Page 1

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

1

Page 2

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

2

Page 3

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

3

Page 4

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-

4

Page 5

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

5

Page 6

continuous local martingale N, so that Rt = R0exp{Nt−1

˜ X = X − ?X,N? for any P-local martingale X.

According to L´ evy’s theorem,˜B = (˜B1,··· ,˜Bd) is a Brownian motion

under Q which is (Ft)t≥0-adapted, but the natural filtration (˜Ft)t≥0of˜B may

not coincide with (Ft)t≥0. It can happen that˜Ftis strictly smaller than Ftfor

some t. But, nevertheless any (Ft)-martingale under Q may be represented

as Itˆ o’s integral of (Ft)-predictable processes against˜B. The starting point

of our approach is the following elementary fact.

2?N?t}. Then

Lemma 2.1 For any local P-martingale X (with respect to the filtration

(Ft)t≥0) DB(X) = D˜B(˜ X). That is, the density process appearing in Itˆ o’s

martingale representation is invariant under the change of equivalent proba-

bility measures.

The lemma follows directly from the definition of density processes and

the Girsanov’s theorem, so its proof is left to the reader. With the help of

this simple fact, we can deal with the question of how a backward stochastic

differential equation is transformed under change of equivalent probability

measures.

Theorem 2.2 Suppose that g is global Lipschitz continuous, ξ ∈ L2(Ω,FT,P),

and (Y,S) is the unique solution pair to BSDE

dYt= −g(t,Yt,DB(S)t)dt + dSt, YT= ξ

on (Ω,F,Ft,P). Let˜S = S − ?S,N?. Then (Y,˜S) solves

(2.1)

dYt= −g(t,Yt,D˜B(˜S)t)dt +

d

?

j=1

D˜B(˜S)j

tDB(N)j

tdt + d˜St

(2.2)

on (Ω,F,Ft,Q), YT= ξ.

Proof. Since (Y,S) is a solution to (2.1), Yiare continuous semimartin-

gale with martingale parts Sj. By Lemma 2.1, DB(Si) = D˜B(˜Si), so that

˜Si

t−˜Si

0=

d

?

j=1

?t

0

D˜B(˜Si)j

sd˜Bj

s=

d

?

j=1

?t

0

DB(Si)j

sd˜Bj

s

6

Page 7

and

˜Si

t= Si

t−

d

?

j=1

?t

0

D˜B(˜Si)j

sDB(N)j

sds.

Therefore

Yi

T− Yi

t

= −

?T

d

?

t

gi(s,Ys,D˜B(˜S)s)ds

+

j=1

?T

t

D˜B(˜Si)j

sDB(N)j

sds +˜Si

T−˜Si

t

(2.3)

which is valid under the probability Q. That is, the pair (Y,˜S) solves BSDE

(2.2) on (Ω,F,Ft,Q), with terminal values Yi

The most interesting case is the following special choice of N.

T= ξi.

Corollary 2.3 Under the same assumptions as in the previous theorem, and

Nt=

d

?

j=1

?t

0

fj(Ys,DB(S)s)dBj

s

where f : Rm×Rmd→ Rdis Borel measurable. Then (Y,˜S) solves the BSDE

dYt= −g(t,Yt,D˜B(˜S)t)dt +

d

?

j=1

D˜B(˜S)j

tfj(Yt,D˜B(˜S)t)dt + d˜St

with the terminal value YT= ξ on (Ω,F,Ft,Q).

One may reformulate Corollary 2.3 in terms of forward-backward stochas-

tic differential equations. Observe that, with the choice of N made in Corol-

lary 2.3, X = x +˜B is the solution to the stochastic differential equation

dXt= −f(Ys,Zt)dt + dBt,Xt= x

while (Y,Z = DB(S)) is the solution of the BSDE

dYt= −g(t,Yt,Zt)dt + ZtdBt, YT= u0(XT),

7

Page 8

thus (X,Y,Z) is a solution to the following forward-backward stochastic dif-

ferential equations (FBSDEs)

dXt= −f(Ys,Zt)dt + dBt,

dYt= −g(t,Yt,Zt)dt + ZtdBt,

Xt= x, YT= u0(XT).

(2.4)

FBSDEs such as (2.4) have been studied by various authors, and are

well presented in the research monograph [10]. By utilizing the fundamental

apriori estimates established in [8], it has been proved in [9] that FBSDE

(2.4) has a unique solution such that Y is bounded, as long as u0is bounded

and Lipschitz, and if f and g are Lipschitz continuous.

Corollary 2.4 Let f and g be Lipschitz continuous, u0be bounded and Lip-

schitz continuous. Let (X,Y,Z) be the unique solution of (2.4) such that Y

is bounded. Define Q on (Ω,FT) by

dQ

dP

FT

0

????

= exp

??T

f(Ys,Zs).dBs−1

2

?T

0

|f(Ys,Zs)|2ds

?

.

Then (Y,˜Z) is the unique solution (such that Y is bounded) to

?

dYt=

?

−g(t,Yt,˜Zt) +?d

j=1fj(Yt,˜Zt)˜Zj

t

?

dt +?d

j=1˜Zj

td˜Bj

t,

YT= u0(XT)

(2.5)

on (Ω,F,Ft,Q).

3 Cameron-Martin formula for PDEs

In this section we apply the machinery developed in the previous section to

the initial value problem of the following quasi-linear parabolic system

∂ui

∂t+

d

?

j=1

fj(u,∇u)∂ui

∂xj=1

2∆ui+ gi(u,∇u) in Rd

(3.1)

for i = 1,··· ,m, where fjand giare global Lipschitz continuous in their

arguments, together with the initial data u(x,·) = u0(x) which is bounded

and global Lipschitz continuous. According to Theorem 7.1 in [8], there is

8

Page 9

a unique bounded solution u(x,t) to the initial value problem of (3.1). By

Itˆ o’s formula, Yt= u(x + Bt,T − t) and Zt= ∇u(x + Bt,T − t) solve the

BSDE

dYi

t=

?

−gi(Yt,Zt) +

d

?

j=1

fj(Yt,Zt)Zi,j

t

?

dt +

d

?

j=1

Zi,j

tdBj

t

(3.2)

with the terminal value YT= u0(x + BT) over the filtered probability space

(Ω,F,Ft,P), which is indeed the unique bounded solution of BSDE (3.2).

On the other hand, since giare global Lipschitz, according to Pardoux

and Peng [14], for any ξ ∈ L2(Ω,FT,P) there is a unique solution pair (Y,Z)

to the backward stochastic differential equation:

dYt= −g(Yt,Zt)dt +

d

?

j=1

Zj

tdBj

t, YT= ξ (3.3)

on (Ω,F,Ft,P), which is the system derived from (3.2) with the quadratic

non-linear term dropped . The solution (Y,Z) depends on the terminal value

ξ, so it is denoted by Y (ξ) and Z(ξ) respectively. For x ∈ Rdand T > 0, let

N(ξ)t=

d

?

j=1

?t

0

fj(Y (ξ)s,Z(ξ)s)dBj

s

(3.4)

and

˜B(ξ)t= Bt−

?t

0

f(Y (ξ)s,Z(ξ)s)ds.(3.5)

Theorem 3.1 Let u be a classical solution of (3.1) on [0,T]×Rdsuch that

u(0,·) = u0(·). Suppose that ξ ∈ L∞(Ω,FT,P) is the solution to the function

equation

ξ = u0(x +˜B(ξ)T),

then

u(x +˜B(ξ)t,T − t) = Y (ξ)t

and

∇u(x +˜B(ξ)t,T − t) = Z(ξ)t

for all t ≤ T almost surely.

9

Page 10

Proof. Define an equivalent probability measure Q by

Then˜B(ξ) is a Brownian motion (up to time T) under Q. Let˜S = S(ξ) −

?N(ξ),S(ξ)?.

BSDE

dQ

dP

??

Ft= E(N(ξ))t.

According to Corollary 2.3, (Y (ξ),˜S) solves the following

dYt=

d

?

j=1

D˜B(˜S)j

tfj(Yt,D˜B(˜S)t)dt − g(t,Yt,D˜B(˜S)t)dt + d˜St

with terminal value YT = ϕ(x +˜B(ξ)T) on (Ω,F,Ft,Q). This system is

exactly the BSDE that Yt = u(x +˜B(ξ)t,T − t) should satisfy, and the

conclusions follow immediately.

By Theorem 3.1, in order to provide a probabilistic representation for

(3.1), the problem is reduced to solve the function equation

ξ = u0

?

x + BT−

?T

0

f(Y (ξ)s,Z(ξ)s)ds

?

:= φ(ξ).(3.6)

The following local existence of the solutions to (3.6) is elementary.

Proposition 3.2 Let Cu0and Cf be the Lipschitz constants for u0 and f

respectively, and

?

8C2

If T ≤ τ, then (3.6) admits a unique fixed point ξ ∈ L∞(Ω,FT,P).

Proof. For ξ and η in L2(Ω,FT,P),

?t

≤ Cu0Cf

0

τ =

1

u0C2

f

+ 1 − 1.

|φ(ξ) − φ(η)| ≤ Cu0

0

|f(Y (ξ)s,Z(ξ)s) − f(Y (η)s,Z(η)s)|ds

?t

|Y (ξ − η)s| + |Z(ξ − η)s|ds

and furthermore,

E?|φ(ξ) − φ(η)|2?≤ C2

u0C2

fE

???T

??T

0

|Y (ξ − η)s| + |Z(ξ − η)s|ds

?2?

≤ 2C2

u0C2

fT

0

(E|Y (ξ − η)s|2+ E|Z(α − β)s|2)ds

?

.

10

Page 11

Now, for any η ∈ L2(Ω,Ft,P),

?Y (η)s?2

2= E[|E(η|Fs)|2] ≤ ?η?2

2,

and

E

??t

0

|Z(η)s|2ds

?

= E?|Y (η)t− Y (η)0|2?

≤ 2?η?2

2,

so that

E?|φ(ξ) − φ(η)|2?≤ 2C2

u0C2

fT

??T

0

?ξ − η?2

2ds + 2?ξ − η?2

2

?

= 2C2

u0C2

fT(T + 2)?ξ − η?2

2.

Hence

||φ(ξ) − φ(η)||2≤

√2Cu0Cf

?

T(T + 2)||ξ − η||2

and the claim follows from a simple application of the fixed point theorem.

Theorem 3.3 Under the previous assumptions, the function equation (3.6)

has a unique solution.

Proof. We try to extend the local solution constructed in the previous

proposition to the case T > τ. To do this we need a uniform gradient estimate

of the solutions to PDE (3.1): there exists a constant Cusuch that

sup

(t,x)∈[0,T]×Rd|∇u|(x,t) ≤ Cu.

For the proof see for example [8] and [4]. Based on such constant Cu we

define

?

8C2

Now we construct the random variable ξ′on the time interval [τ,τ + τ′].

Consider the following function equation:

τ′=

1

uC2

f

+ 1 − 1.

ξ = u

?

x + B′

τ+τ′ −

?τ+τ′

τ

f(s,Y (ξ)s,Z(ξ)s)ds,τ

?

:= φ′(ξ).(3.7)

11

Page 12

where B

s ∈ [τ,τ + τ′]. Analogous to the proof of Proposition 3.2, (3.7) admits a

unique fixed point ξ′∈ L∞(Ω,Fτ+τ′,P) ⊂ L2(Ω,Fτ+τ′,P). Based on such

ξ′, we have the following representation formulae on [τ,τ + τ′]:

u(x +˜B(ξ′)t,2τ + τ′− t) = Y (ξ′)t

and

∇u(x +˜B(ξ′)t,2τ + τ′− t) = Z(ξ′)t

for t ∈ [τ,τ + τ′], where

˜B(ξ′)t= B′

τ

We then move to the next interval [τ + τ′,τ + 2τ′] and repeat the above

procedure until we touch T.

′is a Brownian motion on [τ,τ + τ′] defined by B

′

s= Bs− Bτ for

t−

?t

f(u,Y (ξ′)u,Z(ξ′)u)du.

4Some applications

In this section we establish some explicit gradient estimates for the solution of

(3.1) by using the representation theorem 3.1, to demonstrate the usefulness

of non-linear Cameron-Martin’s formula. Further applications will appear in

a separate paper.

Let us retain the notations and assumptions established in the previous

section. Let T > 0 be fixed, u = (u1,··· ,um) be the unique solution to the

initial value problem (3.1) with initial data u0, where f (which determines

the nature of the quadratic non-linear term) depends only on y, i.e. f(y,z)

does not depend on z and g = 0. That is, u is the solution to the initial value

problem of the following system of quasi-linear parabolic equations

∂ui

∂t+

d

?

j=1

fj(u)∂ui

∂xj=1

2∆ui

in Rd

(4.1)

for i = 1,··· ,m.

Let ξ be the solution to the function equation (3.6) established in Theorem

3.3.

Let Q be the equivalent measure with density process

E(N(ξ))t, where

N(ξ)t=

0

dQ

dP

??

Ft= Rt =

?t

f(Y (ξ))s).dBs.

12

Page 13

Theorem 4.1 1) Let p ∈ [1,2) and (Pt)t≥0the heat semi-group, i.e. Pt=

e

1

2t∆. Then

?t

0

Ps

??∇ui??p(x,T − s)ds ≤ d1−p

for any t ≤ T, i = 1,··· ,m.

2) We have

2?EP?Y (ξ)i?t

?p

2

??t

0

EPR

2

2−p

s

ds

?1−p

2

??∇ui??2(x,T) ≤ limt↓01

tEP?Y (ξ)i?t

for any t ≤ T, i = 1,··· ,m.

Proof. By Theorem 3.1

Y (ξ)i

t− Y (ξ)i

0=

d

?

j=1

?t

0

∂ui

∂xj(x +˜B(ξ)s,T − s)dBj

s

so that

?Y (ξ)i?t=

?t

0

??∇ui??2(x +˜B(ξ)s,T − s)ds

and therefore

EP

?t

0

??∇ui??2(x +˜B(ξ)s,t − s)ds = EP?Y (ξ)i?t.

On the other hand, for 1 ≤ p < 2 we have

??t

=

0

?t

?t

EQ

0

??∇ui??p(x +˜B(ξ)s,T − s)ds

EQ???∇ui??p(x +˜B(ξ)s,T − s)

?

?

?

ds

?t

?

≤

0

EQ

?1

Rs

??∇ui??2(x +˜B(ξ)s,T − s)

EP???∇ui??2(x +˜B(ξ)s,T − s)

??p

2?

2?

EP

EQ?

?

R

p

2−p

s

??1−p

2

2ds

=

0

??p

R

2

2−p

s

??1−p

ds

13

Page 14

Since˜B(ξ) is a Brownian motion under Q, so that

EQ

?t

Ps

0

??∇ui??p(x +˜B(ξ)s,T − s)ds

???∇ui??p(·,T − s)?(x)ds

EP??∇ui??2(x +˜B(ξ)s,T − s)

??t

?p

which is the first estimate. To prove the second one, we write the previous

estimate as

=

?t

?t

0

≤

0

??p

2?

EPR

2

2−p

s

?1−p

2

ds

≤ d1−p

2

0

EP??∇ui??2(x +˜B(ξ)s,T − s)ds

2?EP?Y (ξ)i?t

?p

2??t

0

EPR

2

2−p

s

ds

?1−p

2

= d1−p

2

??t

0

EPR

2

2−p

s

ds

?1−p

2

1

t

?t

0

Ps

??∇ui??p(x,T − s)ds

tEP?Y (ξ)i?t

≤ d1−p

2

?1

?p

2?1

t

?t

0

EPR

2

2−p

s

ds

?1−p

2

.

Letting t → 0 one obtains that

p

?

|∇ui|p(x,T) ≤ d

1

p−1

2limt→0

?1

tEP?Y (ξ)i?t.

then letting p ↑ 2 we obtain 2).

Lemma 4.2 Then for any p ∈ [1,2)

?

EP

R

2

2−p

t

?

≤ exp

?

p

(2 − p)2t max

|y|≤|u0|∞|f(y)|2

?

.

Proof. Let

Ht= exp

?

2

(2 − p)

?t

0

f(Y (ξ)s).dBs−

2

(2 − p)2

?t

0

|f(Y (ξ)s)|2ds

?

14

Page 15

which is exponential martingale, so that EP(Ht) = 1. Then

R

2

2−p

t

= Htexp

?

?

p

(2 − p)2

p

(2 − p)2t max

?t

|y|≤|u0|∞|f(y)|2

0

|f(Y (ξ)s)|2ds

?

≤ Htexp

?

which yields the claim.

Corollary 4.3 We have

?t

0

Ps

??∇ui??p(x,T − s)ds ≤ d1−p

for any p ∈ [1,2), and i = 1,··· ,m, where Ps= e

Proof. Observe that

2|ui

0|p

∞exp

?

p

2(2 − p)t max

|y|≤|u0|∞|f(y)|2

?

(4.2)

1

2s∆the heat semigroup.

EP?Y (ξi

t)?s = EP?EP(ξi

= EP?EP(ξi

≤ ||ξi

≤ |ui

t|Fs) − EP(ξi

t|Fs)2− EP(ξi

t|F0)?2

t|F0)2?

t||2

0|∞

and therefore, the item 1) in Theorem 4.1 together with Lemma 4.2 yields

the gradient estimate (4.2).

5Non-linear flow associated with quasi-linear

PDEs

In this section we construct a non-linear stochastic flow associated with the

quasi-linear system (4.1).

Assume that fj: Rm→ Rd(j = 1,··· ,d) are differentiable with bounded

1st, 2nd and 3rd derivatives, and the initial value ui(x,0) = ui

1,··· ,m) are bounded, differentiable with 1st and 2nd bounded derivatives:

|∇k−1u0| ≤ C0,

for some non-negative constants C0and C1, here ∇kdenote the k-th deriva-

tive in space variables. Of course ∇0u0= u0. In general, we apply ∇ to mean

0(x) (i =

|∇kf| ≤ C1 for k = 1,2,3

15

Page 16

the total derivative operator in space variables. For example, ∇u means (∂ui

but does not include the derivative in time parameter t.

We are going to construct a continuous adapted process ξ = (ξt) with

values in the function space C2

least for small t. The spirit in devising such a formula is quite similar to

those initiated in the seminal works Bismut [1] and Malliavin [12].

Consider the function space L∞(Ω;C([0,T];C2

∂xj)

b(Rd,Rm) such that ∇ju(·,t) = EP(∇jξt) at

b(Rd,Rm))). If

ξ ∈ L∞(Ω;C([0,T];C2

b(Rd,Rm)))

then for any ω, ξ(ω) ∈ C([0,T];C2

and t → ξt(ω) is continuous from [0,T] to C2

has continuous and bounded first and second derivatives. Let HT denote

the space of all functions ξ in L∞(Ω;C([0,T];C2

(ξt)t∈[0,T]is adapted. HT is equipped with the L∞-norm, namely

b(Rd,Rm)), so that ξt(ω) ∈ C2

b(Rd,Rm), and x → ξt(ω,x)

b(Rd,Rm)

b(Rd,Rm))) such that ξ =

||ξ|| = esssup

ω∈Ω

sup

t∈[0,T]

2

?

j=0

sup

x∈Rd|∇jξt(x,ω)|

where, as we have explained,

∇ξt(x,ω) =

?

∂ξi

∂xj

????

(x,ω,t)

?

i=1,···,m

j=1,···,d

etc. HT is a Banach space under || · ||.

In this section, if ζ = (ζi), where ζi∈ L2(Ω,F,P), then we define

Y (ζ)s = EP{ζ|Fs} and Z(ζ) = DB(Y (ζ)). Therefore, for any t > 0, if

ζi∈ L2(Ω,Ft,P), then (Y (ζ)s,Z(ζ)s) is the unique solution of the BSDE

dY (ζ)s= Z(ζ)s.dBs, Y (ζ)t= ζ.

It is easy to see that, if ξ ∈ HT, then Y (∇kξt) = ∇kY (ξt) and Z(∇kξt) =

∇kZ(ξt) for any t ≤ T and k = 0,1,2. This follows from the fact that the

mappings ζ → Y (ζ) and ζ → Z(ζ) are both affine.

Let ξ ∈ HT. Then, according to the non-linear Cameron-Martin formula,

for any fixed t ≤ T, we define a probability Qt,xon (Ω,Ft) by

dQt,x

dP

Ft

0

????

= exp

??t

f(Y (ξt(·,x))s).dBs−1

2

?t

0

|f|2(Y (ξt(·,x))s)ds

?

.

16

Page 17

We will omit the argument · (a sample point) and the space variable x if no

confusion may arise. Under Qt,x,˜B(ξt)s= Bs−?s

X(ξ)t = x +˜B(ξt)t

?t

According to Theorem 3.1, we want to find a fixed point ξ ∈ HT: ξ =

u0(X(ξ)). To this end we define Φ(ξ·) = u0(X(ξ)·) for any ξ ∈ HT. Then

∇Φ(ξ)t= ∇u0(X(ξ)t)∇X(ξ)t

and

0f(Y (ξt)r)dr (0 ≤ s ≤ t)

is Brownian motion up to time t. Define

= x + Bt−

0

f(Y (ξt)s)dsfor t ≤ T.

∇2Φ(ξ)t = ∇2u0(X(ξ)t)(∇X(ξ)t,∇X(ξ)t)

+∇u0(X(ξ)t)∇2X(ξ)t.

Lemma 5.1 1)For any T > 0 and ξ ∈ HT

||Φ(ξ)|| ≤ C0(1 + d)2+ C0C1T {(2d + 1) + (1 + C1T)||ξ||}||ξ||.

2) If K = 2C0(1 + d)2and

T ≤

1

2√C0C1

?

d + 1/2 + C0(1 + C1)(1 + d)2∧ 1,

then ||Φ(ξ)|| ≤ K as long as ξ ∈ HT and ||ξ|| ≤ K.

Proof. By definition for any t ≤ T and any x (but the argument x is

suppressed from the notations for simplicity, and | · |∞denotes the essential

supremum norm)

∇X(ξ)t= IRd −

?t

0

∇f(Y (ξt)s)Y (∇ξt)sds

and

∇2X(ξ)t = −

?t

?t

0

∇2f(Y (ξt)s)(Y (∇ξt)s,Y (∇ξt)s)ds

−

0

∇f(Y (ξt)s)Y (∇2ξt)sds.

17

Page 18

From these equations and the fact that the conditional expectation is a con-

traction on L∞(Ω,Ft,P), one can easily to see the following estimates:

|∇X(ξ)t| ≤ d + C1t|∇ξt|∞,

and

|∇2X(ξ)t| ≤ C1t|∇ξt|2+ C1t|∇2ξt|.

Therefore

|Φ(ξ)| ≤ C0,

|∇Φ(ξ)t| ≤ C0|∇X(ξ)t|

≤ C0{d + C1t|∇ξt|∞}

and

|∇2Φ(ξ)t| ≤ C0|∇X(ξ)t|2+ C0|∇2

≤ C0{d + C1t|∇ξt|∞}2+ C0C1t?|∇ξt|2

which yield the required estimates.

xX(ξ)t|

∞+ |∇2ξt|∞

?

Lemma 5.2 Let T > 0. There is positive constant depending only on C0,C1

and d such that

||Φ(ξ) − Φ(η)|| ≤ K0C1T(1 + T)(1 + ||ξ|| + ||η||)||ξ − η||

for any ξ,η ∈ HT.

Proof. By a simple computation, we have

|X(ξ)t− X(η)t| ≤

????

?t

0

f(Y (ξt− ηt)s)ds

≤ tC1|ξt− ηt|∞,

????

|∇X(ξ)t− ∇X(η)t| ≤

????

+

?t

????

0

(∇f(Y (ξt)s) − ∇f(Y (ηt)s))Y (∇ξt)sds

?t

?t

+C1

0

≤ C1t|ξt− ηt||∇ξt| + C1t|∇ξt− ∇ηt|,

????

0

∇f(Y (ηt)s)Y (∇(ξt− ηt))sds

????

≤ C1

0

Y (ξt− ηt)s||Y (∇ξt)s|ds

?t

|Y (∇(ξt− ηt))s|ds

18

Page 19

and, similarly

??∇2X(ξ)t− ∇2X(η)t

?? ≤ C1t|ξt− ηt||∇ξt|2

+C1t|ξt− ηt||∇2ξt| + C1t|∇2ξt− ∇2ηt|

+C1t(|∇ξt| + |∇ηt|)|∇ξt− ∇ηt|

and the estimate follows easily from these inequalities.Φ(ξ·) = u0(X(ξ)·)

|Φ(ξ)t− Φ(η)t| ≤ C0C1t|ξt− ηt|∞

|∇Φ(ξ)t− ∇Φ(η)t| = C0|X(ξ)t− X(η)t||∇X(ξ)t|

+C0|∇X(ξ)t− ∇X(η)t|

≤ C2

+C0C1t(|ξt− ηt||∇ξt| + |∇ξt− ∇ηt|)

0C1t|ξt− ηt|∞(d + C1t|∇ξt|∞)

Now we are in a position to establish a Bismut type formula (see Bismut

[1] for the linear case) for the solution of quasi-linear system (4.1).

Theorem 5.3 There is T > 0 depending only on d, C0and C2, so that there

is a unique fixed point ξ of Φ in HT. Moreover

∇ju(x,t) = EP(∇jξt(·,x)),

u(x +˜B(ξt)s,t − s) = Y (ξt)s for all s ≤ t ≤ T a.e.

j = 0,1,2.(5.1)

and

∇u(x +˜B(ξt)s,t − s) = Z(ξt)s for all s ≤ t ≤ T a.e.

Proof. According to Theorem 3.1, for t ≤ T we have u(x,t) = EP(ξt(·,x)|F0).

Taking expectation we obtain u(x,t) = EP(ξt(·,x)). Since ξt(ω,·) ∈ C2

so we may take derivatives under integration to obtain (5.1).

b(Rd,Rm)

Acknowledgements. The research was supported in part by EPSRC grant

EP/F029578/1, and by the Oxford-Man Institute.

19

Page 20

References

[1] Jean-Michel Bismut, Large deviations and the Malliavin calculus,

Progress in Mathematics, vol. 45, Birkh¨ auser Boston Inc., Boston, MA,

1984. MR 755001 (86f:58150)

[2] Philippe Briand and Ying Hu, BSDE with quadratic growth and un-

bounded terminal value, Probab. Theory Related Fields 136 (2006),

no. 4, 604–618. MR 2257138 (2007m:60187)

[3]

, Quadratic BSDEs with convex generators and unbounded ter-

minal conditions, Probab. Theory Related Fields 141 (2008), no. 3-4,

543–567. MR 2391164 (2009e:60133)

[4] Fran¸ cois Delarue, Estimates of the solutions of a system of quasi-linear

PDEs. A probabilistic scheme, S´ eminaire de Probabilit´ es XXXVII, Lec-

ture Notes in Math., vol. 1832, Springer, Berlin, 2003, pp. 290–332. MR

2053051 (2005b:60165)

[5] Mark Freidlin, Functional integration and partial differential equations,

Annals of Mathematics Studies, vol. 109, Princeton University Press,

Princeton, NJ, 1985. MR 833742 (87g:60066)

[6] I. V. Girsanov, On transforming a class of stochastic processes by abso-

lutely continuous substitution of measures, Teor. Verojatnost. i Prime-

nen. 5 (1960), 314–330. MR 0133152 (24 #A2986)

[7] Magdalena Kobylanski, Backward stochastic differential equations and

partial differential equations with quadratic growth, Ann. Probab. 28

(2000), no. 2, 558–602. MR 1782267 (2001h:60110)

[8] O. A. Ladyˇ zenskaja, V. A. Solonnikov, and N. N. Ural′ceva, Linear and

quasilinear equations of parabolic type, Translated from the Russian by

S. Smith. Translations of Mathematical Monographs, Vol. 23, American

Mathematical Society, Providence, R.I., 1967. MR 0241822 (39 #3159b)

[9] Jin Ma, Philip Protter, and Jiong Min Yong, Solving forward-backward

stochastic differential equations explicitly—a four step scheme, Probab.

Theory Related Fields 98 (1994), no. 3, 339–359. MR 1262970

(94m:60118)

20

Page 21

[10] Jin Ma and Jiongmin Yong, Forward-backward stochastic differential

equations and their applications, Lecture Notes in Mathematics, vol.

1702, Springer-Verlag, Berlin, 1999. MR 1704232 (2000k:60118)

[11] A. Majda, Compressible fluid flow and systems of conservation laws

in several space variables, Applied Mathematical Sciences, vol. 53,

Springer-Verlag, New York, 1984. MR 748308 (85e:35077)

[12] Paul Malliavin, Stochastic analysis, Grundlehren der Mathematischen

Wissenschaften [Fundamental Principles of Mathematical Sciences], vol.

313, Springer-Verlag, Berlin, 1997. MR 1450093 (99b:60073)

[13] Marie-Am´ elie Morlais, Quadratic BSDEs driven by a continuous mar-

tingale and applications to the utility maximization problem, Finance

Stoch. 13 (2009), no. 1, 121–150. MR 2465489 (2010a:91150)

[14]´E. Pardoux and S. G. Peng, Adapted solution of a backward stochastic

differential equation, Systems Control Lett. 14 (1990), no. 1, 55–61. MR

1037747 (91e:60171)

[15] Shi Ge Peng, Probabilistic interpretation for systems of quasilinear

parabolic partial differential equations, Stochastics Stochastics Rep. 37

(1991), no. 1-2, 61–74. MR 1149116 (93a:35159)

[16] Revaz Tevzadze, Solvability of backward stochastic differential equations

with quadratic growth, Stochastic Process. Appl. 118 (2008), no. 3, 503–

515. MR 2389055 (2009d:60194)

G. Liang, A. Lionnet and Z. Qian

Mathematical Institute and Oxford-Man Institute

University of Oxford

Oxford OX1 3LB, England

Email: liangg@maths.ox.ac.uk, arnaud.lionnet@maths.ox.ac.uk

and qianz@maths.ox.ac.uk

21