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Continuous-Time Principal-Agent Problems

with Hidden Action: The Weak Formulation

Jaksa Cvitanic

Xuhu Wan

Jianfeng Zhang

July 26, 2005

Abstract

We consider the problem of optimal contracts in continuous time, when the agents

actions are unobservable by the principal. We apply the stochastic maximum principle

to give necessary conditions for optimal contracts, both for the case of the utility from

the payoff being separable and not separable from the cost of the agents effort. The

necessary conditions are shown also to be sufficient for the case of quadratic cost

and separable utility, for general utility functions. The solution for the latter case

is almost explicit, and the optimal contract is a function of the final outcome only,

the fact which was known before for exponential and linear utility functions. For

general non-separable utility functions, sufficient conditions are hard to establish, but

we suggest a way to check sufficiency using non-convex optimization. Unlike previous

work on the subject, we use the weak formulation both for the agents problem and

the principals problem, thus avoiding tricky measurability issues.

Key Words and Phrases: Principal-Agent problems, hidden action, optimal contracts

and incentives, stochastic maximum principle, Forward-Backward SDEs.

AMS (2000) Subject Classifications: 91B28, 93E20; JEL Classification: C61, C73, D82,

J33, M52

Research supported in part by NSF grants DMS 00-99549 and DMS 04-03575.Caltech, M/C 228-77, 1200 E. California Blvd. Pasadena, CA 91125. Ph: (626) 395-1784. E-mail:

[email protected] of Information and Systems Management, HKUST Business School , Clear Water Bay,

Kowloon , Hong Kong. Ph: +852 2358-7731. Fax: +852 2359-1908. E-mail: [email protected] of Mathematics, USC, 3620 S Vermont Ave, MC 2532, Los Angeles, CA 90089-1113. Ph:

(213) 740-9805. E-mail: [email protected].


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1 Introduction

This paper is a continuation of our work Cvitanic, Wan and Zhang (2004) on principal-

agent problems in continuous time, in which the principal hires the agent to control a given

stochastic process. The problem of optimal compensation of portfolio managers and the

problem of optimal compensation of company executives are important applications of this

theory.

In the previous paper we study the case in which the actions of the agent are observed

by the principal. Here, we consider the case of hidden actions. More precisely, the

agents control of the drift of the process is unobserved by the principal. For example, the

controlled part of the drift can be due to the agents superior abilities or effort, unknown by

the principal.

The seminal paper in the continuous-time framework is Holmstrom and Milgrom (1987),which showed that if both the principal and the agent have exponential utilities, then the

optimal contract is linear. Schattler and Sung (1993) generalized those results using a

dynamic programming and martingales approach of Stochastic Control Theory, and Sung

(1995) showed that the linearity of the optimal contract still holds even if the agent can

control the volatility, too. A nice survey of the literature is provided by Sung (2001). More

complex models are considered in Detemple, Govindaraj and Loewenstein (2001), Hugonnier

and Kaniel (2001). Ou-Yang (2003), Sannikov (2004), DeMarzo and Sannikov (2004). The

paper closest to ours is Williams (2004). That paper uses the stochastic maximum principle

to characterize the optimal contract in the principal-agent problems with hidden information,in the case of the penalty on the agents effort being separate (outside) of his utility, and

without volatility control. Williams (2004) focuses on the case of a continuously paid reward

to the agent, while we study the case when the reward is paid once, at the end of the contract.

Moreover, we prove our results from the scratch, thus getting them under weaker conditions.

(Williams (2004) also deals with the so-called hidden states case, which we do not discuss

here.) In addition, unlike the existing literature, we study the principals problem in the

same formulation as the agents problem, so-called weak formulation, thus avoiding the

problem of having to check whether the principals strong solution is adapted to the

appropriate -algebra. We also provide a detailed discussion on how to check whether the

controls satisfying necessary conditions are also sufficient.

In particular, under the assumption of quadratic cost function on the agents effort and

separable utility, we find an almost explicit solution in a framework with general utility

functions. To the best of our knowledge, this is the first time that the solution is explicitly

described in a continuous-time Principal-Agent problem with hidden action, other than for

exponential and linear utilities. Moreover, it turns out that the solution depends only on


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the final outcome and not on the history of the controlled process, the fact which was known

before for exponential and linear utilities.

The general case of non-separable utility functions is much harder. If the necessary

conditions determine a unique control process, then if we proved existence of the optimal

control, we would know that the necessary conditions are also sufficient. The existence of anoptimal control is hard because, in general, the problem is not concave. It is related to the

existence of a solution to Forward-Backward Stochastic Differential Equations (FBSDEs),

possibly fully coupled. However, it is not known under which general conditions these

equations have a solution. The stochastic maximum principle is covered in the book Yong

and Zhou (1999), while FBSDEs are studied in the monograph Ma and Yong (1999).

The paper is organized as follows: In Section 2 we set up the model and its weak

formulation. In Section 3, we find necessary conditions for the agents problem and the

principals problem. In Section 4 we discuss how to establish sufficiency. We present some

important examples in Section 5. We conclude in Section 6, mentioning possible further

research topics.

2 The model

2.1 Optimization Problems

Let {Wt}t0 be a standard Brownian Motion on a probability space (, F, P) and denote

by F := {Ft}tT its augmented filtration on the interval [0, T]. The controlled state process

is denoted X = Xu,v and its dynamics are given by

dXt = utvtdt + vtdWt, X0 = x. (2.1)

Here for simplicity we assume all the processes are one-dimensional. We note that in the

literature process X usually takes the form

dXt = b(t, Xt, ut, vt)dt + (t, Xt, vt)dWt. (2.2)

When is nondegenerate, one can always set

vt= (t, Xt, vt), ut = b(t, Xt, ut, vt)

1(t, Xt, vt).

Then (2.2) becomes (2.1). Moreover, under some monotonicity conditions on b, , one can

write u, v as functions of (X, u, v). In this sense, (2.1) and (2.2) are equivalent. In the

following we shall always consider (2.1).

The full information case, in which the principal observes X,u,v and thus also W,

was studied in Cvitanic, Wan and Zhang (2004). In the so-called hidden action case,

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the principal can only observe the controlled process Xt, but not the underlying Brownian

motion W or the agents control u (so the agents action ut is hidden to the principal). We

note that the principal observes the process X continuously, which implies that the volatility

control v can also be observed through the quadratic variation of X, under the assumption

v 0.In this paper we investigate the principal-agent problem with hidden actions. At time

T, the principal gives the agent compensation in the form of a payoff CT = F(X), where

F : C[0, T] IR is a (deterministic) mapping. For a given process v, the principal can

design the payoffF in order to induce (or force) the agent to implement it. In this sense,

we may consider v as a control chosen by the principal instead of by the agent, as is usual

in the literature. We say that the pair (F, v) is a contract.

The agents problem is that, given a contract (F, v), the agent chooses the control u (over

some admissible set which will be specified later) in order to maximize his utility

V1(F, v)= sup

u

V1(u; F, v)= sup

u

E[U1 (F(Xu,v ), G

u,vT )].

Here,

Gu,vt=

t0

g(s, Xs, us, vs)ds (2.3)

is the accumulated cost of the agent, and with a slight abuse of notation we use V1 both for

the objective function and its maximum. We say a contract (F, v) is implementable if there

exists unique uF,v such that

V1(uF,v; F, v) = V1(F, v).

The principal maximizes her utility

V2= max

F,vE[U2(X

uF,v,vT F(X

uF,v,v ))],

where the maximum is over all implementable contracts (F, v) such that the following par-

ticipation constraint or individual rationality (IR) constraint holds:

V1(F, v) R . (2.4)

Function g is a cost or penalty function of the agents effort. Constant R is the reservation

utilityof the agent and represents the value of the agents outside opportunities, the minimum

value he requires to accept the job. Functions U1 and U2 are utility functions of the agent

and the principal. The typical cases studied in the literature are the separable utility case

with U1(x, y) = U1(x) y, and the non-separable case with U1(x, y) = U1(x y), where,

with a slight abuse of notation, we use the same notation U1 also for the function of one

argument only. We could also have the same generality for U2, but this makes less sense

from the economics point of view.

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2.2 The weak formulation

A very important assumption we make, which is standard in the literature, is that u depends

on X, rather than W. That is, ut FXt . In other words, for any t, ut = ut(X) where ut

is a (deterministic) mapping from C[0, t] to IR, and can be interpreted to mean that the

agent chooses his action based on the performance of process X up to the current time. We

note again that vt is always determined by X and thus vt = vt(X) for some deterministic

functional vt. We will consider only those vt such that the SDE Xt = x +t0

vs(X)dWs has

a strong solution. We also note that in 2.1 a contract would be more precisely defined as

(F, v) instead of (F, v), and an action should be u instead of u.

The above assumption enables us to use the weak formulation approach. We note that

in the literature it is standard to use the weak formulation for the agent problem and the

strong formulation (the original formulation in 2.1) for the principals problem. However,

when one applies the optimal action uF,v

, which is obtained from the agents problem byusing the weak formulation, back to the strong formulation of the principals problem, there

is a subtle measurability issue which seems to be ignored in the literature. In this paper

that problem is non-existent, as we use weak formulation for both the agents problem and

the principals problem.

We now reformulate the problem. Let B be a standard Brownian motion under some

probability space with probability measure Q, and FB be the filtration generated by B. For

any FB-adapted process v 0, let

Xt = x +t0

vsdBs. (2.5)

Then vt = vt(X) and obviously it holds that

FXt = FBt , t.

Now for any functional ut, let

ut= ut(X); B

ut

= Bt

t

0

usds; Mut

= exp

t

0

usdBs 1

2 t

0

|us|2ds; (2.6)

and dQu

dQ

= MuT. Then we know by Girsanov Theorem that, under certain conditions, B

u is

a Qu-Brownian motion and

dXt = vtdBt = (utvt)(X)dt + vt(X)dBut .

That is, (X, Bu, Pu) is a weak solution to (2.1) corresponding to actions (u, v).

For any CT FBT , there exists some functional F such that CT = F(X). So a contract

(F, v) is equivalent to a random variable CT FBT and a process v F

B. Also, an action u

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is equivalent to a process u FB. For simplicity, in the following we abuse the notation by

writing ut = ut(X) and vt = vt(X) when there is no danger of confusion.

Now given a contract CT FBT and v F

B, the agents problem is to find an optimal

control uCT,v FB such that

V1(uCT,v; CT, v) = V1(CT, v)

= sup

u

V1(u; CT, v),

where, recalling (2.3),

V1(u; CT, v)= EQ

u

{U1(CT, GT)} = EQ{MuTU1(CT, GT)}. (2.7)

For simplicity from now on we denote E= EQ and Eu

= EQ

u

. The principals problem is

to find optimal (CT, v) such that

V2(CT, v) = V2 = supCT,v

V2(uCT,v; CT, v),

where

V2(u; CT, v)= Eu{U2(XT CT)} = E{M

uTU2(XT CT)}. (2.8)

2.3 Standing assumptions

First we need the following assumptions on the coefficients.

(A1.) Function g : [0, T] IR IR IR IR is continuously differentiable withrespect to x,u,v, gx is uniformly bounded, and gu, gv have uniform linear growth in x,u,v.

In addition, g is jointly convex in (x,u,v), gu > 0 and guu > 0.

(A2.) (i) Functions U1 : IR2 IR, U2 : IR IR are differentiable, with 1U1 > 0, 2U1 0, U1 is jointly concave and U2 is concave.

(ii) Sometimes we will also need U1 K for some constant K.

For any p 1, denote

LpT(Qu) = { FBT : E

u{||p} < }; Lp(Qu) = { FB : Eu{T0

|t|pdt} < },

and define LpT(Q), Lp(Q) in a similar way.

We next define the admissible set for the agents controls.

(A3.) Given a contract (CT, v), the admissible set A(CT, v) of agents controls associated

with this contract is the set of all those u FB such that

(i) Girsanov Theorem holds true for (Bu, Qu);

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(ii) U1(CT, GT), 2U1(CT, GT) L2T(Q

u);

(iii) For any bounded u FB, there exists 0 > 0 such that for any [0, 0),

u satisfies (i) and (ii) at above and |u|4, |g|4, |gu|4, |MT|

4, U21 (CT, GT), 2U

21 (CT, G

T) are

uniformly integrable in L1(Q) or L1T(Q), where

u= u + u, GT

=

T0

g(t)dt, M t= Mu

t , V1

= V1(u

),

and

g(t)= g(t, Xt, u

t , vt), g

u(t)

= gu(t, Xt, u

t , vt).

When = 0 we omit the superscript 0. We note that, for any u A(CT, v) and u, 0

satisfying (A3) (iii), we have u A(CT, v) for any [0, 0). We note also that, under

mild assumptions on (CT, v), all bounded u belong to A(CT, v).

The admissible set for the contracts (CT, v) is more involved. We postpone it to 3.2.

3 Necessary Conditions

Here we use the method of so-called Stochastic Maximum Principle, as described in the

book Yong and Zhou (1999). First we establish a simple technical lemma.

Lemma 3.1 Assumeu FB, Girsanov Theorem holds true for(Bu, Qu), andMuT L2T(Q).

Then for any L2T(Q

u

), there exists a unique pair (Y, Z) L2

(Qu

) such that

Yt =

Tt

ZsdBus . (3.1)

Obviously Yt = Eut {}, and uniqueness also follows immediately. But in general F

Bu =

FB, so we cannot apply the Martingale Representation Theorem directly to obtain Z. On

the other hand, since in general u is unbounded, one cannot claim the well-posedness of the

following equivalent BSDE from the standard literature:

Yt = +Tt

usZsds Tt

ZsdBs.

We prove the existence of Z below.

Proof. We first assume is bounded. Then MuT L2T(Q). Let (Y , Z) be the unique

solution to the BSDE

Yt = MuT

Tt

ZsdBs.

Define

Yt= Yt[M

ut ]1, Zt

= [Zt utYt][M

ut ]1.

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One can check directly that

dYt = ZtdBut , YT = .

Moreover,

Yt = Et{MuT}[M

ut ]1 = Eut {},

which implies that

Eu{ sup0tT

|Yt|2} CEu{||2} < .

Then one can easily get Z L2(Qu).

In general, assume n are bounded and Eu{|n |

2} 0. Let (Yn, Zn) be the solution

to BSDE (3.1) with terminal condition n. Then

Eu

sup

0tT|Ynt Y

mt |

2 +

T

0

|Znt Zmt |

2dt

CEu{|n m|

2} 0.

Therefore, (Yn, Zn) converges to some (Y, Z) which satisfies (3.1).

3.1 The Agents problem

We fix now a contract (CT, v), u A(CT, v), and ut FB bounded. Denote, omitting

arguments of U1, 2U1,

g(t)= gu(t, Xt, ut, vt)ut;

Gt

=t0 g(s)ds;

Mt = Mt[

t0

usdBs

t0

ususds] = Mt

t0

usdBus ;

V1 = E

MTU1 + MT2U1GT

.

Moreover, for any bounded u G and (0, 0) as in (A3)(iii), denote

g(t)=

g(t) g(t)

; GT

=

GT GT

; MT=

MT MT

; V1=

V1 V1

.

Introduce the so-called adjoint processes

YA,1t = U1(CT, GT)

Tt

ZA,1s dBus ;

YA,2t = 2U1(CT, GT)

Tt

ZA,2s dBus .

(3.2)

Theorem 3.1 Under our standing assumptions, we have

lim0

V1 = V1 (3.3)

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and

V1 = EuT

0

At utdt

, (3.4)

where

At= ZA,1t + gu(t, Xt, ut, vt)Y

A,2t . (3.5)

In particular, the necessary condition for u to be an optimal control is:

At 0. (3.6)

Proof: See Appendix.

We see that given (CT, v) (and thus also X), the optimal u should satisfy the following

Forward Backward Stochastic Differential equation (FBSDE)

Gt =

t0

g(s, Xs, us, vs)ds;

YA,1t = U1(CT, GT)

Tt

ZA,1s dBus ;

YA,2t = 2U1(CT, GT)

Tt

ZA,2s dBus ;

(3.7)

with maximum condition (3.6).

Moreover, since guu

> 0, we may assume there exists a function h(t,x,v,z) such that

gu(t,x,h(t,x,v,z), v) = z. (3.8)

Note that 2U1 < 0, so YA,2t < 0. Thus, (3.6) is equivalent to

ut = h(t, Xt, vt, ZA,1t /Y

A,2t ). (3.9)

That is, given (CT, v) and X, one may solve the following (self-contained) FBSDE:

Gt =t0 g(s, Xs, h(s, Xs, vs, Z

A,1

s /YA,2

s ), vs)ds;

YA,1t = U1(CT, GT) +

Tt

ZA,1s h(s, Xs, vs, ZA,1s /Y

A,2s )ds

Tt

ZA,1s dBs;

YA,2t = 2U1(CT, GT) +

Tt

ZA,2s h(s, Xs, vs, ZA,1s /Y

A,2s )ds

Tt

ZA,2s dBs.

(3.10)

Then, as a necessary condition, the optimal control uCT,v should be defined by (3.9).

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3.2 The Principals problem

3.2.1 Admissible contracts

We now characterize the admissible set A of contracts (CT, v). Our first requirement is:

(A4.) (CT, v) is implementable. That is, (3.10) has a unique solution, and uCT,v

definedby (3.9) satisfies uCT,v A(CT, v) and V1(u

CT,v; CT, v) = V1(CT, v).

Note that 3.1 gives only necessary conditions. In 4 there will be some discussion on

when the above uCT,v is indeed the agents optimal control.

Now an implementable contract (CT, v) uniquely determines uCT,v. In fact, for fixed v,

the correspondence between CT and uCT,v is one to one, up to a constant. To see this, we

fix some (u, v) and want to find some CT such that uCT,v = u. For notational convenience,

we denote

XA= YA,1, YA

= YA,2, ZA

= ZA,2.

If u = uCT,v for some CT, then (3.6) holds true for u. That is,

ZA,1t = gu(t, Xt, ut, vt)YAt .

Denote R= XA0 = Y

A,10 . Then (3.7) becomes

Gt =

t0

g(s, Xs, us, vs)ds;

XAt = R

t0

gu(s, Xs, us, vs)YAs dB

us ;

YAt = 2U1(CT, GT) T

t

ZAs dBus ;

(3.11)

where

XAT = U1(CT, GT). (3.12)

Since 1U1 > 0, we may assume there exists a function H(x, y) such that

U1(H(x, y), y) = x. (3.13)

Then (3.12) leads to

CT= H(XAT , GT), (3.14)

Plugging this into (4.4), we get

Xt = x +

t0

vsdBs;

Gt =

t0

g(s, Xs, us, vs)ds;

XAt = R

t0

gu(s, Xs, us, vs)YAs dB

us ;

YAt = 2U1(H(XAT , GT), GT)

Tt

ZAs dBus ;

(3.15)

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Now fix (R,u,v). If FBSDE (3.15) is well-posed, we may define CT by (3.14) and we can

easily see that uCT,v = u. In this sense, for technical convenience, from now on we consider

(R,u,v) (instead of (CT, v)) as a contract, or say, as the principals control. Then (A4)

should be rewritten as

(A4.) (R,u,v) is an implementable contract. That is,(i) FBSDE (3.15) is well-posed;

(ii) For CT defined by (3.14), (CT, v) is implementable in the sense of (A4).

We note that the theory of FBSDEs is far from complete. The well-posedness of (3.15) is

in general unclear (unless we put strict conditions such as u is bounded), mainly due to the

introduction of Bu. In fact, even for linear FBSDEs there is no general result like Lemma

3.1. Instead of adopting too strong technical conditions, in this paper we assume the well-

posedness of the involved FBSDEs directly and leave the general FBSDE theory for future

research. However, in the so called separable utility case, the corresponding FBSDEs willbecome decoupled FBSDEs and thus we can use Lemma 3.1 to establish their well-posedness,

as we will see in 3.4.2.

Now for any (u, v) and any bounded (u, v), denote

ut= ut + ut; v

t

= vt + vt;

Xt = x +

t0

vsdBs; (3.16)

GT=

T

0

g(t, Xt , ut , v

t )dt.

Denote also with superscript all corresponding quantities.

(A5.) The principals admissible set A of controls is the set of all those contracts (R,u,v)

such that, for any bounded (u, v), there exists a constant 1 > 0 such that for for any

[0, 1):

(i) (A4) holds true for (R, u, v);

(ii) The FBSDEs (3.20) and (3.22) below are well-posed for (R, u, v);

(iii) lim0

V2 = YP0 for V

2 , Y

P0 defined in (3.19) and (3.20) below, respectively.

Note again that we will specify sufficient conditions for (A5

) in the separable utility case

in 3.4.2. We also assume that A is not empty.

3.2.2 Necessary conditions

We now derive the necessary condtions for the Principals problem. Our first observation is

that

R = Eu{XAT} = Eu{U1(CT, GT)}

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is exactly the optimal utility of the agent. So condition (2.4) becomes equivalent to

R R.

Intuitively it is obvious that the principal would choose R = R in order to maximize her

utility. Again, due to the lack of satisfactory theory of FBSDEs, here we simply assume the

optimal R = R, and we will prove it rigorously in the separable utility case by using the

comparison theorem of BSDEs.

Given (u, v), let (X,G,XA, YA, ZA) be the solution to (3.15) with R = R. Define CT by

(3.14) and let

YPt = U2(XT CT)

Tt

ZPs dBus ; (3.17)

By Lemma 3.1 (3.17) is well-posed. Then the principals problem is to choose optimal ( u, v)in order to maximize

V2(u, v)= Eu{YPT } = Y

P0 . (3.18)

We now introduce more notation and adjoint processes for the principal. Similarly as

before, for any bounded u FB, v FB, and (0, 1) as in (A5), we denote

X=

X X

; GT

=

GT GT

; V2=

V2 V2

. (3.19)

Also denote, omitting functions arguments,

Xt =

t0

vsdBs;

gu = guuu + guvv + guxX

GT =

T0

[gxXt + guut + gvvt]dt.

Moreover, consider the following FBSDE system

XAt =

t0

guYAs usds

t0

[guYAs + Y

As gu]dB

us ;

YAt = 12U1CT + 22U1GT +

Tt

ZAs usds

Tt

ZAs dBus ;

YPt = U2[XT CT] +

Tt

ZPs usds

Tt

ZPs dBus .

(3.20)

where CT is defined by

XAT = 1U1CT + 2U1GT; (3.21)

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and, noting that 1U1 > 0,

X1t =

t0

guZ1sds;

X2t

= t0

[gux

Z1s

YAs

+ gx

Y2s

]ds;

Y1t =1

1U1[U2 X

1T12U1]

Tt

Z1sdBus ;

Y2t =2U11U1

[U2 X1T12U1] + X

1T22U1

Tt

Z2sdBus ;

Y3t = X2T + U

2

Tt

Z3sdBus .

(3.22)

Theorem 3.2 Under (A5), we have

YP0 = EuT

0P,1t utdt +

T0

P,2t vtdt

, (3.23)

where P,1t

= ZPt guY

1t Y

At + X

1t Z

At + guuZ

1t Y

At + guY

2t ;

P,2t= guvZ

1t Y

At + gvY

2t + Z

3t + u(Y

3t X

2t ).

(3.24)

In particular, the necessary condition for (u, v) to be an optimal control is:

P,1t = P,2t = 0. (3.25)

To prove the theorem, we first recall the following simple lemma (see, e.g., Cvitanic, Wan

and Zhang (2004)):

Lemma 3.2 Assume Wt =t0

sdBs + At is a continuous semimartingale, where B is a

Brownian motion. Suppose that

1)T0

|t|2dt < a.s.

2) Both Wt and At are uniformly (in t) integrable.

Then E[WT] = E[AT].

Proof of Theorem 3.2. The necessity of (3.25) is obvious because (u, v) is arbitrary.

So it suffices to prove (3.23).

Note that

Xt =

t0

vsdBus +

t0

usvsds.

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The proof is complete.

In summary, we have the following system of necessary conditions for the principal:

Xt = x + t

0

vsdBs;

Gt =

t

0

g(s, Xs, us, vs)ds;

XAt = R

t0

guYAs dB

us ;

X1t =

t0

guZ1sds;

X2t =

t0

[guxZ1sY

As + gxY

2s ]ds;


T

t

ZAs dBus ;

YPt = U2(XT H(XAT , GT))

T

t

ZPs dBus ;

Y1t =1

1U1[U2 X

1T12U1]

Tt

Z1sdBus ;

Y2t =2U11U1

[U2 X1T12U1] + X

1T22U1

Tt

Z2sdBus ;

Y3t = X2T + U

2

Tt

Z3sdBs;

(3.26)

with maximum condition (3.25).

In particular, if (3.25) has a unique solution

ut = h1(t, Xt, Y1t Y

At , Y

2t , Z

Pt + X

1ZAt , Z1t Y

At , Z

3t );

vt = h2(t, Xt, Y1t Y

At , Y

2t , Z

Pt + X

1ZAt , Z1t Y

At , Z

3t ),

then, by plugging (h1, h2) into (3.26) we obtain a self contained FBSDE.

14


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3.3 Fixed volatility case

If the principal has no control on v, then both v and X are fixed. In this case, along the

variation one can only choose v = 0. Then (3.26) can be simplified as

Gt =t0

g(s, Xs, us, vs)ds;

XAt = R

t0

guYAs dB

us ;

X1t =

t0

guZ1sds;


Tt

ZAs dBus ;

YPt = U2(XT H(XAT , GT))

Tt

ZPs dBus ;

Y1t =U2

X1T

12

U1

1U1 Tt

Z1sdBus ;

Y2t =2U11U1

[U2 X1T12U1] + X

1T22U1

Tt

Z2sdBus ;

(3.27)

with maximum condition

P,1t= ZPt guY

1t Y

At + X

1t Z

At + guuZ

1t Y

At + guY

2t = 0. (3.28)

3.4 Separable Utilities

In this subsection we assume the agent has a separable utility function, namely,

U1(CT, GT) = U1(CT) GT. (3.29)

Here we abuse the notation U1. We note that if U1 > 0 and U

1 0, then Assumption A.2

(i) still holds true.

3.4.1 The agents problem

In this case obviously we have

YA,2t = 1; ZA,2t = 0.

Then (3.5) becomes

At= ZA,1t gu(t, Xt, ut, vt). (3.30)

Denote YA,1t= YA,1t +

t0

gds. Then (3.7) and (3.10) become

YA,1t = U1(CT) +

Tt

[usZA,1s g]ds

Tt

ZA,1s dBs; (3.31)

15


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and

YA,1t = U1(CT) +

Tt

[ZA,1s h(s, Xs, vs, ZA,1s ) g(s, Xs, h(s, Xs, vs, Z

A,1s ), vs)]ds

Tt

ZA,1s dBs;

(3.32)

respectively.

3.4.2 The principals problem

First one can check straightforwardly that

YA = 1; ZA = 0; Y2 = Y1; Z2 = Z1. (3.33)

Denote

J1= U11 (3.34)

and

XAt= XAt + Gt; Y

3t

= Y3t X

2t .

Then (3.14) and (3.24) become, recalling notation (3.34),

CT = J1(XAT);

P,1t

= ZPt guuZ

1t ;

P,2t

= Z3t + utY

3t gvY

1t guvZ

1t ; (3.35)

Therefore, (3.26) becomes

Xt = x +

t0

vsdBs;

XAt = R +

t0

gds +

t0

gudBus ;

YPt = U2(XT J1(XAT))

Tt

ZPs dBus ;

Y1t =U2(XT J1(X

AT))

U1(J1(XAT))

Tt

Z1sdBus ;

Y3t = U2(XT J1(X

AT))

T

t

[gxY1s + guxZ

1s ]ds

T

t

Z3sdBus ;

(3.36)

with maximum conditions P,1t = P,2t = 0.

As mentioned in 3.2, we shall specify some sufficient conditions for the well-posedness

of the FBSDEs in this case. First, under the integrability conditions in (A5) below, X and

X are well defined. Applying Lemma 3.1 on (YP, ZP), (Y1, Z1) and then on (Y3, Z3), we

see that (3.36) is well-posed. Therefore, FBSDEs (3.11), (3.20), and (3.22) are well-posed

in this case.

Recall (3.16), (3.19), and define other -terms similarly. We now modify A as follows.

16


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(A5.) The principals admissible set A of controls is redefined as the set of all those

contracts (R,u,v) such that, for any bounded (u, v), there exists a constant 1 > 0 such

that for any [0, 1):

(i) u, v, MT, [MT]1, g, gu, g

v, g

x, g

uu, g

uv, g

ux, U

1 , U

2 , [U

2], and [J1]

are uniformly inte-

grable in Lp0(Q) or Lp0

T (Q), for some p0 large enough (where J1 = U11 ).(ii) u A(CT, v) and (CT, v) is implementable in the sense of (A4), where CT is defined

in (3.35);

Note that we may specify p0 as in (A5). But in order to simplify the presentation and

to focus on the main ideas, we assume p0 is as large as we want.

Theorem 3.3 Assume (A5). Then (A5) holds true and the optimal R is equal to R.

Proof: We first show that the principals optimal control R is R. In fact, for fixed (u, v),

let superscriptR

denote the processes corresponding to R. Then obviously XA,Rt X

A,Rt

for any R R. Since

1H(x, y) =1

1U1(H(x, y), y)> 0, U2 > 0,

we get

YP,RT = U2(XT CRT) = U2(XT H(X

A,RT , GT)) U2(XT H(X

A,RT , GT)) = Y

P,RT .

Therefore,

Y

P,R

0 = E

u

{Y

P,R

T } E{Y

P,R

T } = Y

P,R

0 .Thus, optimal R is equal to R.

It remains to prove

lim0

V2 = YP0 . (3.37)

We postpone the proof to the Appendix.

To end this subsection, for future use we note that (3.14) becomes, recalling notation

(3.34),

CT = J1

R +

T

0

gu(t, Xt, ut, vt)dBut +

T

0

g(t, Xt, ut, vt)dt

.

This means that the principals problem is

supu,v

Eu

U2

x +

T0

utvtdt +

T0

vtdBut (3.38)

J1

R +

T0

gu(t, Xt, ut, vt)dBut +

T0

g(t, Xt, ut, vt)dt

.

17


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3.4.3 Fixed volatility case

If we also assume v (hence X) is fixed, then (3.36) becomes

XAt = R + t

0

gds + t

0

gudBus ;


Tt

ZPs dBus ;

Y1t =U2(XT J1(X

AT))

U1(J1(XAT))

Tt

Z1sdBus ;

(3.39)

with maximum condition P,1t = 0.

4 Sufficient conditions

4.1 A general result

If the necessary condition uniquely determines a candidate for the optimal solution, it is

also a sufficient condition, if an optimal solution exists. We here discuss the existence of an

optimal solution. In general, our maximization problems are non-concave, so we have to use

infinite dimensional non-convex optimization methods.

Let H be a Hilbert space with norm and inner product < >. Let F : H IR be a

functional with Frechet derivative f : H H. That is, for any h, h H,

lim0

1

[F(h + h) F(h)] =< f(h), h > .

The following theorem is a direct consequence of the so-called Ekelands variational

principle, see Ekeland (1974).

Theorem 4.4 Assume

(A1) F is continuous;

(A2) There exists unique h H such that f(h) = 0;

(A3) For > 0, > 0 such that F(h) F(h) whenever f(h) .

(A4) V

= suphH

F(h) < .

Then h is the maximum argument of F. That is, F(h) = V.

Remark 4.1 (1) A sufficient condition for (A3) is thatf is invertible andf1 is continuous

at 0.

(2) If H = IR and f is continuous and invertible, then F is either convex or concave,

and thus the result obviously holds true.

(3) If (A4) is replaced by infhH

F(h) > , then h is the minimum argument of F.

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21/36


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4.2.4 Fixed volatility case

In this case v is fixed. Set H to be the admissible set of u with inner product defined by

(4.2). The functional is V2(u) with Frechet derivative P,1(u). We need the following:

(i) Considering u as a parameter, FBSDE (3.27) (without assuming (3.28)) is well-posed;

(ii) FBSDE (3.27) together with (3.28) has a unique solution u;

(iii) For any sequence of u,

P,1(u) 0 YP,u0 YP,u

0 .

Then u is the optimal control of the principals problem.

Similarly, (iii) can be replaced by the following stronger condition:

(iii) For any , FBSDE (3.27) together with condition P,1t = t is well-posed. In

particular,

V2(u) V2(u0), as 0.

5 Examples

We now present an important example, which is quite general in the choice of the util-

ity functions, and thus could be of use in many economic applications. The solution is

semi-explicit, as it boils down to solving a linear BSDE (not FBSDE!), and a nonlinear de-

terministic equation for a constant c. To the best of our knowledge, this is the firstexplicit

description of a solution to a continuous-time Principal-Agent problem with hidden action,other than the case of exponential and linear utility functions. Moreover, as in the latter

case, the optimal contract is still a function only of the final outcome XT, and not of the

history of process X.

5.1 Separable utility with fixed v and quadratic g

Consider the separable utility with fixed v and

g(t,x,u,v) = u

2

/2.

5.1.1 Necessary conditions

First by (3.8) and noting that gu = u, we get h(t,x,v,z) = z. Given (CT, v), the adjoint

process for the agents problem is

YA,1t = U1(CT) +1

2

Tt

|ZA,1s |2ds

Tt

ZA,1s dBs; (5.1)

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and, by (3.6) and section 4.2.1, the necessary and sufficient condition for the optimal control

is uCT,vt = ZA,1t . Denote

YA,1t= exp(YA,1t ); Z

A,1t

= YA,1t Z

A,1t .

Then (5.1) is equivalent to

YA,1t = exp(U1(CT))

Tt

ZA,1s dBs; (5.2)

If Assumption A.2 (ii) holds, exp(U1(CT)) is bounded, then (5.2) is well-posed, and so is

(5.1).

As for the principals problem, we note that (3.39) becomes

X

A

t = R

1

2t0 u

2

sds +t0 usdBs;


Tt

ZPs dBus ;

Y1t =U2(XT J1(X

AT))

U1(J1(XAT))

Tt

Z1sdBus ;

and, recalling (3.35), the maximum condition becomes

P,1t= ZPt Z

1t = 0.

Obviously we getYPt Y

1t c

where c is a constant (equal to YP0 Y10 .) In particular, at time T, we have

U2(XT J1(XAT))

U2(XT J1(XAT))

U1(J1(XAT))

= c.

Denote

F(x, y)= U2(x y)

U2(x y)

U1(y).

By our assumptions,

Fy(x, y) = U2 +

U2

U1+

U2U1

|U1|2

< 0.

Thus we may assume there exists a function (x, c) such that

F(x, (x, c)) = c.

Denote (x, c)= U1((x, c)). We have X

AT = (XT, c) for some constant c. Recall again

that XT FBT is a given fixed random variable.

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We would like to have a solution (XA,ct , uc) to the quadratic BSDE

XA,ct = (XT, c) +

Tt

u2s2

ds

Tt

usdBs. (5.3)

By Itos rule, this is equivalent to the linear BSDE

Xt := eX

A,ct = e(XT,c)

Tt

usXsdBs.

If e(XT,c) is square-integrable, there exists a solution to the BSDE (this is satisfied, in

particular, if U1 is bounded).

Furthermore, assume that we can find a unique constant c = cR so that

XA,c0 = R , or, equivalently, E[e(XT,c)] = eR. (5.4)

Then

XA,cR

=

XA

, and ucR

is a candidate for the optimal u for the principals problem.

5.1.2 Sufficiency

We next show that the above controls are indeed optimal. For the agents problem, this

follows from section 4.2.1. We now consider the principals problem. From (3.38), the

principals problem is

supu

Eu

U2

x +

T0

usvsds +

T0

vsdBus J1

XAT

.

Note thatlog Mut = X

At R.

We can thus rewrite the principals problem as

supu

E[G(MuT)] := supu

E[MuTU2 (XT J1 (R + log(MuT)))] .

Here, G is a random function on positive real numbers, defined by

G(x) = xU2(XT J1(R + log(x)).

We find

G(x) = U2(XT J1(R + log(x)) U2(XT J1(R + log(x))J

1(R + log(x)),

G(x) = 1

xU2(XT J1(R + log(x))J

1(R + log(x))

+1

x[J1(R + log(x))]

2U2 (XT J1(R + log(x))

1

xU2(XT J1(R + log(x))J

1 (R + log(x)).

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Note that

G(x) 0

so that G is a concave function, for every fixed XT().

We define the dual function, for y > 0,

G(y) = maxx>0

[G(x) xy].

The maximum is attained at

x = (G)1(y).

Thus, we get the following upper bound on the principals problem, for any constant c > 0:

E[G(MuT)] E[G(c)] + cE[MuT] = E[G(c)] + c.

The upper bound will be attained if

MuT = (G)1(c)

and c = c is such that

E[(G)1(c)] = 1.

This means that if u satisfies the Backward SDE

Mut = (G)1(c)

T

t

usMus dBs

then u is optimal.

We now show that

(G)1(c) = eRe(XT,c)

in the notation introduced before (5.3). Indeed,

G(eRe(XT,c)) = U2(XT J1((XT, c)) U2(XT J1((XT, c)))J

1((XT, c)) = c

where the last equality comes from the definition of (XT, c). Thus the sufficient Backward

SDE for u to be optimal becomes

Mut = eRe(XT,c)

Tt

usMus dBs

where

E[e(XT,c)] = eR.

We see from (5.4) that we can take c = cR. Also, because log Mt = XAt R, the sufficient

BSDE is the same as the necessary BSDE (5.3).

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27/36

Note that the agent wants to maximize YA,10 = Y0. By the BSDE Comparison Theorem,

the latter is maximized if the drift in (5.7) is maximized. We see that this will be true if

condition (5.5) is satisfied, which is then a sufficient condition.

Denote

J1(y) := U11 (y) = log(y)/1

The principals problem is then to maximize

EQu

[U2(XT J1(YA,1T (u)) GT)] (5.8)

We now impose the assumption (with a slight abuse of notation) that

g(t,x,u,v) = tx + g(t,u,v), (5.9)

for some deterministic function t. Doing integration by parts we get the following repre-

sentation for the first part of the cost GT:T0

sXsds = XT

T0

sds

T0

s0

udu[usvsds + vsdBus ] . (5.10)

If we substitute this into GT =T0

sXsds +T0

g(s, us, vs)ds, and plug the expression

for XT and the expression (5.6) for YA into (5.8), with U2(x) = e

ix, we get that we need

to minimize

Eu

exp

2[1

T0

sds][X0 +

T0

utvtdt] + 21

T0

g2u(s, us, vs)

2ds

+2

T

0

g(s, us, vs)ds 2

T

0

[

s

0

rdr]usvsds 2[1

T

0

sds]

T

0

vsdBus

+2

T0

gu(s, us, vs)dBus 2

T0

[

s0

rdr]vsdBus

. (5.11)

This is a standard stochastic control problem, for which the solution turns out to be de-

terministic processes u, v (as can be verified, once the solution is found, by either verifying

Hamilton-Jacobi-Bellman equation, or by verifying the corresponding maximum principle).

Assuming that u, v are deterministic, the expectation above can be computed by using the

fact thatEu[exp(

T

0

fsdBus )] = exp(

1

2

T

0

f2s ds)

for a given square-integrable deterministic function f. Then, the minimization can be done

inside the integral in the exponent, and boils down to minimizing over (ut, vt) the expression

1

Tt

sds

utvt + 1

g2u(t, ut, vt)

2+ g(t, ut, vt)

+22

1

Tt

sds

vt gu(t, ut, vt)

2. (5.12)

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The optimal contract is found from (3.14), as:

CT = GT 1

1log(YA,1T )

where YA

should be written not in terms of the Brownian Motion Bu

, but in the terms ofthe process X. Since we have

YA,1t = R exp

t0

21g2u(s, us, vs)/2ds +

t0

1usgu(s, us, vs)ds

t0

1gu(s, us, vs)dBs

(5.13)

we get that the optimal contract can be written as (assuming optimal vt is never equal to

zero)

CT = c + T

0

gu(s, us, vs)

vsdXs

for some constant c. If gu(s,us,vs)vs

is a constant, then we get a linear contract.

Let us consider the special case of Holmstrom-Milgrom (1987), with

v 1 , g(t,x,u,v) = u2/2.

Then (5.12) becomes

ut + 1u2t/2 + u

2t/2 +

22

{1 ut}2 .

Minimizing this we get constant optimal u of Holmstrom-Milgrom (1987), given by

u =1 + 2

1 + 1 + 2

The optimal contract is linear, and given by CT = a + bXT, where b = u and a is such that

the IR constraint is satisfied,

a = 1

1log(R) bX0 +

b2T

2(1 1) . (5.14)

5.3 Separable Exponential/Linear Utility

Example 5.1 Let us consider the separable utility case with U1(x) = U2(x) = x, that is,

both the principal and the agent are risk-neutral. Also assume that g(t,x,u,v) = g(t,u,v).

(The case with a linear cost in x could be handled as in the previous example.) From (3.38)

it is clear that we have to maximize

Eu[

T0

[usvs g(s, us, vs)ds]

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Assume that g is such that there is (u, v) which maximizes

usvs g(s, us, vs).

Then (u, v) is optimal pair, and it is deterministic. The optimal contract is

CT = J1(YA,1T + GT) = R +

T0

gu(s, us, vs)

vsdBus +

T0

g(s, us, vs)ds,

which is equivalent to

CT = R +

T0

gu(s, us, vs)

vsdXs +

T0

[g(s, us, vs) usgu(s, us, vs)]ds.

Note that ifgu is not a function of t, then u, v are constant and the contract is linear in XT.

Example 5.2 Let us consider now the case U1(x) = x, U2 = e

2x

. Also assume thatg(t,x,u,v) = g(t,u,v). From (3.38) it is clear that we have to minimize

Eu

exp

2

T0

[usvs g(s, us, vs)]ds +

T0

[vs gu(s, us, vs)]dBus

As in (5.11), the solution will be deterministic, and it minimizes

usvs + g(s, us, vs) +22

[vs g(s, us, vs)]2.

and the optimal contract is of the same form as in the previous example:

CT = R +

T0

gu(s, us, vs)

vsdXs +

T0

[g(s, us, vs) usgu(s, us, vs)]ds.

For example, if v 1 and g(t,u,v) = u2/2, it is easy to see that u 1 and CT = R T2

+

XT X0.

6 Conclusion

We provide a general theory for Principal-Agent problems with hidden action in models

driven by Brownian Motion. We analyze both the agent and the principals problem in

weak formulation, thus having a consistent framework. In the case of separable utility and

quadratic cost function, we find the optimal solution for general utility functions. In general,

however, the question of the existence of an optimal solution remains open.

Important application of the Principal-Agent theory is the problem of optimal compen-

sation of executives. For that application, other, more general forms of compensation should

be considered, such as a possibility for the agent to cash in the contract at a random time.

We leave these problems for future research.

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31/36

Then

E{|MT MT|2} = E

10

MT

T0

utdBt MT

T0

utdBt

M

TT0 (ut + ut)utdt MT

T0 ututds

d2

C

10

E

|MT MT|2|

T0

utdBt|2

+|MT MT|2|

T0

(ut + ut)utdt|2

+|MT|2|

T0

[(ut + ut)ut utut)]dt|2

d

C1

0 E{|MT MT|

4}

+

E{|MT MT|4}

T0

E{|(ut + ut)|4}dt

+E{|MT|2}2

d.

Then by (7.2) and Assumption A3 (iii) we prove the result.

We now show (3.3), that is, we show that

lim0

V1 = EMTU1 + MT2U1GT.Note that we have

V1 =V1 V1

= E

MTU

1 + MT

U1U1

(7.4)

As for the limit of the first term on the right-hand side, we can write

MTU1 MTU1 = [M

T MT]U1 + M

T[U

1 U1].

By Assumption A3 (iii) and the above L2 bounds on MT, this is integrable uniformly with

respect to , so the expected value (under Q) converges to zero, which is what we need.As for the limit of the second term in the right side of (7.4), notice that we have

MT lim0

U1 U1

= MT2U1GT. (7.5)

We want to prove the uniform integrability again. We note thatU1 U1 =

10

2U1 (CT, GT + (GT GT)) d

|GT| {|2U1 (CT, GT)| + |2U1 (CT, G

T)|}|G

T|

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where the last inequality is due to monotonicity of 2U1.

Therefore, we get

MTU1 U1

C|2U1 (CT, GT)|2 + |2U1 (CT, G

T)|

2 + |GT|4 + |MT|

4

Thus, from Assumption A3 (iii), the left-hand side is uniformly integrable, and the expec-

tations of the terms in (7.5) also converge, and we finish the proof of (3.3).

We now want to prove (3.4). We have

V1 = E

MTU1 + MT2U1GT

= E

MTU1

T0

utdBut + MT2U1

T0

guutdt

= Eu

YA,1TT0

utdBut + YA,2T

T0

guutdt

= EuT

0

At utdt +

T0

Bt dBut

, (7.6)

where

At= ZA,1t + gu(t, Xt, ut, vt)Y

A,2t ,

Bt

= YA,1t us + Z

A,1t

t0

usdBus + Z

A,2t Gt

and the last equality is obtained from Itos rule, and definitions of YA,i, ZA,i. We need to

show

EuT0

Bt dBut = 0

We want to use Lemma 3.2 in the last two lines of (7.6), with = B and

Wt = YA,1t

t0

usdBus + Y

A,2t

t0

gu(s)usds

At =t0

A

s usds

From the BSDE theory and our assumptions we have

Eu

sup0tT

(|YA,1t |2 + |YA,2t |

2) +

T0

(|ZA,1t |2 + |ZA,2t |

2)dt

< . (7.7)

From this it is easily verified that T0

Bt

2

dt <

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so that condition 1) of the lemma is satisfied. Next, we have

Eu

sup

0tT|Wt|

CEu

sup

0tT|YA,1t |

2 + |YA,2t |2+

+

T

0

|gu(t)|2dt

C+ CE

M2T +

T

0

|gu(t)|4dt

< ,

thanks to (7.7) and (7.1). Moreover,

Eu

sup0tT

|At|

= Eu

sup0tT

|

t0

[ZA,1s + gu(s)YA,2s ]usds|

CE

MT

T0

|ZA,1t + gu(s)YA,2s |dt

CE

|MT|4

+T0 [|Z

A,1

t |2

+ |gu(t)|4

+ |YA,2

t |2

]dt

<

The last two bounds ensure that condition 2) of the lemma is satisfied, so that the last term

in (7.6) is zero, and we finish the proof of (3.4).

Finally, (3.6) follows directly from (3.4) if u is optimal, as ut is arbitrary.

7.2 Proof of Theorem 3.3

We have already shown that we can set R = R. Recall that

Xt = x +

t

0

vsdBs;

Mt = expt

0

usdBs 1

2

t0

|us|2ds

;

XAt = R +

t0

gds

t0

usguds +

t0

gudBs;

YPt = U2(XT J1(XAT)) +

Tt

usZPs ds

Tt

ZPs dBs;

and that

Xt =t0

vsdBs;

Mt = Mt[

t0

usdBs

t0

ususds];

= uut + vvt + xXt, = g, gu;

XAt =

t0

gds

t0

[guus + usgu]ds +

t0

gudBs;

YPt = U2(XT J1(X

AT))[XT

XATU1(J1(X

AT))

]

+ T

t[ZPs us + usZ

Ps ]ds

T

tZPs dBs;

32


34/36

To prove (3.37), we need the following result. For any random variable and any p > 0,

Eu

{||p} = E{Mu

T ||p}

E{|Mu

T |2}

E{||2p} C

E{||2p};

E{||p} = Eu

{[Mu

T ]1||p} E

u{[Mu

T ]2}E

u{||2p}

=

E{[MuT ]1}

Eu{||2p} C

Eu{||2p}.

(7.8)

Proof of (3.37): In this proof we use a generic constant p 1 to denote the powers,

which may vary from line to line. We assume all the involved powers are always less than

or equal to the p0 in (A5).

First, one can easily show that

lim0

E sup0tT[|Xt Xt|

p + |Mt Mt|p + |XAt X

At |

p] + T

0

[|g g|p + |gu gu|p]dt = 0.

Using the arguments in Lemma 3.1 we have

Eu

[

T0

|ZP,t |2dt]p

C < ,

which, by applying (7.8) twice, implies that

Eu

[

T0

|ZP,t |2dt]p

C < .

Note that

YP,t YPt = U

2 U2 +

Tt

usZ

P,s + us[Z

P,s Z

Ps ]

ds

Tt

[ZP,s ZPs ]dBs

= U2 U2 +

Tt

usZP,s ds

Tt

[ZP,s ZPs ]dB

us .

Using the arguments in Lemma 3.1 again we get

lim0

Eu

sup0tT

|YP,t YPt |

p + [

T0

|ZP,t ZPt |

2dt]p

= 0,

which, together with (7.8), implies that

lim0

E

sup0tT

|YP,t YPt |

p + [

T0

|ZP,t ZPt |

2dt]p

= 0. (7.9)

Next, recall (3.19) one can easily show that

lim0

E

sup0tT

[|Xt Xt|p + |Mt Mt|

p + |XAt XAt |

p]

+

T0

[|g g|p + |gu gu|p]dt

= 0.

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35/36

Then similar to (7.9) one can prove that

lim0

E

sup0tT

|YP,t YPt |

p

= 0.

In particular, lim0

V2 = lim0

YP,0 = YP0 .

The proof is complete.

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