ1 ≤ τ2 ≤ · · · ≤ b be an increasing sequence oflF-stoppingtimes such that

ϑ0 = f01{a} +∞∑j =1

f j 1(τ j ,τ j +1], ϑ1 = g01{a} +∞∑j =1

gj 1(τ j ,τ j +1] .

(i) ⇒ (ii): For t = a, (ii) is trivially satisfied. So let us assumet ∈ (a, b]. Foralmost allω ∈ �, there exists aj ∈ IN, such thatt ∈ (τ j , τ j +1], and

Vϑa +

(ϑ0 · S0

)t+(ϑ1 · S

)t

= f0S0τ1

+ g0Sτ1 +j −1∑i =1

fi(

S0τi +1

− S0τi

)+ f j

(S0

t − S0τ j

)

+j −1∑i =1

gi

(Sτi +1 − Sτi

)+ gj

(St − Sτ j

)

=j∑

i =1

S0τi( fi −1 − fi )+

j∑i =1

Sτi (gi −1 − gi )+ f j S0t + gj St = ϑ0

t S0t + ϑ1

t St ,

where the last inequality follows from (i) and the fact thatf j = ϑ0t , gj = ϑ1

t .(ii) ⇒ (i): Let j ≥ 1, k = 1, . . . , j andl ≥ 0. On{

τ j −k < τ j −k+1 = τ j +l < τ j +l+1}

we have (f j +l − f j −k

)S0τ j

+ (gj +l − gj −k

)Sτ j

=(

f j +l S0τ j +l+1

+ gj +l Sτ j +l+1

)−(

f j −kS0τ j

+ gj −kSτ j

)− f j +l

(S0τ j +l+1

− S0τ j

)− gj +l

(Sτ j +l+1 − Sτ j

)=(ϑ0τ j +l+1

S0τ j +l+1

+ ϑ1τ j +l+1

Sτ j +l+1

)−(ϑ0τ j

S0τ j

+ ϑ1τ j

Sτ j

)

−ϑ0

a S0a + ϑ1

a Sa +j +l∑i =1

fi(

S0τi +1

− S0τi

)+

j +l∑i =1

gi

(Sτi +1 − Sτi

)

+ϑ0

a S0a + ϑ1

a Sa +j −1∑i =1

fi(

S0τi +1

− S0τi

)+

j −1∑i =1

gi

(Sτi +1 − Sτi

) a.s.= 0 ,


where the last inequality follows from (ii).The equivalence of (i) and (iii) is trivial, and the equivalence of (iii) and (iv)can be shown in the same way as the equivalence of (i) and (ii). 2

Remark 2.7 It follows from Proposition 2.6 that for allϑ ∈ 2aSsf (lF),

ϑ0t

a.s.= Vϑa +

(ϑ1 · S

)t− ϑ1

t St , t ∈ [a, b] . (2.2.3)

This shows that if we identify indistinguishable processes, the map

ϑ =(ϑ0, ϑ1

)7→(

Vϑa , ϑ

1)

is a bijection from2aSsf (lF) to L0(Fa) × aS(lF). In particular, there exists for

all(ξ, ϑ1

) ∈ L0(Fa) × aS(lF), a uniqueϑ0 ∈ aS(lF) such thatϑ = (ϑ0, ϑ1

)is in2aS

sf (lF) andVϑa = ξ .

In 2aSsf (lF

S) there exist so called doubling strategies which can create ar-bitrage even in the standard Samuelson model, where

St = S0 exp(νt + σ Bt) , t ∈ [0, T ] ,

for constantsS0 > 0, ν, σ and a Brownian motionB. It was noticed by Har-rison and Pliska (1981) that they can be ruled out by putting an admissibilitycondition on the trading strategies. We use the admissibility condition of Del-baen and Schachermayer (1994). It is more liberal than the one of Harrisonand Pliska (1981) but restrictive enough to exclude arbitrage in the Samuelsonmodel.

Definition 2.8 Let c≥ 0. We callϑ ∈ 2aSsf (lF), c-admissible if

inft∈[a,b]

(Vϑ

t − Vϑa

) = inft∈[a,b]

(ϑ1 · S

)t≥ −c almost surely.

We callϑ admissible if it is c-admissible for some c≥ 0.

2Ssf,adm(lF) :=

{ϑ ∈ 2S

sf(lF) : ϑ is admissible}.

2aSsf,adm(lF) :=

{ϑ ∈ 2aS

sf (lF) : ϑ is admissible}.

2.3. Construction of arbitrage 31

2.3 Construction of arbitrage

Theorem 2.9 Let BH be a fractional Brownian motion. Let T∈ (0,∞),ν ∈ C1[0, T ] andσ > 0. Then in all four cases:

(i) H ∈ (12, 1), St = ν(t)+ σ BH

t , t ∈ [0, T ](ii) H ∈ (1

2, 1), St = exp(ν(t)+ σ BH

t

), t ∈ [0, T ]

(iii) H ∈ (0, 12), St = ν(t)+ σ BH

t , t ∈ [0, T ](iv) H ∈ (0, 1

2), St = exp(ν(t)+ σ BH

t

), t ∈ [0, T ]

there exists for every constant c> 0 and all n ∈ IN, a ϑ1(n) ∈ S(lFS) suchthat

a) P[(ϑ1(n) · S

)T = c

]> 1 − 1

n andb) inft∈[0,T]

(ϑ1(n) · S

)t ≥ − 1

n .

In particular, the strategiesϑ(n) = (ϑ0(n), ϑ1(n)

) ∈ 2Ssf,adm(lF

S), n ∈ IN,

whereϑ0(n) is given by

ϑ0t (n) =

(ϑ1(n) · S

)t− ϑ1

t (n)St , t ∈ [0, T ] , n ∈ IN ,

form a c-FLVR. In the cases (iii) and (iv),ϑ1(n) can be chosen such that also

c)∣∣ϑ1(n)

∣∣ ≤ 1n .

Theorem 2.10 In all four cases (i)-(iv) of Theorem 2.9 there exists for everyconstant c> 0, a 1

c -admissible c-arbitrageϑ ∈ 2aSsf,adm(lF

S). In the cases

(iii) and (iv), ϑ can be chosen such that∣∣ϑ1

∣∣ ≤ 1c .

In order to prove Theorems 2.9 and 2.10 we need the following two lem-mas.

Lemma 2.11 Let (Zt)t∈[a,b] be a continuous stochastic process. If

P [Zb = Za] = 0 , (2.3.1)

and for all ε > 0 there exist deterministic times a= t0 < · · · < tn = b suchthat

P

max

t∈[a,b]

n−1∑j =0

(Zt j +1∧t − Zt j ∧t

)2 ≥ ε

< ε, (2.3.2)

then there exists for all M> 0 a γ ∈ S(lFZ) such that

a) P[(γ · Z)b < M] < 1M and

b) inft∈[a,b] (γ · Z)t ≥ − 1M .


Proof. Let M > 0. It follows from (2.3.1) and (2.3.2) that there exist anε > 0such that

P[(Zb − Za)

2 < ε]<

1

2M(2.3.3)

and a partitiona = t0 < · · · < tn = b, such that

P

max

t∈[a,b]

n−1∑j =0


)2 ≥ ε

M2 + 1

< 1

2M. (2.3.4)

SinceZ is continuous,

ξ = inf

t ∈ [a, b] :

n−1∑j =0


)2 ≥ ε

M2 + 1

(2.3.5)

(we set inf∅ = b)

is anlFZ-stopping time (see e.g. Problem 1.2.7 of Karatzas and Shreve (1988))and (2.3.4) implies

P [ξ < b] <1

2M. (2.3.6)

Furthermore,

γ = 2

ε

(M + 1

M

) n−1∑j =1

(Zt j − Za

)1(t j ,t j +1]1[0,ξ ] (2.3.7)

is in S(lFZ) and a calculation shows that for allt ∈ [a, b],

(γ · Z)t = M + 1M

ε

(Zt∧ξ − Za

)2 −n−1∑j =0

(Zt j +1∧t∧ξ − Zt j ∧t∧ξ

)2 .

(2.3.8)This together with (2.3.5) implies b). From (2.3.8), (2.3.6) and (2.3.3) it fol-lows that

P[(γ · Z)b < M]

= P

M + 1

M

ε

(Zξ − Za

)2 −n−1∑j =0

(Zt j +1∧ξ − Zt j ∧ξ

)2 < M

≤ P[(

Zξ − Za)2< ε

]≤ P [ξ < b] + P

[(Zb − Za)

2 < ε]<

1

M.

This shows a), and the lemma is proved. 2


Lemma 2.12 Let (Zt )t∈[a,b] be a continuous stochastic process. If for allL > 0 there exist deterministic times a= t0 < · · · < tn = b, such that

P

n−1∑

j =0

(Zt j +1 − Zt j

)2< L

< 1

L, (2.3.9)

then there exists for all M> 0 a γ ∈ S(lFZ) such that

a) P[(γ · Z)b < M] < 1M ,

b) inft∈[a,b] (γ · Z)b ≥ − 1M and

c) |γ | ≤ 1M .

Proof. Let M > 0. SinceZ is continuous,

ξN = inf {t ∈ [a, b] : |Zt − Za| ≥ N} (we set inf∅ = b) (2.3.10)

is for all N > 0 an lFZ-stopping time and{ξN < b} → ∅, as N → ∞.Therefore there exists anN ≥ 2, such that

P [ξN < b] <1

2M. (2.3.11)

By assumption (2.3.9) there exists a partitiona = t0 < · · · < tn = b, suchthat

P

n−1∑

j =0

(Zt j +1 − Zt j

)2< N2(M2 + 1)

< 1

2M. (2.3.12)

It is easy to see that

γ = − 2

M N2

n−1∑j =1

(Zt j − Za

)1(t j ,t j +1]1[0,ξN ]

is in S(lFZ) and satisfies c). As in the proof of Lemma 2.11 a calculationshows that for allt ∈ [a, b],

(γ · Z)t = 1

M N2

n−1∑

j =0

(Zt j +1∧t∧ξN − Zt j ∧t∧ξN

)2 − (Zt∧ξN − Za

)2 .

(2.3.13)


This together with (2.3.10) implies b). From (2.3.13), (2.3.11) and (2.3.12)follows that

P[(γ · Z)b < M

]= P

1

M N2

n−1∑j =0

(Zt j +1∧ξN − Zt j ∧ξN

)2 − (ZξN − Sa

)2 < M

≤ P

n−1∑

j =0

(Zt j +1∧ξN − Zt j ∧ξN

)2< M2N2 + N2

≤ P [ξN < b] + P

n−1∑

j =0

(Zt j +1 − Zt j

)2< N2(M2 + 1)

< 1

M.

This shows a) and the lemma is proved. 2

Remark 2.13 The conclusions of Lemmas 2.11 and 2.12 remain true if (2.3.2)or (2.3.9) are satisfied for general stopping timesa = τ0 ≤ · · · ≤ τn = b in-stead of deterministic timesa = t0 < · · · < tn = b. However, for the proof ofTheorems 2.9 and 2.10 the versions with deterministic times are sufficient.

Proof of Theorem 2.9By self-similarity of BH it is enough to prove The-orem 2.9 forT = 1.

(i) H ∈ (12, 1), St = ν(t)+ σ BH

t , t ∈ [0, 1]:It is clear that(St)t∈[0,1] satisfies (2.3.1). It follows from Lemma 1.4 a) andthe fact thatν is Lipschitz that

maxt∈[0,1]

n−1∑j =0

(Sj +1

n ∧t − Sjn ∧t

)2 (n→∞)−→ 0 almost surely. (2.3.14)

This shows that(St)t∈[0,1] satisfies (2.3.2). Thus, it follows from Lemma 2.11that for alln ∈ IN, there exists aγ (n) ∈ S(lFS) such that

a) P[(γ (n) · S)1 < c] < 1n and

b) inft∈[0,1] (γ (n) · S)t ≥ − 1n .

For everyn ∈ IN,

ξn = inf{t : (γ (n) · S)t = c

}(we set inf∅ = 1)


is anlFZ-stopping time and forϑ1(n) = γ (n) · 1[0,ξn] ∈ S(lFZ) we have

a) P[(ϑ1(n) · S)1 = c] > 1 − 1

n andb) inft∈[0,1]

(ϑ1(n) · S

)t ≥ − 1

n .

(ii) H ∈ (12, 1), St = exp

(ν(t)+ σ BH

t

), t ∈ [0, 1] :

It is clear that(St)t∈[0,1] satisfies (2.3.1). That(St)t∈[0,1] satisfies (2.3.2) fol-lows from

|St − Su| ≤(

maxv∈[0,1]

Sv

)|ln St − ln Su| , u, t ∈ [0, 1] ,

and (2.3.14). Now the assertion can be deduced from Lemma 2.11 as before.

(iii) H ∈(0, 1

2

), St = ν(t)+ σ BH

t , t ∈ [0, 1]:To show that(St)t∈[0,1] satisfies (2.3.9) we choose anL > 0. It follows fromLemma 1.4 c) that

1

n

n−1∑j =0

∣∣∣∣BHj +1n

− BHjn

∣∣∣∣ (n→∞)−→ 0 in L1.

Hence,n−1∑j =0

2

∣∣∣∣(ν

(j + 1

n

)− ν

(j

n

))(σ BH

j +1n

− σ BHjn

)∣∣∣∣

≤ 2∥∥ν′∥∥∞

1

nσ

n−1∑j =0

∣∣∣∣BHj +1n

− BHjn

∣∣∣∣ (n→∞)−→ 0 in L1.

In particular, there exists ann1 ∈ IN, such that for alln ≥ n1,

P

n−1∑

j =0

∣∣∣∣2(ν

(j + 1

n

)− ν

(j

n

))(σ BH

j +1n

− σ BHjn

)∣∣∣∣ > L

< 1

2L.

On the other hand, Lemma 1.4 d) implies that there exists ann2 ∈ IN, suchthat for alln ≥ n2,

P

n−1∑

j =0

(σ BH

j +1n

− σ BHjn

)2

< 2L

< 1

2L.


Hence, for alln ≥ max(n1, n2),

P

n−1∑

j =0

(Sj +1

n− Sj

n

)2< L

≤ P

n−1∑

j =0

(σ BH

j +1n

− σ BHjn

)2

+2

(ν

(j + 1

n

)− ν

(j

n

))(σ BH

j +1n

− σ BHjn

)< L

]

≤ P

n−1∑

j =0

(σ BH

j +1n

− σ BHjn

)2

< 2L

+P

n−1∑

j =0

2

∣∣∣∣(ν

(j + 1

n

)− ν

(j

n

))(σ BH

j +1n

− σ BHjn

)∣∣∣∣ > L

< 1

L.

This shows that(St)t∈[0,1] satisfies (2.3.9). By Lemma 2.12 there exists for alln ∈ IN, a γ (n) ∈ S(lFZ) such that

a) P[(γ (n) · S)1 < c] < 1n

b) inft∈[0,1] (γ (n) · S)t ≥ − 1n

c) |γ (n)| ≤ 1n .

Having shown this, we can constructϑ1(n) as in (i). By c) we get∣∣ϑ1(n)∣∣ ≤ 1

n .(iv) H ∈ (0, 1

2), St = exp(ν(t)+ σ BH

t

), t ∈ [0, 1] :

Since(St)t∈[0,1] is positive and continuous, minv∈[0,1] Sv > 0. Therefore,there exists anε > 0 such that

P

[minv∈[0,1] Sv ≤ ε

]<

1

2L.

It follows from what we have shown in the proof of (iii) that there exists apartition 0= t0 < · · · < tn = 1, such that

P

n−1∑

j =0

(ln St j +1 − ln St j

)2<

1

ε2L

< 1

2L.

Since for all j ,

∣∣St j +1 − St j

∣∣ ≥(

minv∈[0,1] Sv

) ∣∣ln St j +1 − ln St j

∣∣ ,


we obtain

P

n−1∑

j =0

(St j +1 − St j

)2< L

≤ P

[minv∈[0,1] Sv ≤ ε

]+ P

n−1∑

j =0

(ln St j +1 − ln St j

)2<

1

ε2L

< 1

L.

This shows that(St)t∈[0,1] satisfies (2.3.9). Thus,ϑ1(n) can be constructed asin (iii). Again

∣∣ϑ1(n)∣∣ ≤ 1

n . This completes the proof of the theorem. 2

Proof of Theorem 2.10SinceBH is self-similar, it is enough to prove thetheorem forT = 1. We split(0, 1] into the subintervals

In = (an = 1 − 21−n, bn = 1 − 2−n] , n ∈ IN .

By Sn we denote the restriction ofS to In and bylFSn = (F Sn

t

)t∈In

the filtra-

tion generated bySn. Note thatF Sn

t ⊂ F St for all n ∈ IN andt ∈ In.

Since BH has stationary increments, it follows from Theorem 2.9 thatthere exists for alln ∈ IN, a γ (n) ∈ S(lFSn

) such that

a) P[(γ (n) · Sn)bn< c + 1

c ] < 1n

b) inft∈In (γ (n) · Sn)t ≥ − 12nc .

For

γ =∞∑

n=1

γ (n)1In ,

ξ = inf{t ∈ [0, 1] : (γ · S)t = c

}(we set inf∅ = 1)

is an lFS-stopping time. a) and b) implyP [ξ < 1] = 1. Therefore,ϑ1 = γ · 1[0,ξ ] belongs toaS(lFS) and(

ϑ0, ϑ1)

withϑ0

t =(ϑ1 · S

)t− ϑ1

t St , t ∈ [0, 1] ,

is a 1c -admissiblec-arbitrage in2aS

sf,adm(lFS). In the cases (iii) and (iv), all

γ (n)’s can be chosen such that|γ (n)| ≤ 1c . Then

∣∣ϑ1∣∣ ≤ 1

c too, and thetheorem is proved. 2


Remarks 2.14

1. In a market model

((S0

t

)t∈[0,T] ,

(St

)t∈[0,T]

)with strong arbitrage it is

possible to super-replicate a European call option with time-T pay-off CT =(ST − K

)+, K > 0, without initial endowment in the following way: At

time 0 one borrows money from the money market account to buy one stockshare. Then one applies a strong arbitrage strategy to generate the amountof money needed to pay back ones debts without selling the stock share. Attime T one owns a stock share and has no debts. This hedges the option.The following example shows that a European call option can have a positive

super-replication price if the model

((S0

t

)t∈[0,T] ,

(St

)t∈[0,T]

)only admits

arbitrage:Let (�,A, P) be a probability space with a Brownian motionB and an inde-pendent fractional Brownian motionBH , H ∈ (0, 1

2) ∪ (12, 1). Furthermore,

let ξ be a random variable on(�,A, P) that is independent ofB andBH andsuch thatP[ξ = 0] = P[ξ = 1] = 1

2. Let r, ν andσ > 0, be constants. The

model

((S0

t

)t∈[0,T] ,

(St

)t∈[0,1]

)with

S0t = exp(r t ) and St = exp

{(r + ν)t + σ

((1 − ξ)Bt + ξBH

t

)},

t ∈ [0, 1] ,has arbitrage but no strong arbitrage in2aS

sf,adm(lFS). Is is clear that the super-

replication ofC1 with a strategy from2aSsf,adm(lF

S) costs at least the Black-Scholes price.2. As we mentioned in the introduction, it is shown in Salopek (1998) thata stochastic processZ which is almost surely continuous and of boundedp-variation for somep < 2, can be integrated path-wise with respect to itself,and ∫ t

02 (Zu − Z0)d Zu = (Zt − Z0)

2 for all t ∈ [0, T ] . (2.3.15)

The process (2.3.7), which is the building block for our arbitrage strategy inthe cases (i) and (ii) of Theorem 2.9, is a multiple of a discrete version of theintegrand in (2.3.15).3. It is clear that Theorem 2.9 cannot only be applied to models((

S0t

)t∈[0,T] ,

(St

)t∈[0,T]

)

2.4. Exclusion of arbitrage 39

withSt = ν(t)+ σ BH

t or St = exp(ν(t)+ σ BH

t

),

but to all models

((S0

t

)t∈[0,T ] ,

(S)

t∈[0,T]

)such that(St)t∈[0,T ] satisfies con-

ditions (2.3.1) and (2.3.2) of Lemma 2.11 or condition (2.3.9) of Lemma2.12. In particular, condition (2.3.2) is fulfilled by all processes with van-ishing quadratic variation, and all processes with infinite quadratic variationsatisfy condition (2.3.9). For different generalizations of Lemma 1.4 see e.g.Shao (1996), Takashima (1989) or Kono and Maejima (1991). Shao (1996)contains results onp-variation of Gaussian processes with stationary incre-ments. Takashima (1989) gives sample path properties of ergodic self-similarprocesses, and in Kono and Maejima (1991), results on Holder continuity ofsample paths of some self-similar stable processes can be found.

2.4 Exclusion of arbitrage

The arbitrage strategies that we constructed in Section 3 act on ever smallertime intervals. They can be excluded by introducing a minimal amount of timeh > 0 that must lie between two consecutive transactions.

Definition 2.15 Let lF = (Ft )t∈[0,T] be a filtration and h> 0.

Sh(lF) :=g01{0} +

n−1∑j =1

gj 1(τ j ,τ j +1] ∈ S(lF) : τ j +1 ≥ τ j + h , ∀ j

.

2hsf(lF) :=

{ϑ ∈ 2S

sf : ϑ0, ϑ1 ∈ Sh(lF)}. (2.4.1)

In the following we will show that none of the models (i)-(iv) of Theorem 2.9has an arbitrage in

⋃h>02

hsf(lF

S).

Lemma 2.16 Let H ∈ (0, 12) ∪ (1

2, 1) and (Bt)t≥0 a one-sided Brownian

motion. Let(Zt )t≥0 be a continuous version of(∫ t

0(t − s)H− 12 d Bs

)t≥0

. Then,

for all c ≥ 0 and all h and T such that0< h ≤ T ,

P

[inf

t∈[h,T] Zt ≥ c

]= P

[sup

t∈[h,T]Zt ≤ −c

]> 0 .


Proof. Let c ≥ 0 and 0< h ≤ T .

P

[inf

t∈[h,T] Zt ≥ c

]= P

[sup

t∈[h,T]Zt ≤ −c

]

follows from the fact that(−Zt )t≥0 has the same distribution as(Zt )t≥0. The-orem 2.9.25 of Karatzas and Shreve (1988) shows that for alln ∈ IN, thereexists a measurable set�n ⊂ � with P[�n] = 1 such that for allω ∈ �n andall t ∈ [0, n],

lims→t

Bt − Bs√|t − s| log(

1|t−s|

) = 0 . (2.4.2)

For � = ⋂∞n=1, P[�] = 1, and (2.4.2) holds for allω ∈ � andt ≥ 0. Hence,

(Bt )t≥0 induces Wiener measureQW on(�,B

), where

� =ω ∈ C[0,∞) : ω(0) = 0 , lim

s→t

ω(t)− ω(s)√|t − s| log

(1

|t−s|) = 0 , ∀t ≥ 0

andB is theσ -algebra of subsets of� generated by the cylinder sets. Notethat for allω ∈ �, ∫ t

0(t − s)H− 1

2 dω(s)

can for allt ≥ 0, be defined as an improper Riemann-Stieltjes integral whichis continuous int . Hence,

P

[inf

t∈[h,T] Zt ≥ c

]= QW

[inf

t∈[h,T]

∫ t

0(t − s)H− 1

2 dω(s) ≥ c

].

Let us first assumeH ∈ (12, 1). In this case we set

m = H + 12

hH+ 12

[c + T H− 1

2

], ωm(t) = ω(t)− mt , t ∈ [0, T ]

and

Am ={ω ∈ � : sup

t∈[0,T]|ωm(t)| ≤ 1

}.

By Girsanov’s Theorem there exists a probability measureQm that is equiva-lent toQW such that(ωm(t))t∈[0,T] is a Brownian motion underQm. It is wellknown thatQm [ Am] > 0. Equivalence ofQW andQm implies that also

QW [ Am] > 0 . (2.4.3)


For allω ∈ � andt ≥ 0,∫ t

0(t − s)H− 1

2 dω(s) =∫ t

0ω(s)(H − 1

2)(t − s)H− 3

2 ds,

= (H − 1

2)

∫ t

0ωm(s)(t − s)H− 3

2 ds+ (H − 1

2)m∫ t

0s(t − s)H− 3

2 ds

= (H − 1

2)

∫ t

0ωm(s)(t − s)H− 3

2 ds+ mt H+ 1

2

H + 12

Forω ∈ Am, we obtain for allt ∈ [h, T ] the following estimates:

(H − 1

2)

∫ t

0ωm(s)(t − s)H− 3

2 ds ≥ −(H − 1

2)

∫ t

0(t − s)H− 3

2 ds

= −t H− 12 ≥ −T H− 1

2

and, by our choice ofm,

mt H+ 1

2

H + 12

=(

t

h

)H+ 12 (

c + T H− 12

)≥ c + T H− 1

2 .

Hence, ∫ t

0(t − s)H− 1

2 dω(s)ds ≥ −T H− 12 + c + T H− 1

2 = c .

It follows that

Am ⊂{

inft∈[h,T]

∫ t

0(t − s)H− 1

2 dω(s) ≥ c

}.

This and (2.4.3) prove the lemma forH ∈ (12, 1).

For H ∈ (0, 12), the proof is slightly more delicate. It follows from

QW

[sup

t∈[0,T ]|ω(t)| ≤ 1

2

]> 0

and Lemma 1.4 a) that there exist constantsε ∈ (0, h) andδ > 0 such that

QW

[A(

1

2, ε, δ)

]> 0 ,


where

A(1

2, ε, δ) =

ω ∈ � : sup

t∈[0,T ]|ω(t)| ≤ 1

2and sup

t,s∈[0,T ];0<t−s<ε

|ω(t)− ω(s)|(t − s)

12− H

2

≤ δ

We set

m = H + 12

hH+ 12

[c + εH− 1

2 + (1

2− H )

2δ

Hε

H2 + 1

2hH− 1

2

],

ωm(t) = ω(t)− mt , t ∈ [0, T ]andQm as before. Furthermore, we define

Am(1

2, ε, δ) =

{ω ∈ � : ωm ∈ A(

1

2, ε, δ)

}.

Since(ωm(t))t∈[0,T ] is a Brownian motion underQm,

Qm

[Am(

1

2, ε, δ)

]= QW

[A(

1

2, ε, δ)

]> 0 .

Hence, also

QW

[Am(

1

2, ε, δ)

]> 0 . (2.4.4)

Forω ∈ � andt ≥ h, we can write∫ t

0(t − s)H− 1

2 dω(s) =∫ t

0(t − s)H− 1

2 d [ω(s)− ω(t)]

= (1

2− H )

∫ t

0[ω(t)− ω(s)] (t − s)H− 3

2 ds+ t H− 12ω(t)

= (1

2− H )

∫ t

0[ωm(t)− ωm(s)] (t − s)H− 3

2 ds

+ (1

2− H )m

∫ t

0(t − s)H− 1

2 ds+ t H− 12ωm(t)+ mtH+ 1

2

= (1

2− H )

∫ t−ε

0[ωm(t)− ωm(s)] (t − s)H− 3

2 ds

+(12

− H )∫ t

t−ε[ωm(t)− ωm(s)] (t − s)H− 3

2 ds+ t H− 12ωm(t)+ m

t H+ 12

H + 12

.


If ω ∈ Am(12, ε, δ) andt ∈ [h, T ], we can estimate the four preceding terms

as follows:

(1

2− H )

∫ t−ε

0[ωm(t)− ωm(s)] (t − s)H− 3

2 ds

≥ −(12

− H )∫ t−ε

0(t − s)H− 3

2 ds = −εH− 12 + t H− 1

2 ≥ −εH− 12 ,

(1

2− H )

∫ t

t−ε[ωm(t)− ωm(s)] (t − s)H− 3

2 ds

≥ −(12

− H )∫ t

t−εδ(t − s)

H2 −1ds = −(1

2− H )

2δ

Hε

H2 ,

t H− 12ωm(t) ≥ −1

2hH− 1

2

and

mt H+ 1

2

H + 12

=(

t

h

)H+ 12[c + εH− 1

2 + (1

2− H )

2δ

Hε

H2 + 1

2hH− 1

2

]

≥ c + εH− 12 + (

1

2− H )

2δ

Hε

H2 + 1

2hH− 1

2 .

Hence, ∫ t

0(t − s)H− 1

2 dω(s) ≥

−εH− 12 −(1

2−H )

2δ

Hε

H2 −1

2hH− 1

2 +c+εH− 12 +(1

2−H )

2δ

Hε

H2 +1

2hH− 1

2 = c .

This and (2.4.4) prove the lemma forH ∈ (0, 12). 2

Theorem 2.17 Let BH be a fractional Brownian motion with H∈ (0, 12) ∪

(12, 1). Let T ∈ (0,∞), σ > 0 andν : [0, T ] → IR be a measurable function

such thatsupt∈[0,T] |ν(t)| < ∞. Consider the two cases

(i) St = ν(t)+ σ BHt , t ∈ [0, T ]

(ii) St = exp(ν(t)+ σ BH

t

), t ∈ [0, T ]

If

ϑ1 = g01{0} +n−1∑j =1

gj 1(τ j ,τ j +1] ∈⋃h>0

Sh(lFS)


and there exists a j∈ {1, . . . , n − 1} with P[gj 6= 0

]> 0,

then in case (i),

P[(ϑ1 · S

)T

≤ −c]> 0 for all c ≥ 0 ,

and in case (ii),

P[(ϑ1 · S

)T< 0

]> 0 .

Proof. For notational simplicity we give the proof forSt = BHt and St =

exp(BH

t

). The generalizations to the cases (i) and (ii) are obvious. To prove

the theorem forSt = BHt we fix anh > 0, and take a

ϑ1 = g01{0} +n−1∑j =1

gj 1(τ j ,τ j +1] ∈ Sh(lFBH) ,

such that there exists aj ∈ {1, . . . , n − 1} with P[gj 6= 0

]> 0. If

k = max{

j ∈ {1, . . . , n − 1} : P[gj 6= 0

]> 0

},

then (ϑ1 · BH

)T

=k∑

j =1

gj

(BHτ j +1

− BHτ j

)almost surely.

Let c ≥ 0. It is clear that

P

k∑

j =1

gj

(BHτ j +1

− BHτ j

)≤ −c

(2.4.5)

≥ P

k−1∑

j =1

gj

(BHτ j +1

− BHτ j

)+ sup

t∈[h,T]gk

(BHτk+t − BH

τk

)≤ −c

.

Let

� =ω ∈ C(IR) : ω(0) = 0 ; lim

s→t

ω(t)− ω(s)√|t − s| log

(1

|t−s|) = 0 , ∀t ≥ IR

,

B theσ -algebra of subsets of� that is generated by the cylinder sets andP the

Wiener measure on(�,B

). Without loss of generality we can assume that


(BH

t

)t≥0 is defined on

(�,B, P

)by the improper Riemann-Stieltjes integrals

BHt (ω) =

∫ t

−∞

[(t − s)H− 1

2 − 1{s≤0}(−s)H− 12

]dω(s) , t ≥ 0 . (2.4.6)

We define the filtrationlF� =(F �

t

)t∈[0,T] by

F �t = σ

{{ω ∈ � : ω(s) ≤ a

}: −∞ < s ≤ t , a ∈ IR

}.

It is clear thatlF� is bigger than the filtrationlFBH =(F BH

t

)t∈[0,T ], which is

given by

F BH

t = σ{

BHs : 0 ≤ s ≤ t

}.

Therefore thelFBH-stopping timesτ1, . . . τk, are alsolF�-stopping times. In

the following we split each functionω ∈ � at the time pointτk(ω). We set

π1ω(s) = ω(s)1(−∞,τk(ω)](s) , s ∈ IR ,

π2ω(s) = ω(τk(ω)+ s)− ω(τk(ω)) , s ≥ 0 ,

and let�1 =

{π1(ω) ∈ IRIR : ω ∈ �

},

B1 theσ -algebra of subsets of�1 that is generated by the cylinder sets,

�2 ={π2(ω) ∈ C[0,∞) : ω ∈ �

}andB2 theσ -algebra of subsets of�2 that is generated by the cylinder sets.It can easily be checked that the mapping

π1 :(�,B

)→ (�1,B1)

is F �τk

-measurable. On the other hand, it follows from Theorem I.32 of Protter(1990) that(π2ω(s))s≥0 is a Brownian motion underP which is independent

of F �τk

. It can be seen from (2.4.6) that for allω ∈ � andt ∈ [h, T ],k−1∑

j =1

gj

(BHτ j +1

− BHτ j

)+ gk

(BHτk+t − BH

τk

) (ω) = Ut (π1ω, π2ω)


where forω1 ∈ �1, ω2 ∈ �2 andt ∈ [h, T ],Ut (ω1, ω2) = U0(ω1)+ gk(ω1)

(U1

t (ω1)+ U2t (ω2)

),

and

U0(ω1) =k−1∑

j =1

gj

(BHτ j +1

− BHτ j

) (ω1) ,

U1t (ω1) =

∫ τk(ω1)

−∞

[(τk(ω1)+ t − s)H− 1

2 − (τk(ω1)− s)H− 12

]dω1(s) ,

U2t (ω2) =

∫ t

0(t − s)H− 1

2 dω2(s) .

Since(Ut )t∈[h,T] is a continuous stochastic process on(�1 ×�2,B1 ⊗ B1),the set

A ={(ω1, ω2) ∈ �1 ×�2 : sup

t∈[h,T]Ut (ω1, ω2) ≤ −c

}

is B1 ⊗ B2-measurable. It follows from Proposition A.2.5 of Lamberton andLapeyre (1996) that for almost everyω ∈ �,

E[1A (π1, π2) |F �

τk

](ω) = φ (π1ω) ,

whereφ : �1 → IR

is defined byφ (ω1) = E[1A (ω1, π2)] , ω1 ∈ �1 .

SinceU1t (ω1) is for all ω1 ∈ �1 continuous int , supt∈[h,T] U1

t (ω1) is for allω1 ∈ �1 finite. Therefore and since(π2ω(t))t≥0 is a Brownian motion underP, it follows from Lemma 2.16 that for allω1 ∈ �1 with gk(ω1) 6= 0,

φ(ω1) = P

[sup

t∈[h,T]Ut (ω1, π2) ≤ −c

]

≥ P

[U0(ω1)+ sup

t∈[h,T]gk(ω1)U

1t (ω1)+ sup

t∈[h,T]gk(ω1)U

2t (π2) ≤ −c

]> 0 .

SinceP [gk ◦ π1 6= 0] > 0 ,


we have

P

k−1∑

j =1

gj

(BHτ j +1

− BHτ j

)+ sup

t∈[h,T ]gk

(BHτk+t − BH

τk

)≤ −c

= E[1A (π1, π2)] = E[E[1A (π1, π2) |F �

τk

]]= E[φ ◦ π1] > 0 .

This and (2.4.5) prove the theorem in the caseSt = BHt .

If St = exp(BH

t

), let us assume there exists anh > 0 and a

ϑ1 = g01{0} +n−1∑j =1

gj 1(τ j ,τ j +1] ∈ Sh(lFBH)

such that(ϑ1 · S

)T ≥ 0 almost surely and there exists aj ∈ {1, . . . , n − 1}

with P[gj 6= 0

]> 0. If

k = min

l : P [gl 6= 0] > 0 and

l∑j =1

gj

(e

BHτ j +1 − e

BHτ j

)≥ 0 a.s.

,

then eitherg1 = · · · = gk−1 = 0 almost surely

or

P

k−1∑

j =1

gj

(e

BHτ j +1 − e

BHτ j

)< 0

> 0 .

In both cases,P [C] > 0 for

C =

k−1∑j =1

gj

(e

BHτ j +1 − e

BHτ j

)≤ 0 , gk 6= 0

.

With the same method that we used in the first part of the proof one can deducefrom Lemma 2.16 that for almost allω ∈ C,

P

k−1∑

j =1

gj

(e

BHτ j +1 − e

BHτ j

)+ sup

t∈[h,T]gk

(eBH

τk+t − eBHτk

)< 0

∣∣∣∣∣ F �τk

(ω) > 0 .


Hence,

P

k∑

j =1

gj

(e

BHτ j +1 − e

BHτ j

)< 0

≥ P

k−1∑

j =1

gj

(e

BHτ j +1 − e

BHτ j

)+ sup

t∈[h,T]gk

(eBH

τk+t − eBHτk

)< 0

= E

P

k−1∑

j =1

gj

(e

BHτ j +1 − e

BHτ j

)+ sup

t∈[h,T]gk

(eBH

τk+t − eBHτk

)< 0

∣∣∣∣∣ F �τk

≥ E

1C P

k−1∑

j =1

gj

(e

BHτ j +1 − e

BHτ j

)

+ supt∈[h,T]

gk

(eBH

τk+t − eBHτk

)< 0

∣∣∣ F �τk

]]> 0 .

This contradicts our assumption and the theorem is proved. 2

It follows from Theorem 2.17 that in both cases

(i) St = ν(t)+ σ BHt , t ∈ [0, T ] , and

(ii) St = exp(ν(t)+ σ BH

t

), t ∈ [0, T ] ,

the model

((S0

t

)t∈[0,T ] ,

(St

)t∈[0,T ]

)has no arbitrage in

⋃h>02

hsf(lF

S). More-

over, in case (i) there exist no non-trivial admissible strategies in⋃

h>02hsf(lF

S).An inspection of the proof of Theorem 2.17 shows that in case (ii), aϑ ∈ ⋃h>02

hsf(lF

S) can only be admissible ifϑ1 is almost surely non-negative.Clearly, the class2S

sf(lFS) is bigger than

⋃h>02

hsf(lF

S). It is an open prob-lem whether or not models of the form (i) and (ii) have arbitrage in2S

sf(lFS)

or2Ssf,adm(lF

S).It follows from similar arguments to the ones in the proof of Theorem 2.17

that in both cases (i) and (ii) the cheapest way to super-replicate a Europeancall option with a strategyϑ ∈ ⋃h>02

hsf(lF

S) is to buy the stock. In particular,

in both cases (i) and (ii) of Theorem 2.17 the model

((S0)

t∈[0,T] ,(

St

)t∈[0,T]

)is incomplete when trading strategies are restricted to

⋃h>02

hsf(lF

S).

Chapter 3

Regularized fractionalBrownian motion and optionpricing

3.1 Introduction

For simplicity we will from now on consider market models((S0

t

)t∈[0,T] ,

(St

)t∈[0,T]

)

with S0t = er t , t ∈ [0, T ], for somer > 0. In this case,lFS = lFS, and the

model is specified if the evolution of the discounted stock priceS is given.A way to make the fractional Brownian motion models

St = S0 + νt + σ BHt , t ∈ [0, T ] , (3.1.1)

St = S0 exp(νt + σ BH

t

), t ∈ [0, T ] , (3.1.2)

arbitrage-free without restricting the trading strategies is indicated in the lastsection of Rogers (1997). Rogers (1997) regularizes fractional Brownian mo-tion by changing the convolution kernelϕH (1.2.4) in the Mandelbrot-VanNess representation (1.2.3) of fractional Brownian motion. He gives a class offunctionsϕ such that the stochastic process

Rϕt =∫ t

−∞[ϕ(t − s)− ϕ(−s)] dWs , t ≥ 0 , (3.1.3)

49

50 Chapter 3. Regularized fBm and option pricing

is a Gaussian semimartingale with the same long-range dependence as frac-tional Brownian motion and proposes to use such a process for modelling adiscounted stock price. However, the semimartingale property of the process(3.1.3) is not enough to ensure that the model

St = S0 exp

(νt + σ

Rϕt∥∥Rϕ1∥∥

2

), t ∈ [0, T ] , (3.1.4)

whereS0 > 0, ν, σ > 0 are constants, is arbitrage-free.

Definition 3.1 Let (C[0, T ],B) be the space of continuous functions withthe σ -algebra generated by the cylinder sets. If(Yt )t∈[0,T ] is an a.s. con-tinuous stochastic process, we denote by QY the measure induced by Y on(C[0, T ],B). We call two a.s. continuous stochastic processes(Yt )t∈[0,T]and(Zt )t∈[0,T] equivalent if QY and QZ are equivalent.

The main result of this chapter is that for a larger class of functionsϕ thanthe one in Rogers (1997), the process

(Rϕt)t∈[0,T], given by (3.1.3), is not only

a semimartingale but also equivalent to Brownian motion. This implies thatthe model (3.1.4) has a unique equivalent martingale measure. Hence, it isarbitrage-free and complete.

In Section 2 we construct for eachH ∈ (0, 1), a class of processes whosefinite-dimensional distributions are close to those ofBH and which have aunique equivalent martingale measure. In Section 3 we use these processesto build regularized fractional Samuelson models. Since these models have aunique equivalent martingale measure, option prices can be obtained by cal-culating conditional expectations. We discuss the pricing of a European calloption in such a framework.

3.2 Regularizing fractional Brownian motion

3.2.1 General idea

In this subsection we give some heuristic arguments that indicate why forH ∈ (0, 1

2) ∪ (12, 1), the behaviour of the functionϕH (1.2.4) near zero is

responsible for the existence of arbitrage in the models (3.1.1), (3.1.2) andhow ϕH can be regularized to yield a process that can be used to build anarbitrage-free stock price model with long-range dependence.

The arbitrage strategies in Section 2.3 consist of combinations of buy andhold strategies that act on ever smaller time intervals. ForH ∈ (0, 1

2), they

3.2. Regularizing fractional Brownian motion 51

exploit the fact thatBH has infinite quadratic variation. ForH ∈ (12, 1) they

use thatBH is a non-constant process with vanishing quadratic variation. Toexclude these arbitrage strategies we vary the local path behaviour of frac-tional Brownian motion in such a way that we obtain a process with non-zero,finite quadratic variation.

To sketch how this can be achieved we first show that the quadratic varia-

tion of BH is related to the rate of convergence of E[(

BHt

)2]to 0, ast ↘ 0.

Since fractional Brownian motion has stationary increments, we have for allt ≥ 0 ands ≥ 0,

E

[(BH

s+t − BHs

)2]

= E

[(BH

t

)2]

= t2H .

For H ∈ (12, 1), we get for every partition

0 = t0 < · · · < tn = T ,

of [0, T ], the estimate

E

n∑

j =1

(BH

t j− BH

t j −1

)2

=

n∑j =1

(t j − t j −1

)2H ≤ maxj

(t j − t j −1

)2H−1T .

This shows that E

[∑nj =1

(BH

t j− BH

t j −1

)2]

converges to zero as the grid size

of the partition goes to zero. Hence,BH has vanishing quadratic variation forH ∈ (1

2, 1). On the other hand, ifH ∈ (0, 12), then

E

n∑

j =1

(BH

j Tn

− BH( j −1) T

n

)2 = n

(T

n

)2H

→ ∞, for n → ∞ .

This indicates thatBH has infinite quadratic variation, forH ∈ (0, 12). We

have shown this rigorously in the proof of Lemma 1.4.To see which part of the functionϕH (1.2.4) accounts for the behaviour of

E[(

BHt

)2]for smallt > 0, we fix a smallδ > 0, and write

t2H = E

[(BH

t

)2]

= c2H

∫ t

−∞[ϕH (t − s)− ϕH (−s)]2 ds

= c2H

∫ −δ

−∞[ϕH (t − s)− ϕH (−s)]2 ds+ c2

H

∫ t

−δ[ϕH (t − s)− ϕH (−s)]2 ds


If H = 12, then

∫ −δ

−∞[ϕH (t − s)− ϕH (−s)]2 ds = 0 .

If H ∈(0, 1

2

)∪(

12, 1

), then

∫ −δ

−∞[ϕH (t − s)− ϕH (−s)]2 ds =

∫ ∞

δ

[(t + x)H− 1

2 − xH− 12

]2dx

≤∫ ∞

δ

[t

(H − 1

2

)xH− 3

2

]2

dx = t2

(H − 1

2

)2

2(1 − H )δ2(H−1) . (3.2.1)

This shows that for allH ∈ (0, 1), for smallt > 0, the essential contribution

to E[(

BHt

)2]comes from the term

c2H

∫ t

−δ[ϕH (t − s)− ϕH (−s)]2 ds.

Hence, the behaviour of the functionϕH near zero determines the rate of con-

vergence of E[(

BHt

)2]to 0, ast ↘ 0. To changeBH into a process with

similar distribution but non-zero, finite quadratic variation, we varyϕH in aneighbourhood of zero so that the resulting functionϕ satisfies

∫ t

−δ[ϕ(t − s)− ϕ(−s)]2 ds ≈ t , as t ↘ 0 ,

where we write for two functionsf andg, f (t) ≈ g(t), ast → t0, if thereexists a constantc ∈ (0,∞) such that limt→t0

f (t)g(t) = c. To give a concrete

example for the sort of functions we have in mind we set forH ∈ (0, 1),a ∈ IR andb > 0,

ϕa,bH (x) :=

{a + ϕH (b)−a

b x x ∈ [0, b]ϕH (x) x ∈ (−∞, 0) ∪ (b,∞)

.

As ϕH , the functionsϕa,bH satisfy

∫IR

[ϕ

a,bH (t − s)− ϕ

a,bH (−s)

]2ds< ∞ , for all t ∈ IR .


1

1.2

2

–5 –4 –3 –2 –1 1 2 3

s

0.8

1

2

–5 –4 –3 –2 –1 1 2 3

s

Figure 3.1: Left: The functionsϕa,bH (t −s) andϕa,b

H (−s) for H = 34, a = 1.2,

b = 1 and t = 3. Right: The functionsϕa,bH (t − s) andϕa,b

H (−s) for H = 14,

a = 0.8, b = 1 and t = 3.

Therefore, they can be used to define the integrals

Rϕ

a,bH

t =∫

IR

[ϕ


a,bH (−s)

]dWs, t ∈ IR , (3.2.2)

in theL2-sense.

It is clear that

(Rϕ

a,bH

t

)t∈IR

is a centred Gaussian process with stationary in-

crements. The same calculation as in (3.2.1) yields∫ −b

−∞

[ϕ


a,bH (−s)

]2dx = O(t2) , for t ↘ 0 .

Similarly, it can be checked that∫ 0

−b

[ϕ


a,bH (−s)

]2dx = O(t2) , for t ↘ 0 .

and∫ t

0

[ϕ


a,bH (−s)

]2dx

{ ≈ t if a 6= 0≈ t3 if a = 0

, for t ↘ 0 .

Hence,∫ t

−∞

[ϕ


a,bH (−s)

]2dx

{ ≈ t if a 6= 0= O(t2) if a = 0

, for t ↘ 0 .

It will follow from Proposition 3.2 and Corollary 3.8 that

(1a R

ϕa,bH

t

)t∈[0,T]

is a

finite variation process ifa = 0 and equivalent to Brownian motion ifa 6= 0.


On the other hand, for smallb,

R

ϕa,bH

t∥∥∥∥∥Rϕ

a,bH

1

∥∥∥∥∥2

t∈[0,T]

is similar to(BH

t

)t∈[0,T ] in

the following sense: It can be checked that for allt ∈ [0, T ∨ 1],∣∣∣∣∫ t

−∞[ϕH (t − s)− ϕH (−s)]2 ds−

∫ t

−∞

[ϕ


a,bH (−s)

]2ds

∣∣∣∣≤∣∣∣∣∣∫ 0

−b[ϕH (t − s)− ϕH (−s)]2 ds−

∫ 0

−b

[ϕ


a,bH (−s)

]2ds

∣∣∣∣∣+∣∣∣∣∫ t

t−b[ϕH (t − s)− ϕH (−s)]2 ds−

∫ t

t−b

[ϕ


a,bH (−s)

]2ds

∣∣∣∣≤∫ 0

−b[ϕH (t − s)− ϕH (−s)]2 ds∨

∫ 0

−b

[ϕ


a,bH (−s)

]2ds

+∫ t

t−b[ϕH (t − s)− ϕH (−s)]2 ds∨

∫ t

t−b

[ϕ


a,bH (−s)

]2ds

≤

2(((T ∨ 1)+ b)H− 1

2 + |a|)2

b if H ∈(

12, 1

)2∫ b

0

(xH− 1

2 + |a|)2

dx if H ∈ (0, 12]

.

This shows that

limb↘0

supt∈[0,T∨1]

∣∣∣∣Var

(BH

t

cH

)− Var

(Rϕ

a,bH

t

)∣∣∣∣ = 0 (3.2.3)

and in particular,

limb↘0

∣∣∣∣∣ 1

c2H

− Var

(Rϕ

a,bH

1

)∣∣∣∣∣ = 0 . (3.2.4)

It follows from (3.2.4) and (3.2.3) that for givenH ∈ (0, 1), a ∈ IR andε > 0there exists ab > 0 such that∣∣∣∣∣∣∣∣∣

c2H − 1∥∥∥∥R

ϕa,bH

1

∥∥∥∥2

2

∣∣∣∣∣∣∣∣∣≤ ε

3

c2H

Var(BH

T

) (3.2.5)


and, for allt ∈ [0, T ],

1∥∥∥∥Rϕ

a,bH

1

∥∥∥∥2

2

∣∣∣∣Var

(BH

t

cH

)− Var

(Rϕ

a,bH

t

)∣∣∣∣ ≤ ε

3. (3.2.6)

From (3.2.5) and (3.2.6) follows that for allt ∈ [0, T ],

∣∣∣∣∣∣∣∣Var

(BH

t

)− Var

R

ϕa,bH

t∥∥∥∥Rϕ

a,bH

1

∥∥∥∥2

∣∣∣∣∣∣∣∣

≤

∣∣∣∣∣∣∣∣∣Var

(BH

t

)− 1∥∥∥∥R

ϕa,bH

1

∥∥∥∥2

2

Var

(BH

t

cH

)∣∣∣∣∣∣∣∣∣

+

∣∣∣∣∣∣∣∣∣1∥∥∥∥Rϕ

a,bH

1

∥∥∥∥2

2

Var

(BH

t

cH

)− Var

R

ϕa,bH

t∥∥∥∥Rϕ

a,bH

1

∥∥∥∥2

∣∣∣∣∣∣∣∣∣

=∣∣∣∣Var

(BH

t

cH

)∣∣∣∣∣∣∣∣∣∣∣∣∣c2

H − 1∥∥∥∥Rϕ

a,bH

1

∥∥∥∥2

2

∣∣∣∣∣∣∣∣∣+ 1∥∥∥∥R

ϕa,bH

1

∥∥∥∥2

2

∣∣∣∣Var

(BH

t

cH

)− Var

(Rϕ

a,bH

t

)∣∣∣∣ ≤ 2ε

3.

By stationarity of the increments ofBH and Rϕa,bH , this implies for allt, s ∈


[0, T ] with s ≤ t,∣∣∣∣∣∣∣∣Cov

(BH

t , BHs

)− Cov

R

ϕa,bH

t∥∥∥∥Rϕ

a,bH

1

∥∥∥∥2

,Rϕ

a,bH

s∥∥∥∥Rϕ

a,bH

1

∥∥∥∥2

∣∣∣∣∣∣∣∣

= 1

2

∣∣∣∣Var(

BHt

)+ Var

(BH

s

)− Var

(BH

t−s

)

−Var

R

ϕa,bH

t∥∥∥∥Rϕ

a,bH

1

∥∥∥∥2

− Var

R

ϕa,bH

s∥∥∥∥Rϕ

a,bH

1

∥∥∥∥2

+ Var

R

ϕa,bH

t−s∥∥∥∥Rϕ

a,bH

1

∥∥∥∥2

∣∣∣∣∣∣∣∣

≤ 1

2

∣∣∣∣∣∣∣∣Var

(BH

t

)− Var

R

ϕa,bH

t∥∥∥∥Rϕ

a,bH

1

∥∥∥∥2

∣∣∣∣∣∣∣∣

+1

2

∣∣∣∣∣∣∣∣Var

(BH

s

)− Var

R

ϕa,bH

s∥∥∥∥Rϕ

a,bH

1

∥∥∥∥2

∣∣∣∣∣∣∣∣

+1

2

∣∣∣∣∣∣∣∣Var

(BH

t−s

)− Var

R

ϕa,bH

t−s∥∥∥∥Rϕ

a,bH

1

∥∥∥∥2

∣∣∣∣∣∣∣∣≤ ε .

3.2.2 Rϕ and its semimartingale decomposition

The largest class of functionsϕ of the form

(R1) ϕ : IR → IR is measurable andϕ(x) = 0 for x < 0 ,

that can be used to define the integrals

Rϕt =∫

IR[ϕ(t − s)− ϕ(−s)] dWs, t ∈ IR, (3.2.7)

in theL2-sense, is the class of functions that besides (R1) also satisfy

(R2)∫

IR [ϕ(t − s)− ϕ(−s)]2 ds< ∞, for all t ∈ IR .


If ϕ satisfies (R1) and (R2), it can easily be seen that the process(Rϕt)t∈IR

defined in (3.2.7) is a centred Gaussian process with stationary increments.But in contrast to fractional Brownian motion it is in general not self-similar.If ϕ is of the form

(R3) ϕ(x) ={ϕ(0)+ ∫ x

0 ψ(y)dy x ≥ 00 x < 0

,

for someψ ∈ L2(IR+), then it also satisfies (R1) and (R2). Hence,(Rϕt)t∈IR

is well-defined.

Proposition 3.2 If ϕ satisfies(R3), then for all t ≥ 0:

Rϕt = ϕ(0)Wt +∫ t

0

∫ s

−∞ψ(s − u)dWuds. (3.2.8)

Proof.

Rϕt =∫ t

−∞[ϕ(t − u)− ϕ(−u)] dWu

=∫ 0

−∞[ϕ(t − u)− ϕ(−u)] dWu +

∫ t

0ϕ(t − u)dWu

=∫ 0

−∞

∫ t

0ψ(s − u)dsdWu +

∫ t

0

[∫ t

uψ(s − u)ds+ ϕ(0)

]dWu

By the stochastic version of Fubini’s theorem (see e.g. Theorem IV.46 ofProtter (1990)), we can change the order of integration. Hence, the aboveequals ∫ t

0

∫ 0

−∞ψ(s − u)dWuds+

∫ t

0

∫ s

0ψ(s − u)dWuds+ ϕ(0)Wt

=∫ t

0

∫ s

−∞ψ(s − u)dWuds+ ϕ(0)Wt ,

i.e. (3.2.8) holds. 2

Corollary 3.3 If ϕ satisfies(R3), then(Rϕt)t≥0 is a continuous semimartin-

gale with respect to the smallest filtrationlFW = (

F Wt

)t≥0 that satisfies the

usual assumptions and contains the filtrationlFW = (F W

t

)t≥0, where for all

t ≥ 0, F Wt = σ {Ws : −∞ < s ≤ t}.

The canonical semimartingale decomposition of Rϕ with respect tolFW

isgiven by (3.2.8).


Proof. The corollary follows immediately from Proposition 3.2. 2

Corollary 3.4 If ϕ satisfies(R3), then(Rϕt)t≥0 is a continuous semimartin-

gale with respect to the smallest filtrationlFRϕ = (

F Rϕt

)t≥0 that satisfies the

usual assumptions and contains the filtrationlFRϕ = (F Rϕ

t

)t≥0, where for all

t ≥ 0, F Rϕt = σ

{Rϕs : 0 ≤ s ≤ t

}.

Proof. It follows from (3.2.7) thatF Rϕt ⊂ F W

t , for all t ≥ 0. Hence,F Rϕ

t ⊂ F Wt , for all t ≥ 0, and the corollary follows from Corollary 3.3

by Stricker’s Theorem (see Protter (1990), p. 45). 2

Remark 3.5 Let ϕ be of the form (R3). The author guesses that in general,

(Wt )t≥0 is not adapted to the filtrationlFRϕ

. In this case, (3.2.8) is not the

lFRϕ

-semimartingale decomposition ofRϕ and it is not obvious how to find it.

3.2.3 Equivalence of(

1ϕ(0)R

ϕ)

t∈[0,T] to Brownian motion

Letϕ be a function that satisfies (R3). It can be seen from (3.2.8) that(Rϕt)t≥0

is a finite variation process if and only ifϕ(0) = 0. In this subsection we show

that(

1ϕ(0) Rϕt

)t∈[0,T] is equivalent to Brownian motion ifϕ(0) 6= 0. The key

to this result is the following theorem.

Theorem 3.6 Let (Wt)t∈IR be a two-sided Brownian motion. Let T∈ (0,∞)

and k∈ L2(G), where

G = {(s, u) ∈ [0, T ] × (−∞, T] : s ≥ u} .Then

exp

{∫ t

0

∫ s

−∞k(s, u)dWudWs − 1

2

∫ t

0

(∫ s

−∞k(s, u)dWu

)2

ds

},

t ∈ [0, T ],is a martingale.

Proof. We show that the Novikov condition is satisfied. By Corollary 3.5.14of Karatzas and Shreve (1988) it is enough to show that there exists a partition0 = t0 < · · · < tl = T such that

E

[exp

{1

2

∫ tn

tn−1

(∫ s

−∞k(s, u)dWu

)2

ds

}]< ∞ for all n = 1, . . . , l .

(3.2.9)


Sincek ∈ L2(G), there exists a partition 0= t0 < · · · < tl = T , such that∫ tn

tn−1

∫ s

−∞(k(s, u))2 duds≤ 1

9for all n = 1, . . . , l . (3.2.10)

To show that (3.2.9) holds for this partition we fixn and set

k(s, u) = k(s, u)1(tn−1,tn](s) , (s, u) ∈ G .

(3.2.10) implies ∥∥∥k∥∥∥

2≤ 1

3. (3.2.11)

For everym ∈ IN, there exists a partitiontn−1 = sm1 < · · · < sm

J(m) = tn, and

akm ∈ L2(G), of the form

km(s, u) =J(m)∑j =1

kmj (u)1(sm

j −1,smj ](s)

with ∥∥∥k − km∥∥∥

2≤ 1

m. (3.2.12)

For allm ∈ IN, let Xm be the centred,J(m)-dimensional Gaussian vector withj-th component

√sm

j − smj −1

∫ smj −1

−∞km

j (u)dWu, j = 1, . . . , J(m).

There exists an orthogonalJ(m) × J(m)-matrix Am such thatYm = AmXm

is a centred Gaussian vector with independent components. SinceAm is or-thogonal, we have

J(m)∑j =1

(Ym

j

)2 = (Ym)T Ym = (

Xm)T (Am)T AmXm

= (Xm)T Xm =

J(m)∑j =1

(Xm

j

)2. (3.2.13)

Together with (3.2.11) and (3.2.12) this implies for allm ≥ 3,

J(m)∑j =1

Var(Ym

j

)=

J(m)∑j =1

E(

Xmj

)2


=J(m)∑j =1

(sm

j − smj −1

)∫ smj −1

−∞

(km

j (u))2

du

= ∥∥km∥∥2

2 ≤(∥∥∥km − k

∥∥∥2+∥∥∥k∥∥∥

2

)2<

1

2. (3.2.14)

Furthermore, it follows from (3.2.13) and the independence of theYmj ,

j = 1, . . . , J(m), that

E

[exp

{1

2

∫ t

0

(∫ s

−∞km(s, u)dWu

)2

ds

}]

= E

exp

1

2

J(m)∑j =1

(Xm

j

)2

= E

exp

1

2

J(m)∑j =1

(Ym

j

)2

=J(m)∏j =1

E

[exp

{1

2

(Ym

j

)2}]

=J(m)∏j =1

(1 − Var

(Ym

j

))− 12.

It can easily be shown by induction onJ(m) that

J(m)∏j =1

(1 − Var

(Ym

j

))≥ 1 −

J(m)∑j =1

Var(Ym

j

).

Therefore, it follows from (3.2.14) that for allm ≥ 3,

J(m)∏j =1

(1 − Var

(Ym

j

))>

1

2.

Hence, for allm ≥ 3,

E

[exp

{1

2

∫ t

0

(∫ s

−∞km(s, u)dWu

)2

ds

}]

≤1 −

J(m)∑j =1

Var(Ym

j

)− 1

2

<√

2 . (3.2.15)

Since

E

[∫ T

0

(∫ s

−∞k(s, u)dWu −

∫ s

−∞km(s, u)dWu

)2

ds

]


=∫ T

0E

[{∫ s

−∞

(k(s, u)− km(s, u)

)dWu

}2]

ds

=∫ T

0

∫ s

−∞

(k(s, u)− km(s, u)

)2duds=

∥∥∥k − km∥∥∥2

2

(m→∞)−→ 0 ,

there exists a subsequence{kmi }∞i =1 such that for almost everyω ∈ �,

∫ T0

{(∫ s−∞ k(s, u)dWu

)(ω)− (∫ s

−∞ kmi (s, u)dWu)(ω)

}2ds

(i →∞)−→ 0 ,

i.e. for almost everyω ∈ �,

∥∥∥(∫ s−∞ k(s, u)dWu

)(ω)− (∫ s

−∞ kmi (s, u)dWu)(ω)

∥∥∥2

L2[0,T](i →∞)−→ 0 .

This implies that for almost everyω ∈ �,

∥∥∥∥(∫ s

−∞kmi (s, u)dWu

)(ω)

∥∥∥∥2

L2[0,T]

(i →∞)−→∥∥∥∥(∫ s

−∞k(s, u)dWu

)(ω)

∥∥∥∥2

L2[0,T].

Hence,

∫ T

0

(∫ s

−∞kmi (s, u)dWu

)2

ds(i →∞)−→

∫ T

0

(∫ s

−∞k(s, u)dWu

)2

ds

almost surely. By Fatou’s lemma and (3.2.15) we obtain

E

[exp

{1

2

∫ tn

tn−1

(∫ s

−∞k(s, u)dWu

)2

ds

}]

= E

[exp

{1

2

∫ T

0

(∫ s

−∞k(s, u)dWu

)2

ds

}]

≤ lim infi

E

[exp

{1

2

∫ T

0

(∫ s

−∞kmi (s, u)dWu

)2

ds

}]≤ √

2 .

This completes the proof of the Theorem. 2


Remark 3.7 Theorem 3.6 is only a slight generalization of Theorem 2 of Hit-suda (1968). But our proof is simpler and does not need results from the theoryof Volterra integral equations.

Corollary 3.8 Let ϕ be a function withϕ(0) 6= 0 that satisfies(R3). Let

lFW = (

F Wt

)t∈[0,T ] be the smallest filtration that satisfies the usual assump-

tions and contains the filtrationlFW = (

F Wt

)t∈[0,T ], whereF W

t = σ (Ws : −∞ < s ≤ t). Then

(1

ϕ(0)Rϕt

)t∈[0,T ]

is a lFW

-Brownian motion with respect to the probability measure

Q = exp

{−∫ T

0

∫ s

−∞ψ(s − u)

ϕ(0)dWudWs

−1

2

∫ T

0

(∫ s

−∞ψ(s − u)

ϕ(0)dWu

)2

ds

}· P .

Proof. It follows from Proposition 3.2 that

1

ϕ(0)Rϕt = Wt +

∫ t

0

∫ s

−∞ψ(s − u)

ϕ(0)dWuds.

Theorem 3.6 implies that

exp

{−∫ t

0

∫ s

−∞ψ(s − u)

ϕ(0)dWudWs − 1

2

∫ t

0

(∫ s

−∞ψ(s − u)

ϕ(0)dWu

)2

ds

},

t ∈ [0, T ] ,is a martingale. Therefore, the corollary follows from Girsanov’s (1960) the-orem. 2

Remark 3.9 It can easily be checked that the functionsϕa,bH from Subsection

3.2.1 satisfy (R3). Fora 6= 0 we call the stochastic process R

ϕa,bH

t∥∥∥∥Rϕ

a,bH

1

∥∥∥∥

t≥0

3.3. Option pricing with regularized fBm 63

a regularized fractional Brownian motion because for smallb > 0 its finite-dimensional distributions are similar to those ofBH and at the same time,

for every T ∈ (0,∞),

(1a R

ϕa,bH

t

)t∈[0,T ]

is equivalent to Brownian motion.

However, note that forH ∈(

12, 1

), the paths of Rϕ

a,bH∥∥∥∥∥Rϕ

a,bH

1

∥∥∥∥∥are less regular than

those ofBH so far as the degree of local Holder continuity is concerned.

3.3 Option pricing with regularized fractionalBrownian motion

As in Section 1.4, we consider a frictionless market that consists of a moneymarket account and a stock. One unit of money in the money market ac-count grows like

(er t)t∈[0,T ] for a constant interest rater . The discounted

stock price follows a stochastic process(St)t∈[0,T ]. The information obtainedby observing the stock is given by the filtrationlFS = (

F St

)t∈[0,T], where

F St = σ (Su : 0 ≤ u ≤ t), t ∈ [0, T ]. By lF = (

F St

)t∈[0,T] we denote the

smallest filtration that containslFS and fulfils the usual assumptions. We areinterested in the price of a European call option on the stock with strike priceK and maturityT . In discounted terms this option pays at timeT a randomamount of

(ST − e−rT K )+ .If S is not a Markov process, it might be useful to know the history ofSwhen pricing the option. Therefore we examine the discounted priceCt0 ofthe European call option at some timet0 ∈ (0, T). To avoid trivial arbitrageopportunities,Ct0 has to lie in the interval((

St0 − e−rT K)+, St0

).

In the Samuelson model

St = S0 exp(νt + σ Bt) , t ∈ [0, T ] , (3.3.1)

whereS0 > 0, ν andσ > 0 are constants andB is a Brownian motion, Blackand Scholes (1973) gave an explicit formula forCt0. For givenr, K andT theBlack-Scholes price only depends ont0, St0 and the volatilityσ but not on thewhole trajectory(St)t∈[0,t0] and not on the parameterν. For givenr, K , T and


fixed t0, St0, the discounted Black-Scholes price dBS(t0, St0, .) of a Europeancall option is a continuous, strictly increasing function ofσ which maps theinterval(0,∞) bijectively to the interval((


).

Alternatively, let us assume that empirical data suggests that the discountedprice of a particular stock should be modelled as a fractional Samuelson pro-cess


t

), t ∈ [0, T ] , (3.3.2)

whereS0 > 0, ν andσ > 0 are constants andBH is a fractional Brownian

motion with H ∈(0, 1

2

)∪(

12, 1

). We have shown in Section 2.4 that this

model is arbitrage-free if one confines the trading strategies to⋃

h>02hsf(lF

S).On the other hand, we have shown that the model admits a FLVR consistingof integrands in2S

sf,adm(lFS) and strong arbitrage in2aS

sf,adm(lFS). However,

we can use the processesRϕa,bH , given in (3.2.2), to regularize the fractional

Samuelson model (3.3.2).

3.3.1 Naive option pricing in regularized fractional Samuel-son models

It is clear from what we have shown in Subsection 3.2.1 that for givenH ∈ (0, 1), there exists for everyε > 0 a continuous function

b : (0,∞) → (0,∞) ,

such that for alla > 0, ∣∣∣∣∣∣∣∣cH − 1∥∥∥∥R

ϕa,b(a)H

1

∥∥∥∥2

∣∣∣∣∣∣∣∣≤ cH

2(3.3.3)

and for allt, s ∈ [0, T ],∣∣∣∣∣∣∣∣Cov

(BH

t , BHs

)− Cov

R

ϕa,b(a)H

t∥∥∥∥Rϕ

a,b(a)H

1

∥∥∥∥2

,Rϕ

a,b(a)H

s∥∥∥∥Rϕ

a,b(a)H

1

∥∥∥∥2

∣∣∣∣∣∣∣∣≤ ε . (3.3.4)


On the other hand, Corollary 3.8 shows that for alla > 0,(1

aRϕ

a,b(a)H

t

)t∈[0,T]

is equivalent to Brownian motion. Therefore there exists for eacha > 0 aunique probability measureQa on (�, F S

T ) which is equivalent toP suchthat the regularized fractional Samuelson process

St = S0 exp

νt + σ

Rϕ

a,b(a)H

t∥∥∥∥Rϕ

a,b(a)H

1

∥∥∥∥2

, t ∈ [0, T ] , (3.3.5)

is a martingale on(�, lFS, Qa

). According to current practice in mathemat-

ical finance, in such a framework discounted option prices are calculated bytaking the conditional expectation of the options discounted pay-off under theequivalent martingale measure. In the model (3.3.5) witha > 0, this leads tothe following discounted price for our European call option at timet0:

Ct0(a) = EQa

S0 exp

νT + σ

Rϕ

a,b(a)H

T∥∥∥∥Rϕ

a,b(a)H

1

∥∥∥∥2

− e−rT K

+∣∣∣∣∣∣∣∣Ft0

= dBS

t0, St0,

σa∥∥∥∥Rϕ

a,b(a)H

1

∥∥∥∥2

(3.3.6)

Hence,Ct0(a) given in (3.3.6), only depends onSt0 and not on the whole past

(Su)u∈[0,t0] even thoughRϕa,b(a)H is not a Markov process underP. Further-

more, it follows from the continuity of the functionb and from (3.3.3) that themapping

a 7→ a∥∥∥∥Rϕ

a,b(a)H

1

∥∥∥∥2

is a continuous surjection from(0,∞) to (0,∞). This shows that althoughfor everya > 0, the model (3.3.5) is close to the model (3.3.2) in the sense of(3.3.4), the discounted option pricesCt0(a) in (3.3.6) fill the whole interval((


),


asa is running through(0,∞).

Remark 3.10 Note that for the special caseH = 12, the model (3.3.2) is the

Samuelson model (3.3.1) and the models (3.3.5) are rather perturbations thanregularizations of (3.3.1). Whereas in the Samuelson model (3.3.1), calculat-ing the discounted option price by taking the conditional expectation underthe equivalent martingale measureQ, leads to the discounted Black-Scholesprice

Ct0 = EQ

[(S0 exp(νT + σ BT) −e−rT K

)+∣∣∣∣ Ft0

]= dBS

(t0, St0, σ

),

in the models (3.3.5), this method yields the pricesCt0(a) given by (3.3.6).A similar result was obtained by Brigo and Mercurio (2000). For a given

finite time-grid0 ⊂ [0, T ] they constructed a class of processes(Ya

t

)t∈[0,T],

a ∈ (0,∞), such that each processYa has the same finite-dimensional distri-bution on 0 as the geometric Brownian motion (3.3.1), the same one-dimensional marginal distributions as (3.3.1) for allt ∈ [0, T ], and a uniqueequivalent martingale measure. As in our case the quadratic variation of theprocessesYa in Brigo and Mercurio (2000) can be very different from that of(3.3.1), and for every constant

c ∈((

S0 − e−rT K)+, S0

)

there exists ana ∈ (0,∞) such that the time zero price of a European call op-tion with maturityT and strike priceK on a stock modelled withYa equalsc.

In contrast to our processesRϕ

a,b(a)12 , a > 0, the processesYa, a > 0, in Brigo

and Mercurio (2000) have exactly the same distribution as (3.3.1) on the finite-time grid0. On the other hand, the log-processes

(logYa

t

)t∈[0,T], a > 0, do

not have stationary increments whereas our log-processes

R

ϕa,b(a)12

t

t∈[0,T],

a > 0, do.

3.3.2 Discussion

In order to understand why (3.3.6) can lead to totally different option prices inmodels that are close to each other in the sense of (3.3.4), we take a closer lookat the mechanism of option pricing by calculating the conditional expectationunder the equivalent martingale measure.


Let us assume that the present timet0 ∈ (0, T) is equal toN0N T , where

N0 < N are two natural numbers, and after observing the discounted stockpricest at times

0, h, 2h, . . . , N0h = t0 ,

whereh = TN , we have come to the conclusion that

ln

(sjh

s( j −1)h

), j = 1, . . . , N0 , (3.3.7)

could be the realisation of a stationary random sequence

(j hν + X j

)N0j =1 ,

whereν is a constant and(X j)N0

j =1 is a centred Gaussian vector that has up tosome statistical toleranceε > 0 the same covariance structure as(

σ BHjh − σ BH

( j −1)h

)N0

j =1,

whereσ > 0 is a constant andBH a fractional Brownian motion with HurstparameterH ∈ (0, 1).If we think that there will be no considerable change in market conditions aftertime t0, it is natural to model the discounted stock price on the time-grid

0, h, . . . , Nh = T ,

asSt = S0 exp

(νt + σ BH

t

), t = 0, h, . . . , Nh = T . (3.3.8)

It can easily be checked that the model (3.3.8) is arbitrage-free if we allow alldiscrete-time predictable processes as trading strategies. But at the same timeit is not possible to replicate the discounted option pay-off

(ST − e−rT K

)+

and the cheapest way to super-replicate it at timet0 is to buy a stock share.In reality transactions are not restricted to a pre-specified time-grid. The

assumption that ln(

StS0

)has stationary increments suggests to extend the model

(3.3.8) to


t

), t ∈ [0, T ] . (3.3.9)


Since the model is based on observations on a time-grid with mesh-widthh,in a first step we only allow trading strategies from the class2h

sf(lFS), which

is given in (2.4.1). We have shown in Theorem 2.17 that this makes the model(3.3.9) arbitrage-free. But then again, as we have seen at the end of Section2.4, the cheapest way to super-replicate the option is to buy the stock. Insteadof super-replicating the option, one could choose an incomplete market crite-rion and try to hedge the option with a strategy from2h

sf(lFS) in an optimal

way according to the chosen criterion.However, forH = 1

2, the process (3.3.9) is the Samuelson process (3.3.1).There exists an equivalent probability measureQ ∼ P under which the pro-cess (3.3.1) is a martingale. Therefore a reasonable option price can be ob-tained by enlarging the class of trading strategies as follows:

2sf,adm

(lF

S)

= {ϑ : ϑ0 andϑ1 arelFS-predictable;∫ T

0

∣∣ϑ0u

∣∣du< ∞ and∫ T

0

(ϑ1

u

)2du< ∞ a.s.;

Vϑt = Vϑ

0 + ∫ t0 ϑ

1udSu for all t ∈ [0, T ];

and there exists a constantc ≥ 0 such thatinft∈[0,T]

∫ t0 ϑ

1t dSu ≥ −c a.s.} .

(3.3.10)

It can be shown (see one of the many textbooks on mathematical finance) that

in a market model

((S0

t

)t∈[0,T ] ,

(St

)t∈[0,T]

)where

(S0

t

)t∈[0,T ] is a contin-

uous, finite variation process andS = S/S0 is as in (3.3.1), there exists no

arbitrage in the class2sf,adm

(lF

S), and there is a uniqueϑ =

(ϑ0, ϑ1

)∈

2sf,adm

(lF

S)

such that (∫ t

0ϑ1

udSu

)t∈[0,T]

is a square-integrable martingale underQ with

V ϑt0 +

∫ T

t0ϑ1

udSu = V ϑT =

(ST − e−rT K

)+.

ϑ is an optimal hedging strategy in the sense thatV ϑt0 ≤ Vϑ

t0 a.s., for allϑ ∈2sf,adm(lF

S) that satisfy

Vϑt0 +

∫ T

t0ϑ1

udSu = VϑT ≥

(ST − e−rT K

)+.


Hence,V ϑt0 is the minimal discounted amount of money needed at timet0 to

produce a perfect replication of the option pay-off in the model (3.3.1) with a

trading strategy from2sf,adm

(lF

S). Therefore

Ct0 = V ϑt0 = EQ

[(ST − e−rT K

)+ |Ft0

]= dBS(t0, St0, σ ) . (3.3.11)

Since the Black-Scholes hedging strategyϑ is almost surely of unboundedvariation, it is not possible to perform it in practice. It remains to be checkedwhether the discounted Black-Scholes price (3.3.11) can be justified by anapproximation ofϑ with a strategy from2h

sf(lFS) for fixed h > 0.

If the discounted stock is modelled with a fractional Samuelson process

(3.3.9) with aH ∈(0, 1

2

)∪(

12, 1

), an extension of the trading strategies

beyond2hsf(lF

S) is problematic. We have shown in Section 2.3 that the mod-els (3.1.1) and (3.1.2) admit FLVR consisting of strategies in2S

sf,adm(lFS)

and strong arbitrage in2aSsf,adm(lF

S). However, we can regularize the pro-

cess (3.3.9) by replacing fractional Brownian motionBH with a regularized

fractional Brownian motionRϕa,b(a)H for somea > 0, whereb(a) is chosen so

that (3.3.3) and (3.3.4) are satisfied for some statistical toleranceε > 0. Asthe model (3.3.9), the model

St = S0 exp

νt + σ

Rϕ

a,b(a)H

t∥∥∥∥Rϕ

a,b(a)H

1

∥∥∥∥2

, t ∈ [0, T ] , (3.3.12)

is for all a > 0, consistent with our observation of the past stock prices (3.3.7).It is clear that for a fixed trading strategyϑ ∈ 2h

sf(lFS) the discounted gain

process∫ϑdS is probabilistically similar in the model (3.3.9) and all mod-

els (3.3.12),a > 0. On the other hand, if strategies from2sf,adm

(lF

S)

are

considered, the space of discounted trading outcomes

{∫ T

0ϑ1

udSu : ϑ ∈ 2sf,adm

(lF

S)}

varies considerably in models of the form (3.3.12) with different parame-ters a > 0. For instance, since for eacha > 0, the model (3.3.12) is aSamuelson model under the equivalent martingale measureQa, there exists a


ϑ(a) ∈ 2sf,adm

(lF

S)

such that

(ST − K e−rT

)+ = EQa

[(ST − K e−rT

)+ | Ft0

]+∫ T

t0ϑ1

t (a)dSt (3.3.13)

almost surely,

and

EQa

[(ST − K e−rT

)+ |Ft0

]

= dBS

t0, St0,

σa∥∥∥∥Rϕ

a,b(a)H

1

∥∥∥∥2

→

{ (St0 − e−rT K

)+for a → 0

St0 for a → ∞ ,

Hence, if strategies from2sf,adm

(lF

S)

are permitted, it is possible to replicate

the option perfectly. But the replication costs increase with the quantity

σa∥∥∥∥Rϕ

a,b(a)H

1

∥∥∥∥2

.

Whereas the parametera does not have a big influence on the probabilistic

properties of the processRϕa,b(a)H , it can be seen from Proposition 3.2 that the

quadratic variation ofRϕa,b(a)H over a fixed time interval is proportional to

a2∥∥∥∥Rϕ

a,b(a)H

1

∥∥∥∥2

2

.

This shows that in the models (3.3.12) option prices obtained by calculatingthe conditional expectation under the equivalent martingale measure heavilydepend on the local path behaviour of the stochastic process that models thestock price, whereas the finite-dimensional distributions of the process do notseem to have an essential influence.

It is not clear whether for givenH ∈ (0, 12) ∪ (1

2, 1), there exists an

a > 0 such that the strategyϑ(a) ∈ 2sf,adm

(lF

S)

that replicates the option in

(3.3.13) can be interpreted as an idealisation of strategies from2hsf(lF

S) and


therefore for thisa,

dBS

t0, St0,

σa∥∥∥∥Rϕ

a,b(a)H

1

∥∥∥∥2

is the “right” option price in this situation.

Chapter 4

Mixed fractional Brownianmotion

4.1 Introduction

By mixed fractional Brownian motion we mean a linear combination of differ-ent fractional Brownian motions. In this chapter we examine whether a mixedfractional Brownian motion is a semimartingale when it is of the special form

M H,αt := Bt + αBH

t , t ∈ [0, T ] ,whereB is a Brownian motion,BH an independent fractional Brownian mo-tion,α ∈ IR \ {0} andT ∈ (0,∞) .

It follows from self-similarity of fractional Brownian motion that the pro-cess (

Bt + αBHt

)t∈[0,T ]

has the same distribution as(T

12 B t

T+ αT H BH

tT

)t∈[0,T]

= T12

(B t

T+ αT H− 1

2 BHtT

)t∈[0,T]

.

This shows that there is no loss of generality in assumingT = 1.

Remark 4.1 Let H ∈(0, 1

2

)∪(

12, 1

)and define the filtrationlF = (Ft )t∈[0,1]

by

Ft = σ

((Bs)0≤s≤t ,

(BH

s

)0≤s≤t

), t ∈ [0, 1] .

73

74 Chapter 4. Mixed fractional Brownian motion

SinceB is anlF-Brownian motion and therefore also anlF-weak semimartin-gale andBH is not anlF-weak semimartingale,M H,α = B + αBH cannotbe anlF-weak semimartingale. This does not imply thatM H,α is not a weaksemimartingale, that is, not a weak semimartingale with respect to its ownfiltration lFM H,α

.

The problem of determining whetherM H,α is a semimartingale is easiest

whenH ∈{

12, 1

}. It is clear that

1√1 + α2

M12 ,α

is a Brownian motion. In particular, it is anlFM

12 ,α

-semimartingale. Hence,

M12 ,α is a semimartingale.M1,α can be represented as

M1,αt = Bt + αtξ, t ∈ [0, 1] ,

where B is a Brownian motion andξ an independent standard normal ran-dom variable. This shows thatM1,α is a semimartingale with respect tolF = (

Ft)t∈[0,1], where

Ft = σ(ξ, (Bs)0≤s≤t

), t ∈ [0, 1] .

With the help of Girsanov’s (1960) theorem we can show even more. Unlike

M12 ,α, M1,α is not a multiple of a Brownian motion under the measureP. But

it is a Brownian motion under an equivalent measureQ. It can be deducedfrom Fubini’s theorem that

E

[exp

(−αξB1 − 1

2(αξ)2

)]= 1 .

Therefore,

Q = exp

(−αξB1 − 1

2(αξ)2

)· P

is a probability measure that is equivalent toP, and it follows from Girsanov’s(1960) theorem thatM1,α is a Brownian motion underQ. Hence,M1,α isequivalent to Brownian motion in the sense of Definition 3.1.

It can be seen from Definition 1.5 that the weak semimartingale propertyis invariant under a change of the probability measure within the same equiv-alence class. The same is true for the semimartingale property. Hence, allprocesses that are equivalent to Brownian motion are semimartingales.

We express the main results of this chapter in the following theorem.

4.2. Proof of Theorem 4.2 forH ∈(0, 1

2

)75

Theorem 4.2(M H,α

)t∈[0,1] is not a weak semimartingale if H∈ (0, 1

2) ∪(1

2,34], it is equivalent to

√1 + α2 times Brownian motion if H= 1

2 and

equivalent to Brownian motion if H∈ (34, 1].

For H ∈{

12, 1

}, we have already proved Theorem 4.2. ForH ∈

(0, 1

2

)∪(

12, 1

), we have shown in Section 1.3 thatBH is not a weak semimartingale.

For H ∈(0, 1

2

), the proof was based on the fact that the quadratic variation

of BH is infinite. The same argument can be used to show thatM H,α is not a

weak semimartingale forH ∈(0, 1

2

), because, as we will show in Section 2,

in this caseM H,α has also infinite quadratic variation. ForH ∈(

12, 1

), BH

is not a weak semimartingale because it is a stochastic process with vanishingquadratic variation and paths of infinte variation. This reasoning cannot be

applied to treatM H,α for H ∈(

12, 1

), because then,M H,α has the same

quadratic variation as Brownian motion. In this case we need more refinedmethods to see whetherM H,α is a semimartingale. Surprisingly,M H,α isnot a weak semimartingale ifH ∈ (1

2,34] and it is equivalent to Brownian

motion if H ∈ (34, 1]. In Section 3 we prove Theorem 4.2 forH ∈ ( 1

2,34].

The proof depends on a theorem of Stricker (1984) on Gaussian processes.In Section 4 we prove Theorem 4.2 forH ∈ (3

4, 1]. In this case we use theconcept of relative entropy and the fact that two Gaussian measures are eitherequivalent or singular. In Section 5 we discuss the price of a European calloption on a stock that is modelled as an exponential mixed fractional Brownianmotion with drift. In Section 6 we discuss general results of Shepp (1966) andHitsuda (1968) on represetations of Gaussian processes that are equivalent toBrownian motion. In Section 7 we solve a linear integral equation to obtainthe Radon-Nikodym derivative ofM H,α with respect to Wiener measure forcertain values ofH andα. In Section 8 we solve a quadratic integral equationto obtain the canonical semimartingale decomposition ofM H,α for certainvalues ofH andα.

4.2 Proof of Theorem 4.2 forH ∈ (0, 12

)From now on we use the following notation. For a stochastic process(Xt)t∈[0,1] andn ∈ IN, we set for j = 1, . . . n, 1n

j X = X jn

− X j −1n

.

Like BH , M H,α cannot be a weak semimartingale forH ∈(0, 1

2

), be-


cause it has infinite quadratic variation. To show this we write forn ∈ IN,

n∑j =1

(1n

j M H,α)2 =

n∑j =1

(1n

j B)2 + 2α

n∑j =1

1nj B1n

j BH + α2n∑

j =1

(1n

j BH)2.

It is known thatn∑

j =1

(1n

j B)2 (n→∞)−→ 1 in L2

(see e.g. Theorem I.28 of Protter (1990)). From

E

n∑

j =1

1nj B1n

j BH

2 =

n∑j ,k=1

E[1n

j B1nj BH 1n

k B1nk BH

]

=n∑

j =1

E

[(1n

j B)2]

E

[(1n

j BH)2]

= n1

n

(1

n

)2H

it follows thatn∑

j =1

1nj B1n

j BH (n→∞)−→ 0 in L2 .

On the other hand, it follows from Lemma 1.4 d) that

n∑j =1

(1n

j BH)2 (n→∞)−→ ∞ in probability.

Hence,n∑

j =1

(1n

j M H,α)2 (n→∞)−→ ∞ in probability.

In particular,

n∑j =1

(1n

j M H,α)2 : n ∈ IN

is unbounded inL0 and M H,α is not a weak semimartingale by Proposition1.9.

2

4.3. Proof of Theorem 4.2 forH ∈ (12,

34] 77

4.3 Proof of Theorem 4.2 forH ∈ (12,

34]

For H ∈ (12,

34], the key in the proof of Theorem 4.2 is Lemma 4.4 below. It

is based on Theorem 1 of Stricker (1984). Before we can formulate Lemma4.4, we must specify our notion of a quasimartingale. We call a stochasticprocess(Xt)t∈[0,1] a quasimartingale if it is a quasimartingale with respect tolFX = (

F Xt

)t∈[0,1], whereF X

t = σ((Xs)0≤s≤t

), t ∈ [0, 1] .

Definition 4.3 A stochastic process(Xt)t∈[0,1] is a quasimartingale if

Xt ∈ L1 for all t ∈ [0, 1], and

supτ

n−1∑j =0

∥∥∥E[Xt j +1 − Xt j |F X

t j

]∥∥∥1< ∞ ,

whereτ is the set of all finite partitions

0 = t0 < t1 < · · · < tn = 1 , n ∈ IN , of [0, 1] .Lemma 4.4 If M H,α is not a quasimartingale, it is not a weak semimartin-gale.

Proof. Let us assume thatM H,α is a weak semimartingale. Then Theorem1 of Stricker (1984) implies thatIM H,α (β(lFM H,α

)) is bounded inL2 (for the

definition of IM H,α (β(lFM H,α)) see Section 1.3). Therefore it is also bounded

in L1. For any finite partition

0 = t0 < t1, · · · < tn = 1 , n ∈ IN ,

n−1∑j =0

sgn(E[M H,α

t j +1− M H,α

t j|Ft j

])1(t j ,t j +1] ∈ β(lFM H,α

) ,

and ∥∥∥∥∥∥IM H,α

n−1∑

j =0

sgn(E[M H,α

t j +1− M H,α

t j|F M H,α

t j

])1(t j ,t j +1]

∥∥∥∥∥∥

1

≥ E

IM H,α

n−1∑

j =0

sgn(E[M H,α

t j +1− M H,α

t j|F M H,α

t j

])1(t j ,t j +1]


=n−1∑j =0

∥∥∥E[M H,α

t j +1− M H,α

t j|F M H,α

t j

]∥∥∥1.

It follows that M H,α is a quasimartingale. Hence, ifM H,α is not a quasi-martingale, it cannot be a weak semimartingale. 2

It remains to prove thatM H,α is not a quasimartingale ifH ∈ (12,

34]. We

do this in the next two lemmas.

Lemma 4.5 For H ∈ (12,

34), MH,α is not a quasimartingale.

Proof. Since conditional expectation is a contraction with respect to theL1-norm, we have for alln ∈ IN and all j = 1, . . . , n − 1,∥∥∥∥E

[1n

j +1M H,α|F M H,α

jn

]∥∥∥∥1

≥∥∥∥E[1n

j +1M H,α|1nj M H,α

]∥∥∥1. (4.3.1)

Moreover,

∥∥∥E[1n


]∥∥∥1

=√

2

π

∥∥∥E[1n


]∥∥∥2

(4.3.2)

because E[1n


]is a centred Gaussian random variable. Us-

ing (4.3.1) and (4.3.2) we obtain

n−1∑j =0

∥∥∥∥E

[1n

j +1M H,α|F M H,α

jn

]∥∥∥∥1

≥√

2

π

n−1∑j =1

∥∥∥E[1n


]∥∥∥2

=√

2

π

n−1∑j =1

∥∥∥∥∥∥Cov

(1n

j +1M H,α,1nj M H,α

)Cov

(1n

j M H,α,1nj M H,α

) 1nj M H,α

∥∥∥∥∥∥2

=√

2

π

n−1∑j =1

Cov(1n

j +1M H,α,1nj M H,α

)√

Cov(1n

j M H,α,1nj M H,α

)

=√

2

π

n−1∑j =1

α2n−2H(

22H

2 − 1)

√1n + α2n−2H

≥√

2

πα2

(22H

2− 1

)n−1∑j =1

n−2H√1n + α2 1

n


34] 79

=√

2

π

(22H

2− 1

)α2

√1 + α2

n−1∑j =1

n12−2H

=√

2

π

(22H

2− 1

)α2

√1 + α2

(n − 1)n12−2H → ∞ , asn → ∞ .

This proves the lemma. 2

Lemma 4.6 M34 ,α is not a quasimartingale.

Proof. For H = 34, the estimate (4.3.1) is not good enough. Now we need

that, for alln ∈ IN and all j = 1, . . . , n − 1,∥∥∥∥E

[1n

j +1M34 ,α|F M

34 ,α

jn

]∥∥∥∥1

≥∥∥∥E[1n

j +1M34 ,α|1n

j M34 ,α, . . . , 1n

1M34 ,α]∥∥∥

1,

which follows, like (4.3.1) from the fact that conditional expectation is a con-traction with respect to theL1-norm. Since

E[1n

j +1M34 ,α|1n

j M34 ,α, . . . , 1n

1M34 ,α]

is centred Gaussian,∥∥∥E[1n

j +1M34 ,α|1n

j M34 ,α, . . . , 1n

1M34 ,α]∥∥∥

1

=√

2

π

∥∥∥E[1n

j +1M34 ,α|1n

j M34 ,α, . . . , 1n

1M34 ,α]∥∥∥

2.

Hence,n−1∑j =0

∥∥∥∥E

[1n

j +1M34 ,α|F M

34 ,α

jn

]∥∥∥∥1

≥√

2

π

n−1∑j =1

∥∥∥E[1n

j +1M34 ,α|1n

j M34 ,α, . . . , 1n

1M34 ,α]∥∥∥

2,

and the lemma is proved if we can show that

n−1∑j =1

∥∥∥E[1n

j +1M34 ,α|1n

j M34 ,α, . . . , 1n

1M34 ,α]∥∥∥

2

(n→∞)−→ ∞ . (4.3.3)


For n ∈ IN and j ∈ {1, . . . , n − 1},(1n

j +1M34 ,α,1n

j M34 ,α, . . . , 1n

1M34 ,α)

is a Gaussian vector. Therefore,

E[1n

j +1M34 ,α|1n

j M34 ,α, . . . , 1n

1M34 ,α]

=j∑

k=1

bk1nk M

34 ,α , (4.3.4)

where the vectorb = (b1, . . . , bj )T solves the system of linear equations

m = Ab, (4.3.5)

in whichm is a j -vector whosek-th componentmk is

Cov(1n

j +1M34 ,α,1n

k M34 ,α)

andA is the covariance matrix of the Gaussian vector(1n

1M34 ,α, . . . , 1n

j M34 ,α).

Note thatA is symmetric and, since the random variables

1n1M

34 ,α, . . . , 1n

j M34 ,α

are linearly independent, also positive definite. It follows from (4.3.4) and(4.3.5) that ∥∥∥E

[1n

j +1M34 ,α|1n

j M34 ,α, . . . , 1n

1M34 ,α]∥∥∥2

2(4.3.6)

= bT Ab = mT A−1m ≥ ‖m‖22 λ

−1 ,

whereλ is the largest eigenvalue of the matrixA. Since

A = 1

nid + α2C ,

whereC is the covariance matrix of the increments of fractional Brownianmotion (

1n1 B

34 , . . . , 1n

j B34

),

we have

λ = 1

n+ α2µ ,


34] 81

whereµ is the largest eigenvalue ofC. As

Ckl = n− 32

1

2

((|k − l | + 1)

32 − 2 |k − l | 3

2 + ||k − l | − 1| 32

),

k, l = 1, . . . , j , it follows from the Gershgorin Circle Theorem (see e.g.Golub and Van Loan (1989)) and the special form ofC that

µ ≤ maxk=1,..., j

j∑l=1

|Ckl | ≤ 2j∑

l=1

|C1l |

= 2n− 32

1

2

j −1∑l=0

((l + 1)

32 − 2l

32 + |l − 1| 3

2

)= n− 3

2

(1 + j

32 − ( j − 1)

32

).

Furthermore,

n− 32

(1 + j

32 − ( j − 1)

32

)≤ 1

n+ n− 3

2∂

∂ jj

32

= 1

n+ n− 3

23

2j

12 ≤ 1

n+ n− 3

23

2n

12 ≤ 3

1

n.

Hence,

λ ≤ 1

n+ α23

1

n=(1 + 3α2

) 1

n.

andλ−1 ≥ n

1 + 3α2 . (4.3.7)

On the other hand,

‖m‖22 =

j∑k=1

(Cov

(1n

j +1M34 ,α,1n

k M34 ,α))2

= α4j∑

k=1

(Cov

(1n

j +1B34 ,1n

k B34

))2

= α4 1

4n−3

j∑k=1

((k + 1)

32 − 2k

32 + (k − 1)

32

)2.

Since the functionx 7→ x32 is analytic on

{x ∈ Cl : Rex > 0

},

(k + 1)32 − 2k

32 + (k − 1)

32 =

∞∑m=1

1

m!∂m

∂kmk

32 + (−1)m

1

m!∂m

∂kmk

32


≥ ∂2

∂k2 k32 = 3

4k− 1

2 , k = 2, . . . j .

That

(k + 1)32 − 2k

32 + (k − 1)

32 ≥ 3

4k− 1

2

also holds fork = 1, can be checked directly. It follows that

‖m‖22 ≥ α4 1

4n−3 9

16

j∑k=1

1

k≥ α4 9

64n−3

∫ j

1

1

xdx = α4 9

64n−3 log j . (4.3.8)

Putting (4.3.6), (4.3.7) and (4.3.8) together, we obtain

n−1∑j =1

∥∥∥E[1n

j +1M34 ,α|1n

j M34 ,α, . . . , 1n

1M34 ,α]∥∥∥

2

≥ 3

8

α2

√1 + 3α2

1

n

n−1∑j =1

√log j → ∞ , asn → ∞ .

Hence, (4.3.3) holds and the lemma is proved. 2

4.4 Proof of Theorem 4.2 forH ∈ (34, 1]

To show that forH ∈ (34, 1], M H,α is equivalent to Brownian motion we use

the concept of relative entropy. The following definition and all results onrelative entropy that we need in this section can be found in chapter 6 of Hidaand Hitsuda (1976).

Definition 4.7 Let Q1 and Q2 be probability measures on a measurable space(�, E) and denote byP all finite partitions

� =n⋃

j =1

Ej , where Ej ∈ E , Ej ∩ Ek = ∅ if j 6= k ,

of�. The entropy H(Q1|Q2) of Q1 relative to Q2 is given by

H (Q1|Q2) := supP

n∑j =1

log

(Q1[Ej ]Q2[Ej ]

)Q1[Ej ] ,

where we assume00 = 0 log 0= 0.

4.4. Proof of Theorem 4.2 forH ∈ (34, 1] 83

For alln ∈ IN, we defineYn : C[0, 1] → IRn by

Yn(ω) =(ω

(1

n

)− ω (0) , . . . , ω (1)− ω

(n − 1

n

))T

,

and setBn = σ(Yn). Note that∨∞

n=1 Bn equals theσ -algebraB generatedby the cylinder sets. We denote byQM H,α the measure induced byM H,α on(C[0, 1],B) and byQW Wiener measure on(C[0, 1],B). Further, we let forall n ∈ IN, Qn

M H,α andQnW be the restrictions ofQM H,α andQW, respectively,

to Bn.To show thatM H,α is equivalent to Brownian motion, we make use of the

following lemma.

Lemma 4.8 Ifsup

nH(Qn

M H,α |QnW

)< ∞, (4.4.1)

then QM H,α and QW are equivalent.

Proof. From (4.4.1) it follows by Lemma 6.3 of Hida and Hitsuda (1976)that QM H,α is absolutely continuous with respect toQW. But two Gaussianmeasures on(C[0, 1],B) can only be equivalent or singular (see e.g. Theorem6.1 of Hida and Hitsuda). ThereforeQMα,H andQW must be equivalent. 2

In the following lemma we show that (4.4.1) holds.

Lemma 4.9sup

nH(Qn

M H,α |QnW

)< ∞

Proof. For all n ∈ IN, Yn is a centred Gaussian vector under both measuresQn

M H,α andQnW. The covariance matrices ofYn underQn

M H,α andQnW are

EQnM H,α

[YnYT

n

]= 1

nid + α2Cn ,

whereCn is the covariance matrix of the increments of fractional Brownianmotion (

1n1BH , . . . , 1n

n BH)

and

EQnW

[YnYT

n

]= 1

nid .

SinceCn is symmetric, there exists an orthogonaln × n-matrix Un such thatUnCnU T

n is a diagonal matrixDn = diag(λn

1, . . . λnn

). Xn = √

nUnYn is still a


centred Gaussian vector under both measuresQnM H,α andQn

W. The covariancematrices ofXn under these two measures are

EQnM H,α

[XnXT

n

]= id + nα2Dn

andEQn

W

[XnXT

n

]= id .

ThroughXn, QnM H,α andQn

W induce measuresRnM H,α and Rn

W on IRn. It caneasily be seen from Definition 4.7 that

H(Qn

M H,α |QnW

) = H(Rn

M H,α |RnW

).

Since both measuresRnM H,α and Rn

W are non-degenerate Gaussian measureson IRn, they are equivalent. We denote byξn the Radon-Nikodym derivativeof Rn

M H,α with respect toRnW. Lemma 6.1 of Hida and Hitsuda (1976) and a

calculation show that

H(Rn

M H,α |RnW

) = ERnM H,α

[logξn

] = 1

2

n∑j =1

(nα2λn

j − log(1 + nα2λn

j

)).

For all x ≥ 0, we have

x − log(1 + x) =∫ x

0

u

1 + udu ≤

∫ x

0udu = 1

2x2 .

Therefore,

H(Rn

M H,α |RnW

) ≤ 1

4n2α4

n∑j =1

(λn

j

)2.

Hence, the lemma is proved if we can show that

supn

n2n∑

j =1

(λn

j

)2< ∞ , (4.4.2)

whereλn1, . . . , λ

nn are the eigenvalues of the covariance matrix of the incre-

ments of fractional Brownian motion(1n

1 BH , . . . , 1nn BH

).

Since orthogonal transformation leaves the Hilbert-Schmidt norm of a matrixinvariant,

n∑j =1

(λn

j

)2 =n∑

j ,k=1

Cov(1n

j BH ,1nk BH

)2.

4.4. Proof of Theorem 4.2 forH ∈ (34, 1] 85

As fractional Brownian motion has stationary increments,

n∑j ,k=1

Cov(1n

j BH ,1nk BH

)2 ≤ 2nn∑

k=1

Cov(1n

k BH ,1n1 BH

)2

= 2nn−4H

1 +

(22H

2− 1

)2+ 2n

n∑k=3

Cov(1n

k BH ,1n1 BH

)2.

Since, forH ∈ (34, 1],

n22nn−4H

1 +

(22H

2− 1

)2 → 0 , asn → ∞ ,

it is enough to show

supn

n3n∑

k=3

Cov(1n

k BH ,1n1BH

)2< ∞ (4.4.3)

to prove (4.4.2). For allk ≥ 3, we have

Cov(1n

k BH ,1n1 BH

)= n−2H 1

2

(k2H − 2 (k − 1)2H + (k − 2)2H

)≤ n−2H 1

2

(∂

∂kk2H − ∂

∂k(k − 2)2H

)

= Hn−2H(k2H−1 − (k − 2)2H−1

)≤ Hn−2H 2

∂

∂k(k − 2)2H−1

= 2H (2H − 1)n−2H (k − 2)2H−2 .

Using this, we obtain

n3n∑

k=3

Cov(1n

k BH ,1n1 BH

)2 ≤ 4H2 (2H − 1)2 n3−4Hn−2∑k=1

k4H−4

≤ 4H2 (2H − 1)2 n3−4H∫ n−2

0x4H−4dx

= 4H2 (2H − 1)2

4H − 3n3−4H (n − 2)4H−3

≤ 4H2 (2H − 1)2

4H − 3.

Hence, (4.4.3) holds, and the lemma is proved. 2


Remark 4.10 In this section we have shown that forH ∈ ( 34, 1], QM H,α and

QW are equivalent. But our method of proof has not given us the Radon-Nikodym derivative nor have we found the semimartingale decomposition ofM H,α. These problems will be addressed in Sections 6, 7 and 8.

4.5 Option pricing with mixed fractional Brown-ian motion

In this section we examine the priceC0 of a European call option with strikeprice K and maturityT = 1. If money in the money market account growslike exp(r t ), t ∈ [0, 1], for a constantr , the option’s discounted pay-off isgiven by (

S1 − e−r K)+.

Let us assume that empirical data suggests that the discounted price of thestockSshould be modelled as


t

), t ∈ [0, 1] , (4.5.1)

for constantsS0 > 0, ν, σ > 0, and a fractional Brownian motionBH . We

have shown in Section 2.3 that forH ∈(0, 1

2

)∪(

12, 1

), such a model admits

arbitrage. However, ifH ∈(

34, 1

), we can exclude all arbitrage strategies by

regularizing fractional Brownian motion in the following way:If (Bt)t∈[0,1] is a Brownian motion independent ofBH , Theorem 4.2 im-

plies that for allε > 0,(εBt + BH

t

)t∈[0,1] is equivalent to (εBt)t∈[0,1] .

We observe that

Cov(εBt + BH

t , εBs + BHs

)= ε2 (t ∧ s)+ Cov

(BH

t , BHs

), t, s ∈ [0, 1] .

Hence,(εBt + BH

t

)t∈[0,1] is a continuous centred Gaussian process that has

up toε2 the same covariance function as(BH

t

)t∈[0,1]. This shows that if the

model (4.5.1) fits empirical data, then so does

St = S0 exp{νt + σ

(εBt + BH

t

)}, t ∈ [0, 1] , (4.5.2)

4.5. Option pricing with mixed fBm 87

for ε > 0 small enough. But in contrast to (4.5.1), (4.5.2) has, like the Samuel-son model, a unique equivalent martingale measureQε. This implies that themodel (4.5.2) is arbitrage-free and also complete if we allow all strategies of

2sf,adm(lFS), given in (3.3.10). According to current practice in mathematical

finance, in such a framework options are priced by taking the expected valueunder the equivalent martingale measure of the option’s discounted pay-off.In the model (4.5.2) this leads to the following option price:

C0(ε) = EQε

[(S0 exp

{ν + σ

(εB1 + BH

1

)}− e−r K

)+]

= BS(0, S0, εσ ) , (4.5.3)

where BS is the Black-Scholes price. Since the function BS(0, S0, .) is contin-uous, strictly increasing and bijective from the interval(0,∞) to the interval((S0 − e−r K )+, S0), C0(ε) in (4.5.3) is close to(S0 − e−r K )+ whenε > 0 issmall. The deeper reason whyC0(ε) is so low in this situation, is that (4.5.3)gives the initial capital necessary to replicate the pay-off of the call option with

a trading strategy from2sf,adm(lFS), and this strategy seems to exploit small

movements of the stochastic process (4.5.2) over very short time intervals.In reality a seller of the option can only carry out finitely many transactions

to hedge the option. Moreover, he cannot buy and sell within nanoseconds.Therefore he will demand a higher price than BS(0, S0, εσ ) ' (

S0 − e−r K)+.

To find a reasonable option price, one should introduce a waiting timeh > 0 and restrict trading strategies to the class2h

sf(lFS) (2.4.1) of self-

financing strategies that can buy and sell atlFS-stopping times but after eachtransaction there must be a waiting period of minimal lengthh before the next.For smallε > 0, the discounted gains process of such a strategy is similar inboth models (4.5.1) and (4.5.2), as should be the case. Moreover (4.5.1) has noarbitrage in2h

sf

(lFS). Hence, if we confine the strategies to the class2h

sf

(lFS),

we can return to the model (4.5.1) to value the option. Since (4.5.1) with thestrategies2h

sf

(lFS) is an incomplete model, one has to decide in which sense

the pay-off of the option should be approximated and then search for an opti-mal strategy. It is not clear whether the regularization (4.5.2) is of any use insuch a procedure.


4.6 Representations of Gaussian processes thatare equivalent to Brownian motion

In this section we discuss general results of Shepp (1966) and Hitsuda (1968)on representations of Gaussian processes that are equivalent to Brownian mo-tion.

Let 0< T ≤ ∞. For 0< T < ∞, we setIT = [0, T ], and forT = ∞,I∞ = [0,∞). Let C(IT ) be the space of real-valued, continuous functions onIT . The coordinates process(Xt )t∈IT

on C(IT ) is given by

Xt(ω) = ω(t) , ω ∈ C(IT ) , t ∈ IT .

It generates theσ -algebra

BT := σ{

X−1t (B) : t ∈ IT , B an open subset of IR

}.

By QW we denote Wiener measure on(C(IT ),BT ). Every almost surely con-tinuous process(Yt )t∈IT

on a probability space(�,A, P) has a distributionQY on (C(IT ),BT ). It is given by

QY [B] = P [Y ∈ B] , B ∈ BT .

4.6.1 The representations of Shepp and Hitsuda

Let (Yt )t∈ITbe a Gaussian process on a probability space(�,A, P), that is,

for all n ∈ IN and {t1, t2, . . . , tn} ⊂ IT , the distribution of(Yt1,Yt2, . . . ,Ytn

)with respect toP is n-dimensional Gaussian. We set

MYt = EP [Yt ] , t ∈ IT ,

and0Y

ts = EP

[(Yt − MY

t )(Ys − MYs )], t, s ∈ IT .

We call (Yt )t∈ITcentred ifMY

t = 0, t ∈ IT . It follows from Theorem 1 ofShepp (1966) that an a.s. continuous Gaussian process(Yt )t∈IT

is equivalentto Brownian motion if and only if

MYt =

∫ t

0mY(u)du , t ∈ IT ,

for a mY ∈ L2(IT ) and(Yt − MY

t

)t∈IT

is equivalent to Brownian motion.Therefore, we will henceforth only treat a.s. continuous, centred Gaussianprocesses.

4.6. Representions of Gaussian processes equivalent to Bm 89

Before we start discussing representations of a.s. continuous, centredGaussian processes(Yt )t∈IT

that are equivalent to Brownian motion, we col-lect some properties of integral operators induced byL2-kernels. The proofsof all these facts can be found in Smithies (1958) or Dunford and Schwartz(1963).

An L2-kernel is ak ∈ L2(I 2T ). It induces a Hilbert-Schmitd operator

kop : L2(IT ) → L2(IT ) ,

given by

kop f (t) =∫ T

0k(t, s) f (s)ds, t ∈ IT , f ∈ L2(IT ) .

The spectrumσ(kop) consists of at most countably many points. Every non-zero value inσ(kop) is an eigenvalue of finite multiplicity. If

{λ j}N

j =1, N ∈IN ∪ {∞}, is the family of non-zero eigenvalues ofkop, repeated according totheir multiplicity, then

N∑j =1

∣∣λ j∣∣2 < ∞ .

The Carleman-Fredholm determinantδk : Cl → Cl is defined by the conver-gent product

δk(λ) =N∏

j =1

(1 − λλ j )eλλ j , λ ∈ Cl .

kop is said to be of trace class if∑N

j =1

∣∣λ j∣∣ < ∞. If kop is of trace class, its

trace is defined to be

tr(kop) =N∑

j =1

λ j .

If k, l ∈ L2(I 2T ), thenk ∗ l given by

k ∗ l (t, s) =∫ T

0k(t, u)l (u, s)du , t, s ∈ IT ,

is again inL2(I 2T ), and

(k ∗ l )op = kopl op .


Moreover,kopl op is always of trace class and

tr(kopl op) =∫ T

0

∫ T

0k(t, s)l (s, t)dsdt.

If k ∈ L2(I 2T ) and 1 /∈ σ(kop), then there exists a unique kernelnk ∈ L2(I 2

T )

such that (id − kop)−1 = id − nop

k .

We callnk the negative resolvent kernel ofk because−nk is usually called theresolvent kernel ofk for the value 1. Ifk is continuous, then so isnk. If k isreal-valued and symmetric,kop is self-adjoint. Therefore, all eigenvaluesλ j

are real, the corresponding eigenfunctionsej can be chosen orthonormal, andk can be represented as

k(t, s) =N∑

j =1

λ j ej (t)ej (s) ,

where the series converges inL2(I 2T ). It follows that

nk(t, s) =N∑

j =1

−λ j

1 − λ jej (t)ej (s) .

In particular,nk is again real-valued and symmetric.The proof of the following theorem can be found in Shepp (1966).

Theorem 4.11 (Shepp)a) A 0 : I 2

T → IR is the covariance function of an a.s. continuous, centredGaussian process equivalent to Brownian motion if and only if it is of the form

0ts = t ∧ s −∫ t

0

∫ s

0k(u, v)dvdu , t, s ∈ IT ,

where k is in L2(I 2T ), real-valued, symmetric andσ(kop) ⊂ (−∞, 1).

b) Let (Yt )t∈ITbe an a.s. continuous, centred Gaussian process that is equiv-

alent to Brownian motion. Then

d QY

d QW(X) = cexp

(∫ T

0

∫ s

0nk(s, u)d Xud Xs

), (4.6.1)

where nk is the negative resolvent kernel of the L2-kernel k that satisfies

0Yts = t ∧ s −

∫ t

0

∫ s



and

c = 1√δk(1) exp

{tr(−nop

k kop)} .

Remarks 4.121. We call (4.6.1) the Shepp-representation ofQY.2. nk is the uniqueL2-kernel that solves the equation

nk(t, s)+ k(t, s) =∫ T

0nk(t, u)k(u, s)du , t, s ∈ IT .

It is also the uniqueL2-kernel that solves the equation

nk(t, s)+ k(t, s) =∫ T

0k(t, u)nk(u, s)du , t, s ∈ IT .

3. Let 0 : I 2T → IR be of the form

0ts = t ∧ s −∫ t

0

∫ s

0k(u, v)dvdu , t, s ∈ IT , (4.6.2)

for a real-valued, symmetricL2-kernelk. 0 is the covariance function of acentred Gaussian process if and only if it is positive semi-definite, that is,

n∑j ,k=1

zj0t j tk zk ≥ 0 ,

for all n ∈ IN, {t1, . . . , tn} ⊂ IT andz ∈ Cl n. But (4.6.2) can be written as

0ts = (1[0,t], 1[0,s]

)− (1[0,t], kop1[0,s]

), t, s ∈ IT ,

where( f, g) = ∫ T0 f (t)g(t)dt, f, g ∈ L2(IT ). This and the fact that the

functions of the form

n∑j =1

zj 1[0,t j ] , n ∈ IN , {t1, . . . , tn} ⊂ IT , z ∈ Cl n ,

are dense inL2(IT ), imply that (4.6.2) is the covariance function of a centredGaussian process if and only ifσ(kop) ⊂ (−∞, 1].


Corollary 4.13a) Let (Yt )t∈IT

be an a.s. continuous, centred Gaussian process equivalentto Brownian motion on a probability space(�,A, P) and k the real-valued,symmetric L2-kernel that satisfies

0Yts = t ∧ s −

∫ t

0

∫ s

0k(u, v)dvdu , t, s ∈ IT .

Then the following two are equivalent:(i) There exists a probability space

(�′,A′, P′) with a Brownian mo-

tion (Bt )t∈ITand an independent, a.s. continuous, centred Gaussian

process(Zt )t∈ITsuch that

(Yt )t∈IT= (Bt + Zt )t∈IT

in distribution.

(ii) σ(kop) ⊂ (−∞, 0].b) Let (Bt)t∈IT

be a Brownian motion and(Zt )t∈ITan independent, a.s. con-

tinuous, centred Gaussian process. Then the following two are equivalent:(i) (Bt + Zt )t∈IT

is equivalent to Brownian motion.(ii) There exists a real-valued, symmetric L2-kernel k such that

0Zts = −

∫ t

0

∫ s


Proof.a) Let (Bt)t∈IT

be a Brownian motion and(Zt )t∈ITan independent, a.s. con-

tinuous, centred Gaussian process on a probability space(�′,A′, P′) such

that(Yt )t∈IT

= (Bt + Zt )t∈ITin distribution.

Then

0Zts = 0Y

ts − t ∧s = −∫ t

0

∫ s

0k(u, v)dvdu = − (

1[0,t], kop1[0,s]), t, s ∈ IT ,

where( f, g) = ∫ T0 f (t)g(t)dt, f, g ∈ L2(IT ). Since0Z

ts is a covariancefunction and therefore positive semi-definite and the functions of the form

n∑j =1

zj 1[0,t j ] , n ∈ IN , {t1, . . . , tn} ⊂ IT , z ∈ Cl n ,

are dense inL2(IT ), it follows thatσ(kop) ⊂ (−∞, 0].


On the other hand, ifσ(kop) ⊂ (−∞, 0], then

−∫ t

0

∫ s


is positive semi-definite. It follows that there exists a probability space(�′,A′, P′) with a Brownian motion(Bt)t∈IT

and an independent centredGaussian process(Zt)t∈IT

such that0Zts = − ∫ t

0

∫ s0 k(u, v)dvdu , t, s ∈ IT .

Hence,

0B+Zts = t ∧ s −

∫ t

0

∫ s

0k(u, v)dvdu = 0Y

ts , t, s ∈ IT ,

and therefore,

(Bt + Zt )t∈IT= (Yt )t∈IT

in distribution.

Since(Bt )t∈0,T and(Yt )t∈0,T are a.s. continuous,(Zt )t∈0,T can also be chosento be a.s. continuous.b) If (Bt + Zt )t∈IT

is equivalent to Brownian motion, then it follows fromTheorem 4.11 a) that there exists a real-valued, symmetricL2-kernelk suchthat

t ∧ s + 0Zts = 0B+Z

ts = t ∧ s −∫ t

0

∫ s


Hence,

0Zts = −

∫ t

0

∫ s


On the other hand, if there exists a real-valued, symmetricL2-kernel ksuch that

0Zts = −

∫ t

0

∫ s

0k(u, v)dudv , t, s ∈ IT ,

thenσ(kop) ⊂ (−∞, 0] because0Z is positive semi-definite. Since

0B+Zts = t ∧ s −

∫ t

0

∫ s

0k(u, v)dudv , t, s ∈ IT ,

it follows from Theorem 4.11 a) that(Bt + Zt )t∈ITis equivalent to Brownian

motion. 2


Remark 4.14 Let 0 < T < ∞, (Bt)0≤t≤T a Brownian motion and(BH

t

)t∈[0,T] an independent fractional Brownian motion with Hurst parameter

H ∈ (0, 1]. Since

0BH

ts = 1

2

(t2H + s2H − |t − s|2H

)

= H (2H − 1)∫ t

0

∫ s

0|u − v|2H−2 dvdu , t, s ∈ [0, T ] ,

it follows from Corollary 4.13 b) that for everyα ∈ IR \ {0}, the mixed frac-tional Brownian motion

M H,αt = Bt + αBH

t , t ∈ [0, T ] ,is equivalent to Brownian motion if and only ifH ∈ ( 3

4, 1]. This assertionis the part of Theorem 4.2 which we proved in Section 4.4. Note that for themixed fractional Brownian motionM H,α, condition (ii) of Corollary 4.13 b)is equivalent to condition (4.4.2).

For H ∈ (34, 1], let λH be the largest eigenvalue ofH (2H − 1) f op

2H−2,where

f p(t, s) = |t − s|p , t, s ∈ [0, T ] , p ∈ (−1

2,∞) .

SinceH (2H − 1) f op2H−2 is positive semi-definite,λH is equal to the operator

norm∥∥H (2H − 1) f op

2H−2

∥∥ > 0 of H (2H − 1) f op2H−2. Hence, ifc ∈ (0, 1

λH),

thent ∧ s − c

2

(t2H + s2H − |t − s|2H

), t, s ∈ [0, T ] ,

is the covariance function of an a.s. continuous, centred Gaussian processequivalent to Brownian motion which cannot have the same distribution as thesum of a Brownian motion and an independent Gaussian process.

t ∧ s − 1

2λH

(t2H + s2H − |t − s|2H

), t, s ∈ [0, T ] ,

is the covariance function of a centred Gaussian process that is neither equiv-alent to Brownian motion nor equal in distribution to the sum of a Brownianmotion and an independent Gaussian process.

A kernelh ∈ L2(I 2T ) is called a Volterra kernel ifh(t, s) = 0 for all s > t . In

this casehop is quasi-nilpotent, that is, the spectral radius

sup{|λ| : λ ∈ σ(hop)

} = lim infn→∞

∥∥(hop)n∥∥ 1n


is zero. Hence, the negative resolvent kernelnh of h always exists, and itcan be shown that it is also a Volterra kernel. Moreover, ifh is a real-valuedVolterra kernel, then so isnh.

The proof of the following theorem can be found in Hitsuda (1968).

Theorem 4.15 (Hitsuda)a) Let(Yt )t∈IT

be an a.s. continuous, centred Gaussian process on a probabil-ity space(�,A, P) that is equivalent to Brownian motion. Then there existsa unique real-valued Volterra kernel h∈ L2(I 2

T ) such that

Wt = Yt −∫ t

0

∫ s

0h(s, u)Yuds, t ∈ IT , (4.6.3)

is a Brownian motion on(�,A, P). Furthermore,

Yt = Wt −∫ t

0

∫ s

0nh(s, u)dWuds, t ∈ IT , (4.6.4)

where nh ∈ L2(I 2T ) is the negative resolvent kernel of h, and the representa-

tion (4.6.4) is unique in the following sense: If(Bt )t∈ITis a Brownian motion

on (�,A, P) and l ∈ L2(I 2T ) a real-valued Volterra kernel such that

Yt = Bt −∫ t

0

∫ s

0l (s, u)d Buds, t ∈ IT ,

then B= W and l= nh.b) Let (Bt)t∈IT

be a Brownian motion on a probability space(�,A, P) andl ∈ L2(I 2

T ) a real-valued Volterra kernel. Then

EP

[exp

(∫ T

0

∫ s

0l (s, u)d Bud Bs − 1

2

∫ T

0

(∫ s

0l (s, u)d Bu

)2

ds

)]= 1 ,

(4.6.5)and, by Girsanov’s (1960) theorem,

Bt −∫ t

0

∫ s

0l (s, u)d Buds, t ∈ IT ,

is a Brownian motion on(�,A, P

), where

P = exp

(∫ T

0

∫ s

0l (s, u)d Bud Bs − 1

2

∫ T

0

(∫ s

0l (s, u)d Bu

)2

ds

)· P .


Remarks 4.161. It follows from (4.6.3) and (4.6.4) that

F Wt = σ {Ws : 0 ≤ s ≤ t} = σ {Ys : 0 ≤ s ≤ t} = F Y

t , t ∈ IT .

Therefore, (4.6.4) is the canonical semimartingale decomposition ofY on(�,A,

(F Y

t

)t∈IT

, P

).

We call it the Hitsuda representation ofY.2. nh is the unique Volterra kernel that solves the equation

nh(t, s)+ h(t, s) =∫ t

snh(t, u)h(u, s)du , t, s ∈ IT , s ≤ t .

It is also the unique Volterra kernel that solves the equation

nh(t, s)+ h(t, s) =∫ t

sh(t, u)nh(u, s)du , t, s ∈ IT , s ≤ t .

4.6.2 The Girsanov-Hitsuda representation and relationsbetween different representations

Let (Yt )t∈ITbe an a.s. continuous, centred Gaussian process that is equivalent

to Brownian motion andk, nk, h, nh the kernels from Subsection 4.6.1.

Proposition 4.17d QY

d QW(X) = (4.6.6)

exp

(∫ T

0

∫ s

0h(s, u)d Xud Xs − 1

2

∫ T

0

(∫ s

0h(s, u)d Xu

)2

ds

).

Proof. It follows from Theorem 4.15 a) that

W = Xt −∫ t

0

∫ s

0h(s, u)d Xuds, t ∈ IT ,

is a Brownian motion on(C(IT ),BT , QY). On the other hand, Theorem

4.15 b) implies thatW is also a Brownian motion on(C(IT ),BT , Q

), where

Q = exp

(∫ T

0

∫ s


2

∫ T

0

(∫ s

0h(s, u)d Xu

)2

ds

)· QW .


SinceF WT = F X

T = BT , it follows that QY = Q on (C(IT ),BT ). Thisproves the proposition. 2

Remark 4.18 Since the second part of Theorem 4.15 b) follows from (4.6.5)by Girsanov’s (1960) theorem, we call (4.6.6) the Girsanov-Hitsuda represen-tation of QY.

Corollary 4.19

nk(t, s) = h(t, s)−∫ T

th(v, t)h(v, s)dv , t, s ∈ IT , s ≤ t . (4.6.7)

logc = −1

2

∫ T

0

∫ s

0h2(s, u)duds. (4.6.8)

Proof. By comparing (4.6.1) with (4.6.6) we get

log c +∫ T

0

∫ s

0nk(s, u)d Xud Xs

=∫ T

0

∫ s


2

∫ T

0

(∫ s

0h(s, u)d Xu

)2

ds

=∫ T

0

∫ s

0h(s, u)d Xud Xs

−1

2

∫ T

0

[2∫ s

0

∫ v

0h(s, u)d Xuh(s, v)d Xv +

∫ s

0h2(s, v)dv

]ds

=∫ T

0

∫ s

0h(s, u)d Xud Xs −

∫ T

0

∫ s

0

∫ T

sh(v, s)h(v, u)dvd Xud Xs

−1

2

∫ T

0

∫ s

0h2(s, v)dvds.

Now (4.6.8) follows by taking expectation with respect toQW. The fact thatfor every real-valued Volterra kernell ∈ L2(I 2

T ),

EQW

[(∫ T

0

∫ s

0l (s, u)d Xud Xs

)2]=∫ T

0

∫ s

0l 2(s, u)duds,

entails that the linear mapping

l 7→∫ T

0

∫ s

0l (s, u)d Xud Xs


is an injection from the real-valued Volterra kernels toL2(C(IT ),BT , QW).Thogether with (4.6.8), this implies (4.6.7). 2

Proposition 4.20

k(t, s) = nh(t, s)−∫ s

0nh(t, v)nh(s, v)dv , t, s ∈ IT , s ≤ t . (4.6.9)

Proof. Recall thatnh(t, s) = 0 for s> t . We definenh by

nh(t, s) ={

nh(t, s) for s ≤ tnh(s, t) for s> t

.

For all t, s ∈ IT with s ≤ t , we have∫ t

0

∫ s

0k(u, v)dvdu = s − 0Y

ts

= s − EP

[(Wt −

∫ t

0

∫ u

0nh(u, v)dWvdu

)(

Ws −∫ s

0

∫ u

0nh(u, v)dWvdu

)]

= s − EP

[(Wt −

∫ t

0

∫ t

v

nh(u, v)dudWv

)(

Ws −∫ s

0

∫ s

v

nh(u, v)dudWv

)]

=∫ s

0

∫ t

v

nh(u, v)dudv +∫ s

0

∫ s

v

nh(u, v)dudv

−∫ s

0

(∫ t

v

nh(u, v)du

)(∫ s

v

nh(u, v)du

)dv

=∫ t

0

∫ s

0nh(u, v)dvdu −

∫ t

0

∫ s

0

∫ u∧v

0nh(u, x)nh(v, x)dxdvdu .

Hence,

k(t, s) = nh(t, s)−∫ t∧s

0nh(t, v)nh(s, v)dv for all t, s ∈ IT ,

which implies (4.6.9). 2

Remark 4.21 Sincek andnh are the negative resolvent kernels ofnk andh,respectively, it should in principle be possible to derive (4.6.9) from (4.6.7)directly and vice versa.

4.7. The Shepp-represention of mfBm 99

4.7 The Shepp-represention of mixed fractionalBrownian motion

It follows from Theorem 4.2 and Theorem 4.11 b) that for allH ∈ ( 34, 1] and

α ∈ IR \ {0} there exists a real-valued, symmetricq ∈ L2([0, 1]2) and ac > 0such that

d QM H,α

d QW(X) = cexp

(∫ 1

0

∫ t

0q(t, s)d Xsd Xt

).

As we have seen in Remark 4.12.2,q is the uniqueL2-kernel that solves theequation

q(t, s)+ α2H (2H − 1)∫ 1

0q(t, x) |x − s|2H−2 dx (4.7.1)

= α2H (2H − 1) |t − s|2H−2 , (t, s) ∈ [0, 1]2 ,and

c =(

E

[exp

(∫ 1

0

∫ t

0q(t, s)d Xsd Xt

)])−1

.

For H = 1, equation (4.7.1) reduces to

q(t, s)+ α2∫ 1

0q(t, x)dx = α2 , (t, s) ∈ [0, 1]2 ,

which is solved by

q(t, s) = α2

1 + α2.

It follows that

d QM1,α

d QW(X) = 1√

1 + α2exp

(α2

1 + α2

1

2X2

1

). (4.7.2)

To treat the caseH ∈ (34, 1) we introduce the following notation. We let

p = 2H − 2 ∈(

−1

2, 0

), µ = α2H (2H − 1) ≥ 0 ,

and we define the bounded linear operatorA : L2[0, 1]2 → L2[0, 1]2 by

Al(t, s) =∫ 1

0l (t, x)dx , l ∈ L2[0, 1]2 .


Theorem 4.22 Let

µ ≥ 0 and ρ = min

{1

2,

1 + µ

1 + 3µ+ 4µ2

}.

Then for all p∈ (−ρ, ρ), the unique qp ∈ L2[0, 1]2 that solves

qp(t, s)+µ∫ 1

0qp(t, x) |x − s|p dx = µ |t − s|p , (t, s) ∈ [0, 1]2 , (4.7.3)

is given by

qp(t, s) = µ |t − s|p − µ

∞∑n=0

qn(t, s)pn , (4.7.4)

whereq0(t, s) = µ

1 + µ(4.7.5)

and for n≥ 1,

qn(t, s) = µ

(id − µ

1 + µA

)(4.7.6)

∫ 1

0

1

1 + µ

lnn |x − s|n! dx +

n∑j =1

∫ 1

0

ln j |t − x|j !

ln(n− j ) |x − s|(n − j )! dx

−n−1∑j =1

∫ 1

0qj (t, x)

ln(n− j ) |x − s|(n − j )! dx

.

In particular, all qn are continuous, bounded, symmetric and for allp ∈ [0, ρ),

∞∑n=0

‖qn‖∞ pn < ∞ ,

(It follows that for all p∈ (−ρ, ρ),∞∑

n=0

qn(t, s)pn

converges uniformly in(t, s) ∈ [0, 1]2, to a continuous, bounded and symmet-ric functionqp.)


Proof. It follows inductively from (4.7.6) that allqn are continuous and boun-ded on[0, 1]2. Next we show that for allp ∈ [0, ρ),

∞∑n=0

‖qn‖∞ pn < ∞ .

For n ≥ 1,

1

1 + µ

lnn |x − s|n! +

n∑j =1

ln j |t − x|j !

ln(n− j ) |x − s|(n − j )!

has for allt, s, x ∈ [0, 1] with x 6= s andx 6= t , the same sign and its absolutevalue is everywhere smaller or equal to the absolute value of

n∑j =0

ln j |t − x|j !

ln(n− j ) |x − s|(n − j )! = 1

n! (ln |t − x| + ln |x − s|)n .

Moreover,

|ln |t − x| + ln |x − s||n ≤ 2n−1∣∣lnn |t − x| + lnn |x − s|∣∣ .

This implies for all(t, s) ∈ [0, 1]2,∣∣∣∣(

id − µ

1 + µA

)∫ 1

0

(1

1 + µ

lnn |x − s|n! +

n∑j =1

ln j |t − x|j !

ln(n− j ) |x − s|(n − j )!

dx

∣∣∣∣∣∣≤

∫ 1

0

∣∣∣∣ 1

1 + µ

lnn |x − s|n! +

n∑j =1

ln j |t − x|j !

ln(n− j ) |x − s|(n − j )!

∣∣∣∣∣∣ dx

∨∫ 1

0

∫ 1

0

∣∣∣∣ 1

1 + µ

lnn |x − s|n! +

n∑j =1

ln j |t − x|j !

ln(n− j ) |x − s|(n − j )!

∣∣∣∣∣∣dxds

≤ 2n−1

n!∫ 1

0

∣∣lnn |t − x| + lnn |x − s|∣∣ dx

∨2n−1

n!∫ 1

0

∫ 1

0

∣∣lnn |t − x| + lnn |x − s|∣∣dxds≤ 2n+1 ,


where the last inequality follows from the fact that

∫ 1

0

∣∣∣lnk |t − x|∣∣∣dx ≤ 2k! for all t ∈ [0, 1] and allk ∈ IN .

Furthermore,∣∣∣∣∣∣(

id − µ

1 + µA

)n−1∑j =1

∫ 1

0qj (t, x)

ln(n− j ) |x − s|(n − j )! dx

∣∣∣∣∣∣

≤(

1 + µ

1 + µ

)2

n−1∑j =1

∥∥qj∥∥∞ = 2 + 4µ

1 + µ

n−1∑j =1

∥∥qj∥∥∞ .

Hence, forqn as in (4.7.6), we obtain for alln ≥ 1,

‖qn‖∞ ≤ 2µ2n + 2µ+ 4µ2

1 + µ

n−1∑j =1

∥∥qj∥∥∞ .

Therefore, we can estimate the‖qn‖∞ as follows:

‖qn‖∞ ≤ an , n ≥ 1 ,

where the sequence(an)∞n=1 is recursively defined by

a1 = 4µ (4.7.7)

and

an = 2µ2n +(

2µ+ 4µ2

1 + µ

)n−1∑j =1

aj , n ≥ 2 . (4.7.8)

The radius of convergence of∑∞

n=0 ‖qn‖∞ pn is at least as big as the radiusof convergence of

∑∞n=1 anzn. It follows from (4.7.7) and (4.7.8) that for

sufficiently smallz,

∞∑n=1

anzn = 2µ∞∑

n=1

(2z)n + 2µ+ 4µ2

1 + µ

∞∑n=1

anzn∞∑

n=1

zn

= 2µ∞∑

n=1

(2z)n + 2µ+ 4µ2

1 + µ

∞∑n=1

anzn z

1 − z.


Hence,1 − 1+3µ+4µ2

1+µ z

1 − z

∞∑n=1

anzn = 2µ∞∑

n=1

(2z)n ,

and ∞∑n=1

anzn = 1 − z

1 − 1+3µ+4µ2

1+µ z2µ

∞∑n=1

(2z)n .

Since the right-hand side of the above equation is analytic inz ∈ Cl on anopen disc with center 0 and radius

ρ = min

{1

2,

1 + µ

1 + 3µ+ 4µ2

},

ρ is the radius of convergence of∑∞

n=1 anzn. This shows that for allp ∈ [0, ρ),

∞∑j =0

‖qn‖∞ pn < ∞ .

To show that

(4.7.4) qp(t, s) = µ |t − s|p − µ

∞∑n=0

qn(t, s)pn

solves equation (4.7.3), we note that for all(t, s) ∈ [0, 1]2 with t 6= s, we canwrite

|t − s|p =∞∑

n=0

lnn |t − s|n! pn . (4.7.9)

Plugging (4.7.4) and (4.7.9) into (4.7.3) yields

µ

∞∑n=0

qn(t, s)pn + µ2

∫ 1

0

∞∑n=0

qn(t, x)pn∞∑

n=0

lnn |x − s|n! pndx (4.7.10)

= µ2∫ 1

0

∞∑n=0

lnn |t − x|n! pn

∞∑n=0

lnn |x − s|n! pndx , (t, s) ∈ [0, 1]2 .

Since for allp ∈ [0, ρ) and all(t, x) ∈ [0, 1]2,

∞∑n=0

|qn(t, x)| pn ≤∞∑

n=0

‖qn‖∞ pn < ∞ ,


and for allp ≥ 0 and all(x, s) ∈ [0, 1]2 with x 6= s,

∞∑n=0

∣∣∣∣ lnn |x − s|n!

∣∣∣∣ pn =∞∑

n=0

lnn |x − s|n! (−p)n = |x − s|−p ,

we obtain for allp ∈ (−ρ, ρ) and allt, x, s ∈ [0, 1] with x 6= s,

∞∑n=0

qn(t, x)pn∞∑

n=0

lnn |x − s|n! pn

=∞∑

n=0

pnn∑

j =0

qj (t, x)ln(n− j ) |x − s|(n − j )!

and

∞∑n=0

∣∣∣∣∣∣pnn∑

j =0

qj (t, x)ln(n− j ) |x − s|(n − j )!

∣∣∣∣∣∣≤

∞∑n=0

|p|nn∑

j =0

∣∣qj (t, x)∣∣∣∣∣∣∣ ln

(n− j ) |x − s|(n − j )!

∣∣∣∣∣=

∞∑n=0

|qn(t, x)| |p|n∞∑

n=0

∣∣∣∣ lnn |x − s|n!

∣∣∣∣ |p|n

≤( ∞∑

n=0

‖qn‖∞ |p|n)

|x − s|−|p| .

Hence, it follows from Lebesgue’s Dominated Convergence Theorem that forall (t, s) ∈ [0, 1]2,

∫ 1

0

∞∑n=0

qn(t, x)pn∞∑

n=0

lnn |x − s|n! pndx

=∫ 1

0

∞∑n=0

pnn∑

j =0

qj (t, x)ln(n− j ) |x − s|(n − j )! dx

=∞∑

n=0

pnn∑

j =0

∫ 1

0qj (t, x)

ln(n− j ) |x − s|(n − j )! dx .


Similarly, it can be shown that for all(t, s) ∈ [0, 1]2,

∫ 1

0

∞∑n=0

lnn |t − x|n! pn

∞∑n=0

lnn |x − s|n! pndx

=∞∑

n=0

pnn∑

j =0

∫ 1

0

ln j |t − x|j !

ln(n− j ) |x − s|(n − j )! dx .

Hence, (4.7.10) becomes

∞∑n=0

pn

µqn(t, s)+ µ2

n∑

j =0

∫ 1

0qj (t, x)

ln(n− j ) |x − s|(n − j )! dx

(4.7.11)

=∞∑

n=0

pnµ2

n∑

j =0

∫ 1

0

ln j |t − x|j !

ln(n− j ) |x − s|(n − j )! dx

, (t, s) ∈ [0, 1]2 .

which is satisfied if, for everyn ∈ IN0, the following linear equation is ful-filled:

qn(t, s)+ µ

n∑j =0

∫ 1

0qj (t, x)

ln(n− j ) |x − s|(n − j )! dx (4.7.12)

= µ

n∑j =0

∫ 1

0

ln j |t − x|j !

ln(n− j ) |x − s|(n − j )! dx , (t, s) ∈ [0, 1]2 .

For n = 0, (4.7.12) reduces to

q0(t, s)+ µ

∫ 1

0q0(t, x)dx = µ , (t, s) ∈ [0, 1]2 ,

which has the solution

(4.7.5) q0(t, s) = µ

1 + µ, (t, s) ∈ [0, 1]2 .

For n ≥ 1, we plug (4.7.5) into (4.7.12) and obtain

(id + µA) qn(t, s) (4.7.13)

= µ

∫ 1

0

1

1 + µ

lnn |x − s|n! dx +

n∑j =1

∫ 1

0

ln j |t − x|j !

ln(n− j ) |x − s|(n − j )! dx


−n−1∑

j =1

∫ 1

0qj (t, x)

ln(n− j ) |x − s|(n − j )! dx

, (t, s) ∈ [0, 1]2 .

That this equation is solved forqn given by (4.7.6) follows from the fact that(id − µ

1 + µA

)= (id + µA)−1 .

It remains to be shown that allqn are symmetric. Reasoning as above, one canshow that for allp ∈ (−ρ, ρ),

gp(t, s)+ µ

∫ 1

0|t − x|p gp(x, s)dx = µ |t − s|p , (t, s) ∈ [0, 1]2 (4.7.14)

has the unique solution

gp(t, s) = µ |t − s|p − µ

∞∑n=0

gn(t, s)pn , (t, s) ∈ [0, 1]2 ,

whereg0(t, s) = µ

1 + µ

and forn ≥ 1,

gn(t, s) = µ

(id − µ

1 + µA

)(4.7.15)

∫ 1

0

lnn |t − x|n!

1

1 + µdx +

n∑j =1

∫ 1

0

ln(n− j ) |t − x|(n − j )!

ln j |x − s|j ! dx

−n−1∑

j =1

∫ 1

0

ln(n− j ) |t − x|(n − j )! gj (x, s)dx

,

where

Al(t, s) =∫ 1

0l (x, s)dx for l ∈ L2[0, 1]2 .

A comparison of (4.7.15) with (4.7.6) shows that for alln ∈ IN0,

gn(t, s) = qn(s, t) , (t, s) ∈ [0, 1]2 .But since for allp ∈ (−ρ, ρ), qp also solves (4.7.14), we haveqp = gp forall p ∈ (ρ, ρ). Hence,qn = gn for all n ∈ IN0, i.e. all qn are symmetric. 2


Remarks 4.231. Let f p(t, s) = |t − s|p, t, s ∈ [0, t], p ∈ IR. For all(µ, p) ∈ IR2, such that

µ ≥ 0 and p ∈ (−ρ, ρ) , (4.7.16)

where

ρ = min

{1

2,

1 + µ

1 + 3µ+ 4µ2

},

theL2-kernelqp from Theorem 4.22 solves

id − qopp = (

id + µ f opp)−1

.

Alternatively, (id + µ f op

p)−1

can be expressed as the Neumann series

∞∑n=0

(−µ f opp )

n (4.7.17)

whenever ∥∥µ f opp∥∥ < 1 . (4.7.18)

The operator norm off opp can be estimated as follows:

∥∥ f opp∥∥ ≤

(∫ 1

0

∫ 1

0|t − s|2p dsdt

) 12

= [(2p + 1)(p + 1)]−12 .

Hence, (4.7.18) holds if

µ2 < (2p + 1)(p + 1) . (4.7.19)

It can be seen in Figure 4.1 that for(µ, p) ∈ IR+ ×(−12, 0], condition (4.7.19)

is more restrictive than condition (4.7.16).2. For H = 1, we can use the formula

(4.7.2)d QM1,α

d QW(X) = 1√

1 + α2exp

(α2

1 + α2

1

2X2

1

)

to compute for alld > 0,

P[M1,α

t < d , t ∈ [0, 1]]

= 8

(d√

1 + α2

)− e2d2α2

8

(−d

1 + 2α2

√1 + α2

),

(4.7.20)


–0.4

–0.2

0.2

0.4

–2 –1 1 2µ

p

–0.4

–0.2

0.2

0.4

–2 –1 1 2µ

p

Figure 4.1: Left: All (µ, p) ∈ IR × [−12,∞) that satisfy (4.7.16). Right: All

(µ, p) ∈ IR × [−12,∞) that satisfy (4.7.19).

where8 is the standard normal distribution function.To prove (4.7.20) we let

D = {X : Xt < d , t ∈ [0, 1]}and note that

P[M1,α

t < d , t ∈ [0, 1]]

= QM1,α [D] = EQW

[1D(X)

d QM1,α

d QW(X)

]

= 1√1 + α2

EQW

[1D(X) exp

(α2

1 + α2

1

2X2

1

)].

The reflection principle gives that for all Borel setsB ⊂ IR,

QW [D ∩ {X1 ∈ B}] =∫

IR1B(x) f (x)dx ,

where

f (x) = 1{x≤d} [ϕ(x)− ϕ(2d − x)] and ϕ(x) = 1√2π

e− x22 .

It follows that

1√1 + α2

EQW

[1D(X) exp

(α2

1 + α2

1

2X2

1

)]

4.8. The Hitsuda-representation of mfBm 109

= 1√1 + α2

∫ d

−∞exp

(α2

1 + α2

1

2x2

)f (x)dx .

Now (4.7.20) follows from a simple calculation.For H ∈ (3

4, 1), we are not able to deduce an explicit formula for

P[M H,α

t < d , t ∈ [0, 1]]

becaused QM H,α

d QW(X)

depends not only onX1 but on the whole path ofX.

4.8 The Hitsuda-representation of mixed frac-tional Brownian motion

It follows from Theorem 4.2 and Theorem 4.15 a) that for allH ∈ ( 34, 1] and

α ∈ IR \ {0} there exists a real-valued Volterra kernelr ∈ L2([0, 1]2) and aBrownian motion(Wt )0≤t≤1 such that

M H,αt = Wt +

∫ t

0

∫ s

0r (s, u)dWuds, t ∈ [0, 1] .

It follows from Proposition 4.20 thatr is the unique Volterra kernel that solvesthe equation

α2H (2H − 1)(t − s)2H−2 = r (t, s)+∫ s

0r (t, x)r (s, x)dx , (t, s) ∈ 4 ,

(4.8.1)where we set4 = {

(t, s) ∈ [0, 1]2 : s< t}. For H = 1, (4.8.1) reduces to

α2 = r (t, s)+∫ s

0r (t, x)r (s, x)dx , (t, s) ∈ 4 ,

and is easy to solve. Its solution is given by

r (t, s) = α2

1 + α2s, (t, s) ∈ 4 .

Hence, for allt ∈ [0, 1],

M1,αt = Wt +

∫ t

0

∫ s

0

α2

1 + α2udWuds = Wt +

∫ t

0

∫ t

u

α2

1 + α2udsdWu


=∫ t

0

(1 + (t − u)

α2

1 + α2u

)dWu =

(1 + α2t

)∫ t

0

dWu

1 + α2u.

As in Section 4.7, it is much harder to solve (4.8.1) forH ∈ (34, 1). We set

p = 2H − 2 ∈(

−1

2, 0

)and µ = α2H (2H − 1) ≥ 0 .

Further, forµ ≥ 0, we define the bounded linear operator

Aµ : L2(4) → L2(4)by

Aµl (t, s) = µ

1 + µs

∫ s

0[l (t, x)+ l (s, x)] dx

− 2µ2

(1 + µs)2

∫ s

0

∫ x

0l (x, y)dydx, l ∈ L2(4) .

Theorem 4.24 Let

µ ≥ 0 and ρ = 1

5.08+ 26.88µ + 13.952µ2 . (4.8.2)

Then for all p∈ (−ρ, ρ), the equation

µ(t − s)p = r p(t, s)+∫ s

0r p(t, x)r p(s, x)dx , (t, s) ∈ 4 , (4.8.3)

is solved by the L2(4)-function

r p(t, s) = µ(t − s)p − µ

∞∑n=0

rn(t, s)pn , (t, s) ∈ 4 , (4.8.4)

wherer0(t, s) = µs

1 + µs, (4.8.5)

and for n≥ 1,

rn(t, s) = Aµlnn(t − s)

n! + µ(id − Aµ

)(4.8.6)

n−1∑

j =1

∫ s

0

(ln j (t − x)

j ! − r j (t, x)

)(ln(n− j )(s − x)

(n − j )! − rn− j (s, x)

)dx

.


In particular, all rn are continuous and bounded on4 and for all p∈ [0, ρ),∞∑

n=0

‖rn‖∞ pn < ∞ .

(It follows that∞∑

n=0

rn(t, s)pn

converges for all p∈ (−ρ, ρ) uniformly in (t, s) ∈ 4 to a bounded andcontinuous functionr p.)

–0.2

–0.1

0

0.1

0.2

1 2

µ

p

Figure 4.2: All (µ, p) ∈ IR+ × [−12,∞) such that p∈ (−ρ, ρ), whereρ is

given by (4.8.2).

Proof. It follows inductively from (4.8.6) that allrn are continuous and boundedon4. In order to prove that for allp ∈ [0, ρ),

∞∑n=0

‖rn‖∞ pn < ∞ ,


we construct recursively for alln ≥ 1, a polynomial

an(s) =n−1∑j =0

an, j sj (4.8.7)

with non-negative coefficientsan,0, . . . , an,n−1 such that for all(t, s) ∈ 4,

|rn(t, s)| ≤ an(s) .

Let n ≥ 1. Since ∫ s

0

[lnn(t − x)

n! + lnn(s − x)

n!]

dx

and ∫ s

0

∫ x

0

lnn(x − y)

n! dydx

have for all(t, s) ∈ 4 the same sign, we have for all(t, s) ∈ 4,∣∣∣∣Aµ lnn(t − s)

n!∣∣∣∣ (4.8.8)

≤ µ

1 + µs

∣∣∣∣∫ s

0

[lnn(t − x)

n! + lnn(s − x)

n!]

dx

∣∣∣∣∨ 2µ2

(1 + µs)2

∣∣∣∣∫ s

0

∫ x

0

lnn(x − y)

n! dydx

∣∣∣∣ ≤ 2µ ,

where the second inequality follows from

µ

1 + µs

∣∣∣∣∫ s

0

[lnn(t − x)

n! + lnn(s − x)

n!]

dx

∣∣∣∣= µ

1 + µs

∣∣∣∣∫ t

t−s

lnn(x)

n! dx +∫ s

0

lnn(x)

n! dx

∣∣∣∣≤ µ2

∣∣∣∣∫ t

0

lnn(x)

n! dx

∣∣∣∣ ≤ 2µ ,

and

2µ2

(1 + µs)2

∣∣∣∣∫ s

0

∫ x

0

lnn(x − y)

n! dydx

∣∣∣∣ ≤ 2µ2

(1 + µs)2

∣∣∣∣∫ s

0dx

∣∣∣∣


= 2µ2s

(1 + µs)2≤ max

s≥0

2µ2s

(1 + µs)2= µ

2.

In particular,|r1(t, s)| ≤ 2µ for all (t, s) ∈ 4 .

Hence, we can seta1 = 2µ . (4.8.9)

Let us now suppose that forn ≥ 2, there exist polynomialsa1, . . . , an−1 ofthe form (4.8.7) such that for all(t, s) ∈ 4,

|rm(t, s)| ≤ am(s) =m−1∑j =0

am, j sj , m = 1, . . . , n − 1 .

To obtain an estimate on|rn(t, s)|, we write

n−1∑j =1

∫ s

0

(ln j (t − x)

j ! − r j (t, x)

)(ln(n− j )(s − x)

(n − j )! − r(n− j )(s, x)

)dx

asn−1∑j =1

∫ s

0

ln j (t − x)

j !ln(n− j )(s − x)

(n − j )! dx (4.8.10)

−n−1∑j =1

∫ s

0

(r j (t, x)

ln(n− j )(s − x)

(n − j )! + r j (s, x)ln(n− j )(t − x)

(n − j )!

)dx

+n−1∑j =1

∫ s

0r j (t, x)r(n− j )(s, x)dx .

and estimate the three terms separately. For all(t, s) ∈ 4,∣∣∣∣∣∣n−1∑j =1

∫ s

0

ln j (t − x)

j !ln(n− j )(s − x)

(n − j )! dx

∣∣∣∣∣∣≤

∣∣∣∣∣∣n∑

j =0

∫ s

0

ln j (t − x)

j !ln(n− j )(s − x)

(n − j )! dx

∣∣∣∣∣∣= 1

n!∣∣∣∣∫ s

0[ln(t − x)+ ln(s − x)]n dx

∣∣∣∣≤ 1

n!∫ s

0|2 ln(s − x)|n dx ≤ 2n .


Furthermore,∣∣∣∣∣∣n−1∑j =1

∫ s

0

(r j (t, x)

ln(n− j )(s − x)

(n − j )! + r j (s, x)ln(n− j )(t − x)

(n − j )!

)dx

∣∣∣∣∣∣≤

n−1∑j =1

∫ s

0aj (x)

∣∣∣∣∣ ln(n− j )(t − x)

(n − j )! + ln(n− j )(s − x)

(n − j )!

∣∣∣∣∣dx

≤ 2n−1∑j =1

∫ s

0aj (x)

∣∣∣∣∣ ln(n− j )(s − x)

(n − j )!

∣∣∣∣∣dx ≤ 2n−1∑j =1

aj (s) ,

where the last inequality follows from the fact that for allm, l ∈ IN0,

∫ s

0xm

∣∣∣∣∣ lnl (s − x)

l !

∣∣∣∣∣ dx ≤ sm∫ s

0

∣∣∣∣∣ lnl (s − x)

l !

∣∣∣∣∣ dx ≤ sm .

The last term of (4.8.10) can be estimated as follows:∣∣∣∣∣∣n−1∑j =1

∫ s

0r j (t, x)r(n− j )(s, x)dx

∣∣∣∣∣∣≤

n−1∑j =1

∫ s

0aj (x)an− j (x)dx .

Combining the three preceding estimates we obtain∣∣∣∣∣∣n−1∑j =1

∫ s

0

(ln j (t − x)

j ! − r j (t, x)

)(ln(n− j )(s − x)

(n − j )! − r(n− j )(s, x)

)dx

∣∣∣∣∣∣≤ 2n + 2

n−1∑j =1

aj (s)+n−1∑j =1

∫ s

0aj (x)an− j (x)dx, , (t, s) ∈ 4 . (4.8.11)

Next observe that forl ∈ L2(4), such that there exists anm ∈ IN0 with

l (t, s) ≤ sm for all (t, s) ∈ 4 ,we have ∣∣(id − Aµ

)l (t, s)

∣∣


≤ sm + µ

1 + µs

∫ s

02xmdx + 2µ2

(1 + µs)2

∫ s

0

∫ x

0ymdydx

≤ sm + µ

1 + µs2

sm+1

m + 1+ 2µ2

(1 + µs)2sm+2

(m + 1)(m + 2)

≤ sm + 2sm + sm = 4sm .

This together with (4.8.6), (4.8.8) and (4.8.11) implies

|rn(t, s)| ≤ 2µ+ 4µ

2n + 2

n−1∑j =1

aj (s)+n−1∑j =1

∫ s

0aj (x)a(n− j )(x)dx

.

Therefore, forn ≥ 2 ands ∈ [0, 1], we can set

an(s) = µ

2 + 2n+2 + 8

n−1∑j =1

aj (s)+ 4n−1∑j =1

∫ s

0aj (x)a(n− j )(x)dx

.

(4.8.12)Since

|rn(t, s)| ≤ an(1) for all n ≥ 1 and all(t, s) ∈ 4 ,the radius of convergence of

∞∑n=0

‖r ‖∞ pn

is at least as big as the radius of convergence of

∞∑n=1

an(1)zn .

To obtain an estimate on the radius of convergence of

∞∑n=1

an(1)zn ,

we set fors ∈ [0, 1] andz ∈ Cl sufficiently small

a(z, s) =∞∑

n=1

an(s)zn .


It follows from (4.8.9) and (4.8.12) that

a(z, s) = 2µ

[ ∞∑n=1

zn + 2∞∑

n=2

(2z)n + 4a(z, s)∞∑

n=1

zn + 2∫ s

0a2(z, x)dx

],

which for small enoughz can be written as

4µ∫ s

0a2(z, x)dx + (8µ+ 1)z − 1

1 − za(z, s) = 2µz

8z2 − 6z − 1

(1 − z)(1− 2z).

(4.8.13)The unique solution of (4.8.13) is given by

a(z, s) = 2µz(8µz+ z − 1)(8z2 − 6z − 1)

8µ2z(1− z)(8z2 − 6z − 1)s + (8µz + z − 1)2(1 − 2z).

In particular,

a(z, 1) = 2µz(8µz+ z − 1)(8z2 − 6z − 1)

8µ2z(1 − z)(8z2 − 6z − 1)+ (8µz + z − 1)2(1 − 2z).

This shows that the radius of convergence of

∞∑n=1

an(1)zn = a(z, 1)

is the radiusr of the largest open discDr (0) ⊂ Cl with center 0 on which thedenominator

1− (4+16µ+8µ2)z+ (5+48µ+24µ2)z2 − (2+32µ+16µ2)z3 −64µ2z4

= 8µ2z(1 − z)(8z2 − 6z − 1)+ (8µz+ z − 1)2(1 − 2z) (4.8.14)

of the rational functiona(z, 1) does not vanish. For|z| ≤ 15, one can estimate

the terms of expression (4.8.14) that depend onz as follows:∣∣∣−(4 + 16µ+ 8µ2)z+ (5 + 48µ+ 24µ2)z2

−(2 + 32µ+ 16µ2)z3 − 64µ2z4∣∣∣

≤ |z|[4 + 16µ+ 8µ2 + 1

5

(5 + 48µ+ 24µ2

)

+ 1

52

(2 + 32µ+ 16µ2

)+ 1

5364µ2

]

= |z|(5.08+ 26.88µ+ 13.952µ2

).


This shows that for

|z| < ρ = 1

5.08+ 26.88µ+ 13.952µ2

the denominator (4.8.14) does not vanish. It follows that

∞∑n=0

‖rn‖∞ pn < ∞ ,

for all p ∈ (−ρ, ρ). To show that

(4.8.4) r p(t, s) = µ(t − s)p − µ

∞∑n=0

rn(t, s)pn , (t, s) ∈ 4 ,

is a solution of (4.8.3), we write it as

r p(t, s) = µ

∞∑n=0

(lnn(t − s)

n! − rn(t, s)

)pn

and plug it into (4.8.3). This yields the equation

µ

∞∑n=0

rn(t, s)pn = (4.8.15)

µ2∫ s

0

∞∑n=0

(lnn(t − x)

n! − rn(t, x)

)pn

∞∑n=0

(lnn(s − x)

n! − rn(s, x)

)pndx ,

(t, s) ∈ 4 .As in the proof of Theorem 4.22, one can show that for all(t, s) ∈ 4,

∫ s

0

∞∑n=0

(lnn(t − x)

n! − rn(t, x)

)pn

∞∑n=0

(lnn(s − x)

n! − rn(s, x)

)pndx =

∞∑n=0

pnn∑

j =0

∫ s

0

(ln j (t − x)

j ! − r j (t, x)

)(ln(n− j )(s − x)

(n − j )! − r(n− j )(s, x)

)dx .

Hence, (4.8.15) becomes

µ

∞∑n=0

rn(t, s)pn = (4.8.16)


µ2∞∑

n=0

pnn∑

j =0

∫ s

0

(ln j (t − x)

j ! − r j (t, x)

)(ln(n− j )(s − x)

(n − j )! − r(n− j )(s, x)

)dx ,

(t, s) ∈ 4 ,which is satisfied if, for alln ≥ 0, the equation

rn(t, s) = (4.8.17)

µ

n∑j =0

∫ s

0

(ln j (t − x)

j ! − r j (t, x)

)(ln(n− j )(s − x)

(n − j )! − r(n− j )(s, x)

)dx ,

(t, s) ∈ 4 ,holds. Forn = 0, (4.8.17) reduces to

r0(t, s) = µ

∫ s

0(1 − r0(t, x)) (1 − r0(s, x)) dx , (t, s) ∈ 4 ,

and is solved by

(4.8.5) r0(t, s) = µs

1 + µs, (t, s) ∈ 4 .

For n ≥ 1, we plug (4.8.5) into (4.8.17). Then (4.8.17) becomes

(id + Bµ)rn(t, s) = Bµlnn(t − s)

n! (4.8.18)

+µ(n−1)∑j =1

∫ s

0

(ln j (t − x)

j ! − r j (t, x)

)(ln(n− j )(s − x)

(n − j )! − r(n− j )(s, x)

)dx ,

(t, s) ∈ 4 ,where the bounded linear operator

Bµ : L2(4) → L2(4)is given by

Bµl (t, s) =∫ s

0

µ

1 + µx[l (t, x)+ l (s, x)] dx , l ∈ L2(4) .

It can be checked by performing integration by parts several times that

BµAµ = Bµ − Aµ on L2(4) .This implies

(id + Bµ)Aµ = Bµ and (id + Bµ)(id − Aµ) = id

and hence shows that thern given in (4.8.6) solve (4.8.18). 2

Bibliography

Bachelier, L. (1900). Theorie de la speculation.Ann. Sci. Ecole Norm. Sup.17, 21-86.

Black, F. and Scholes, M. (1973). The pricing of options and corporate liabil-ities. J. Polit. Econom.81, 637-659.

Brigo, D. and Mercurio, F. (2000). Option pricing impact of alternative con-tinuous-time dynamics for discretely-observed stock prices.Finance andStochastics4, 147-159.

Cutland, N.J., Kopp P.E. and Willinger, W. (1995). Stock price returns andthe Joseph effect: a fractional version of the Black-Scholes model.Progressin Probability36, 327-351.

Delbaen, F. and Schachermayer, W. (1994). A general version of the funda-mental theorem of asset pricing.Math. Ann.300, 463-520.

Dellacherie, C. and Meyer, P.A. (1980).Probabilites et Potentiel. Chap. V -VIII. Hermann.

Doleans-Dade, C. and Meyer, P.A. (1970). Integrales stochastiques par rap-port aux martingales locales.Seminaire Proba. IV. Lecture Notes in Mathe-matics124, 77-107.

Dunford, N. and Schwartz, J.T. (1963).Linear Operators Part II.IntersiencePublishers, New York.

Girsanov, I.V. (1960). On transforming a certain class of stochastic processesby absolutely continuous substitution of measures.Theory of Probability andits Applications5, 285-301.

Golub, G.H. and Van Loan, C.F. (1989).Matrix Computations.The JohnsHopkins University Press.

119

120 Bibliography

Hida, T. and Hitsuda, M. (1976).Gaussian Processes.American Mathemati-cal Society: Translations of Mathematical Monographs.

Hitsuda, M. (1968). Representation of Gaussian processes equivalent toWiener process.Osaka Journal of Mathematics5, 299-312.

Ito, K. (1944). Stochastic integral.Proc. Imp. Acad. Tokyo20, 519-524.

Karatzas, I. and Shreve, S.E. (1988).Brownian Motion and Stochastic Calcu-lus. Springer: Graduate Texts in Mathematics.

Kolmogorov, A.N. (1940). Wienersche Spiralen und einige andere interes-sante Kurven im Hilbertschen Raum.C.R. (Doklady) Acad. Sci. URSS (N.S.)26, 115-118.

Kono, N. and Maejima, M. (1991). Holder continuity of sample paths of someself-similar stable processes.Tokyo Journal of MathematicsVol. 14, No. 1,93-100.

Lamberton, D. and Lapeyre, B. (1996).Stochastic Calculus Applied to Fi-nance.Chapman & Hall.

Lin, S.J. (1995). Stochastic analysis of fractional Brownian motions.Stochas-tics and Stochastics Reports55, 121-140.

Lipster, R.Sh. and Shiryaev, A.N. (1989).Theory of Martingales.KluwerAcad. Publ., Dordrecht.

Mandelbrot, B.B. and Van Ness, J.W. (1968). Fractional Brownian motions,fractional noises and applications.SIAM Review10, 422-437.

Merton, R.C. (1973). Theory of rational option pricing.Bell J. Econom.Management Sci.4, 141-183.

Protter, P. (1990).Stochastic Integration and Differential Equations.Berlin:Springer-Verlag.

Rogers, L.C.G. (1997). Arbitrage with fractional Brownian motion.Mathe-matical Finance7, 95-105.

Salopek, D.M. (1998). Tolerance to arbitrage.Stochastic Processes and theirApplications76, 217-230.

Samuelson, P.A. (1965). Rational theory of warrant pricing.Industrial Man-agement ReviewVol. 6, No. 2, 13-31.

Shao, Q.M. (1996). p-variation of Gaussian processes with stationary incre-ments.Studia Sci. Math. Hungar.31, 237-247.

Bibliography 121

Shepp, L.A. (1966). Radon-Nikodym derivatives of Gaussian measures.Ann.Math. Statist.37, 321-354.

Shiryaev, A.N. (1984).Probability.Springer-Verlag, New York.

Shiryaev, A.N. (1998). On arbitrage and replication for fractal models.Re-search Report no. 20, MaPhySto, Department of Mathematical Sciences, Uni-versity of Aarhus, Denmark.

Smithies, F. (1958).integral equations.Cambridge Univ. Press, London andNew York.

Stricker, C. (1984). Quelques remarques sur les semimartingales Gaussienneset le probleme de l’innovation. Lecture Notes in Control and InformationSciences61, 260-276.

Takashima, K. (1989). Sample path properties of ergodic self-similar pro-cesses.Osaka J. Math.26, 159-189.

Wheeden, R.L. and Zygmund, A. (1977).Measure and Integral.MarcelDekker, Inc.

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