Prof. Dr. T. Meyer-Brandis M. Bauer, D. Ritter Winter term 2018/19

Mathematical Finance IIProblem Sheet 11

Exercise∗ 1 (9 points): Let f : [0,∞)×R→ R be bounded and continuous. Assume that f istwice continuously differentiable in the interior (0,∞) × R of its domain and fulfills there theheat equation

∂f

∂t(t, x) =

1

2

∂2f

∂x2(t, x).

Let B be standard Brownian Motion and Mt := maxBs : 0 ≤ s ≤ t its running maximum.

i) Show that for any t > 0 and x ∈ R, the process (f(t− s, x−Bs))0≤s≤t is a martingale.

ii) Show that E[Mt ∨m] = g(t,m) for t > 0 and m ≥ 0 with

g(t,m) := 2mΦ(t−12m)−m+ 2

√tϕ(t−

12m),

where ϕ(z) = 1√2πe−

z2

2 denotes the density of the standard normal distribution and Φ(x) =∫ x−∞ ϕ(z)dz its cumulative distribution function.

Hint: You may use here that P(Mt ≥ m) = P(|Bt| ≥ m) = 2P(Bt ≥ m) for all t ≥ 0 andm ≥ 0.

iii) Show that (g(t− s, x−Bs))0≤s≤t is a martingale.

iv) Let T > 0 be a finite time horizon. Show that for 0 ≤ t < T we have P-a.s.

E[MT |Ft] = Bt + g(T − t,Mt −Bt),

where (Ft)t∈[0,T ] is the natural filtration of (Bt)t∈[0,T ].

Hint: You may use the Markov property of the Brownian motion.

Exercise 2: Let B = (Bt)t∈R+ be a real-valued Brownian motion and T > 0. For each X ∈L2([0, T ]), find the process Φ ∈ Λ2([0, T ]) such that

X = E[X]+

∫ T

0

Φs dBs,

holds.

i) X = B3T

ii) X = eBT

iii) X =∫ T0Bt dt

iv) X = sinBT

Exercise 3: Let B = (Bt)t∈[0,1] be a real-valued (Ft)t∈[0,1]-Brownian motion on a filtered prob-ability space (Ω,F , (Ft)t∈[0,1],P). Consider the process

(Xa,bt

)t∈[0,1) which is given by

Xa,bt = a(1− t) + bt+ (1− t)

∫ t

0

1

1− sdBs for all 0 ≤ t < 1, (∗∗)

where a, b ∈ R.

i) Is the process Xa,bt well-defined? Explain your answer.

ii) Show that Xa,bt solves on [0, 1) the stochastic differential equation

dXt = b−Xt

1−t dt+ dBt

X0 = a.

iii) Show that Xa,bt converges to b in L2(Ω) as t→ 1.

iv) Show thatX0,0t has the same distribution as the Brownian BridgeBt−tB1 for all 0 ≤ t < 1.

Exercise 4: Consider the SDE given bydXt = f(t,Xt)dt+ σ(t)XtdBt

X0 = x,

where f : R+ × R → R and σ : R+ → R are continuous functions, σ ∈ Λ2 and B is a1-dimensional Brownian motion. Moreover, define

Zt := exp

(−∫ t

0

σ(s)dBs +1

2

∫ t

0

σ2(s)ds

).

i) Show thatd(ZtXt) = Ztf(t,Xt)dt,

and thus by setting Yt(ω) = Zt(ω)Xt(ω) for each ω ∈ Ω, we get

dYt(ω)

dt= Zt(ω)f

(t, Z−1t (ω)Yt(ω)

), Y0 = x.

Note that for each ω ∈ Ω this is a deterministic differential equation.

ii) Use i) to solve the SDE

dXt =1

Xt

dt+ αXtdBt, X0 = x > 0,

where α ∈ R.Hint: Use the separation of variables.