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arXiv:1011.3228v1 [math.PR] 14 Nov 2010

On Girsanov’s transform for backward

stochastic differential equations

By G. Liang, A. Lionnet and Z. Qian

Oxford-Man Institute, Oxford University, England

Abstract. By using a simple observation that the density pro-

cesses appearing in Itˆ o’s martingale representation theorem are

invariant under the change of measures, we establish a non-linear

version of the Cameron-Martin formula for solutions of a class of

systems of quasi-linear parabolic equations with non-linear terms

of quadratic growth. We also construct a local stochastic flow

and establish a Bismut type formula for such system of quasi-

linear PDEs. Gradient estimates are obtained in terms of the

probability representation of the solution. Another interesting

aspect indicated in the paper is the connection between the non-

linear Cameron-Martin formula and a class of forward-backward

stochastic differential equations (FBSDEs).

Key words. Brownian motion, backward SDE, SDE, non-linear equations

AMS Classification. 60H10, 60H30, 60J45

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1Introduction

The goal of the article is to present a unified approach for Girsanov’s tech-

niques of changing probability measures used in the recent literature on back-

ward stochastic differential equations (BSDE). The framework we formulate

has an advantage allowing us to bring together seemingly different sorts of re-

sults on BSDE, forward-backward stochastic differential equations (FBSDE),

and function equations on a probability space.

Consider the following system of quasi-linear parabolic equations

∂

∂tui=1

2∆ui+

d

?

j=1

fj(u,∇u)∂ui

∂xj

in Rd

(1.1)

i = 1,··· ,m, where the drift vector field involves a solution u and its to-

tal derivative ∇u = (∂ui

fj: Rd×Rm×d→ R (j = 1,··· ,d) are Lipschitz continuous but unbounded,

and therefore the non-linear term (often called the convection term) appear-

ing in (1.1) is of quadratic growth. This is a kind of non-linear feature which

appears in many physical PDEs, see for example [5] and [11]. If m = d = 3

and fj(u) = uj, then (1.1) is a modification of the Navier-Stokes equations

with the pressure term and the divergence-free condition dropped altogether,

while the same non-linear convection term has been retained. This kind of

PDEs has been used as simplified models for phenomena such as turbulence

flows. Due to the special structure of the system (1.1), the maximum princi-

ple applies to |u(x,t)|2, thus a bounded solution exists as long as the initial

data u(x,0) is regular and bounded, which makes a distinctive difference

from the Navier-Stokes equations. According to Theorem 7.1 on page 596 in

[8], if the initial data u0is smooth and bounded with bounded derivatives,

then a bounded, smooth solution u to the initial value problem of (1.1) ex-

ists for all time. Our main interest in this article is to establish probabilistic

representations for the solution u by applying Girsanov’s theorem to BSDEs.

To this end, it will be a good idea to look at Peng’s non-linear Feymann-

Kac formula (see [15]) for general quasi-linear parabolic equations with Lip-

schitz non-linear term. The main idea in [14] and [15] can be described as

following. Thanks to Itˆ o’s calculus, Brownian motion B =?B1,··· ,Bd?may

with the Wiener measure, and it is a much tested idea that one may “read

out” solutions of quasi-linear partial differential equations in terms of B.

∂xj) through f = (fj)j≤d. In the interesting cases,

be considered as “coordinates” on the space of continuous paths equipped

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Suppose u(x,t) = (u1(x,t),··· ,um(x,t)) is a sufficiently smooth solution to

the Cauchy problem of the system of quasi-linear parabolic equations

?1

2∆ −∂

∂t

?

ui+ hi(u,∇u) = 0in Rd× [0,∞)

with the initial data ui(x,0) = ui

Bt,T − t) (and set Zi,j

0(x). Applying Itˆ o’s formula to Yi

∂xj(x + Bt,T − t)) to obtain

t= ui(x+

t

=∂ui

Yi

T− Yi

t

=

d

?

j=1

?T

?T

?T

t

∂ui

∂xj(x + Bs,T − s)dBj

s

+

t

?1

2∆ −∂

∂s

?

ui(x + Bs,T − s)ds

=

d

?

j=1

t

Zi,j

sdBj

s−

?T

t

hi(Ys,Zs)ds

for t ∈ [0,T]. The pair (Y,Z) is a solution to the stochastic differential

equations

dYi

t= −hi(Yt,Zt)dt +

d

?

j=1

Zi,j

tdBj

t

(1.2)

on (Ω,F,Ft,P) with the terminal data Yi

Peng [15] and Pardoux-Peng [14] made two crucial observations: firstly

the process Ztmay be recovered from the Itˆ o representation of the martingale

Si

0

?d

solved by specifying a terminal value YT. It was proved in Pardoux-Peng [14]

that if hiare global Lipschitz continuous, then (1.2) may be solved as long

as YT∈ L2(Ω,FT,P).

In the case that hi(y,z) has a special form such as?d

change the probability measure, even for non-Lipschitz non-linear term h as

long as f is Lipschitz continuous. We are thus able to extend the Cameron-

Martin formula to a class of systems of quasi-linear parabolic equations with

quadratic growth. The non-linear version of the Cameron-Martin formula,

which may be considered as our contribution of this article, is of independent

interest.

T= ui

0(x + BT).

t= Si

continuous semimartingale Y , therefore (1.2) is a closed system and may be

0+?t

j=1Zi,j

sdBj

s. Secondly S is indeed the martingale part of the

j=1fj(y,z)zij, one is

t=?d

able to solve (1.2) by firstly solve the trivial BSDE dYi

j=1Zi,j

tdBj

tthen

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Suppose that the initial data u0is Lipschitz continuous and bounded. Let

(Ft)t≥0be Brownian filtration, i.e. the filtration over Wdof all continuous

paths in Rdgenerated by the coordinate process {Bt: t ≥ 0}, augmented by

the Wiener measure. Let T > 0 and x ∈ Rdbe fixed but arbitrary. For each

ξ = (ξ1,··· ,ξm) where ξi∈ L∞(Wd,FT,Px). Define

?t

where Y (ξ)t= EP(ξ|Ft) are bounded martingales and Z(ξ) = (Z(ξ)i,j) are

determined by Itˆ o’s martingale representation:

˜B(ξ)t= Bt−

0

f(Y (ξ)s,Z(ξ)s)ds

ξi− EP(ξi|F0) =

d

?

j=1

?T

0

Z(ξ)i,j

tdBj

t.

We prove that there is a unique ξ ∈ L∞(Wd,FT,Px) such that ξ = u0(˜B(ξ)T),

and u(x,T) = EP(Y (ξ)0), which can be considered as the non-linear version

of the Cameron-Martin formula. More information about the solution u may

be obtained as we can represent the derivative

(Z(ξ)i,j

Martin formula may be reformulated in terms of FBSDE as well, see (2.4)

and Corollary 2.4.

The main reason we are interested in representations of solutions to phys-

ical PDEs in terms of Brownian motion or in general in terms of Itˆ o’s diffu-

sions, lies in the fact that it is then possible to employ probabilistic methods

such as Itˆ o’s calculus, Malliavin’s calculus of variations and path integra-

tion method to the study of non-linear PDEs. We demonstrate this point

by deriving explicit gradient estimates for solutions of a class of systems of

quasi-linear parabolic equations.

Finally we would like to point out that the type of BSDEs such as (1.2)

with quadratic growth non-linear terms has been well studied in Kobylanski

[7]. Her work has been extended and generalized substantially in Briand

and Hu [3] and [2]. BSDEs with quadratic growth driven by martingales

are solved by Morlais [13] and Tevzadze [16]. However their methods can be

applied to scalar BSDEs, not systems in general.

∂ui

∂xj in terms of the process

t)t≤T, see Theorem 3.1 below. The non-linear version of the Cameron-

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2Girsanov’s theorem, martingale represen-

tation and BSDE

Let B = (B1,··· ,Bd) be a Brownian motion in Rdstarted at 0 on a com-

plete probability space (Ω,F,P), (Ft)t≥0 be the filtration generated by B

augmented by probability zero sets in F and F∞= σ{Ft: t ≥ 0}. Since

we are interested in F∞-measurable random variables only, for simplicity, we

assume that F = F∞. According to Itˆ o’s martingale representation theorem,

any martingale S on (Ω,F,Ft,P) is continuous and has a unique represen-

tation in terms of Itˆ o’s integration

St− S0=

d

?

j=1

?t

0

DB(S)j

sdBj

s

where DB(S)jare predictable processes, called the density processes of S with

respect to Brownian motion B. Since we will deal with several equivalent

measures on (Ω,F) at the same time, it is desirable to have labels associated

with notations which involve a probability measure.

convention if confusions may arise. Therefore, EPand EP{·|Ft} denote the

expectation and conditional expectation with respect to P respectively.

Let Q be a probability measure on (Ω,F) equivalent to P, whose Radon-

Nikodym’s derivative with respect to P restricted on Ftis denoted by Rt,

that is,

dP

??

tween local P-martingales and local Q-martingales. If X = (Xt)t≥0is a local

martingale under probability P, then X is a continuous semi-martingale un-

der Q and˜ Xt= Xt−?t

invariant under the change of equivalent probability measure.

It is convenient to formulate the Girsanov’s transform in terms of expo-

nential martingales. The exponential martingale E(N)t= exp{Nt−1

of a continuous local martingale N (under P) is the unique solution to the

stochastic exponential equation

We will follow this

dQ

Ft= Rtfor t ≥ 0. Then R is a positive martingale on (Ω,F,Ft,P)

with EP(Rt) = 1. Girsanov’s theorem [6] establishes a correspondence be-

0

1

Rsd?X,R?sis a local martingale under Q, where the

bracket process ?X,R?tis defined under the probability P, which is however

2?N?t}

E(N)t= 1 +

?t

0

E(N)sdNs.

Up to an initial data R0, the Radon-Nikodym derivative R = (Rt)t≥0of Q

with respect to the measure P must be an exponential martingale of some

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