Representation theorems for backward doubly stochastic differential equations

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Representation theorems for backward doubly stochastic differential equations

Auguste Aman

To cite this version:

Auguste Aman. Representation theorems for backward doubly stochastic differential equations. 2008.

�hal-00196841v4�

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doubly stohasti dierential equations

Auguste Aman

∗

UFR de Mathématiques et Informatique,

22 BP 582 Abidjan 22, Cte d'Ivoire

Abstrat

In this paperwe study the lass of bakward doubly stohasti dierential equations

(BDSDEs, for short) whose terminal value depends on the history of forward diusion.

We rst establisha probabilisti representation for the spatial gradient of the stohasti

visositysolution to a quasilinear paraboli SPDE inthe spirit of the Feynman-Ka for-

mula,withoutusingthederivativesoftheoeientsoftheorrespondingBDSDE.Then

suha representation leadsto a losed-formrepresentation ofthemartingaleintegrand of

BDSDE,underonly standard Lipshitz onditionon theoeients.

Key Words: Adapted solution, antiipatingstohasti alulus, bakward doubly SDEs,

stohasti partial dierential equation,stohasti visosity solutions.

MSC:60H15; 60H20

1 Introdution

Bakward stohasti dierential equations (BSDEs, for short) were rstly been onsidered in

itlinear formby Bismut[1,2℄inthe ontextof optimalstohasti ontrol. However, nonlinear

BSDEs and theirtheory have been introduedby Pardoux and Peng [12℄. Ithas been enjoying

a great interest in the last ten year beause of its onnetion with applied elds. We an ite

stohasti ontrol and stohasti games (see [8℄) and mathematial nane (see [6℄). BSDEs

alsoprovidea probabilistiinterpretationforsolutionstoelliptiorparabolinonlinearpartial

dierential equations generalizing the lassial Feynman-Ka formula [13, 14℄. A new lass

∗

augusteaman5yahoo.fr

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onsideredby PardouxandPeng[15℄. ThisnewkindofBSDEspresenttwostohastiintegrals

driven by two independent Brownian motions

B

^and

W

ând îsôf ^the ^form

Y s = ξ +

Z T s

f (r, Y r , Z r ) dr + Z T

s

g(r, Y r , Z r ) ↓ dB r

− Z T

s

Z r dW r , s ∈ [t, T ],

^(1.1)

where

ξ

îs â ^square întegrable ^variable. ^Let ûs ^remark ^that ⁱⁿ ^the ^sens ôf ^Pardoux ^Peng, ^the

integral driven by

{B _r } _r≥0

îs â ^bakward Ît întegral ând ^the ôther ône ^driven ^by

{W _r } _r≥0

is the standard forward It integral. Further, bakward doubly SDEs seem to be suitable

giving a probabilisti representation for a system of paraboli stohasti partial dierential

equations (SPDEs, in short). We refer to Pardoux and Peng [15℄ for the link between SPDEs

andBDSDEsinthepartiularasewheresolutionsofSPDEsareregular. Thegeneralsituation

ismuh deliatetotreat beauseof diulties ofextending the notionof visosity solutionsto

SPDEs. The stohasti visosity solution for semi-linear SPDEs was introdued for the rst

timeinLionsandSouganidis[10℄. Theyusedtheso-alled"stohastiharateristi"toremove

the stohasti integrals froman SPDE.Another way ofdening astohastivisosity solution

of SPDE is via an appealto the Doss-Sussman transformation. Bukdahn and Ma [3, 4℄ were

the rst to use this approah in order to onnet the stohasti visosity solutions of SPDEs

with BDSDEs.

In this paper we onsider the approah of dening stohasti visosity solution of SPDEs

given by Bukdahn and Ma

[

³

,

⁴

]

^whih, ⁱⁿ ôur ^mind îs ^natural ând ôinide ^(if

g ≡ 0

⁾ ^with

the well-knowvisosity solution ofPDEs introduedby Crandallet

al [

⁵

]

^. ^In ^this ^fat, ^we ^will

workinthesequel ofthis paperwith theversionofbakward doubly SDEsintroduedin[3,4℄,

whih is in fat a time reversal of that onsidered by Pardoux and Peng [15℄. Indeed, for

l, f

beLipshitz ontinuous funtionsintheirspatialvariablesand

g ∈ C _b ^0,2,3 ([0, T ] ×

^I^R

^d ×

^I^R

;

^I^R

^d )

^,

they onsider a lass of bakward doubly SDEs is of this followingform:

Y _s ^t,x = l(X ₀ ^t,x ) + Z s

0 f (r, X _r ^t,x , Y _r ^t,x , Z _r ^t,x ) dr + Z s

0 g(r, X _r ^t,x , Y _r ^t,x ) dB r

− Z s

0 Z _r ^t,x ↓ dW r , s ∈ [0, t].

^(1.2)

The diusion proess

X ^t,x

îs ^the ûnique ^solutionôf ^the ^forward ^SDE

X _s ^t,x = x +

Z t s

b(r, X _r ^t,x ) dr + Z t

s

σ(r, X _r ^t,x ) ↓ dW _r s ∈ [0, t],

^(1.3)

where

b

^and

σ

^are ^some ^measurable ^funtions. ^Here ^the ^supersript

(t, x)

^indiates^the ^depen-

deneof the solutiononthe initialdate

(t, x)

^,ând ît^will^beômitted^when ^the ôntext îs^lear.

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Bukdahn and Ma proved in their two works [3, 4℄,among other things, that

u(t, x) = Y _t ^t,x

^is

a stohasti visosity solutionof nonlinear paraboliSPDE:

du(t, x) = [Lu(t, x) + f (t, x, u(t, x), (∇uσ)(t, x))] dt +g(t, x, u(t, x)) dB t , (t, x) ∈ (0, T ) ×

^I^R

^d , u(0, x) = l(x), x ∈

^I^R

^d ,

(1.4)

where

L

^dened ^by

L = 1 2

n

X

i,j=1 k

X

l=1

σ il σ lj (x)∂ _x ² _i _x _j +

n

X

j=1

b j (x)∂ x j ,

isthe innitesimaloperator generatedbythe diusion proess

X ^t,x

^. ^More ^preisely,^they ^show

thank to the Blumenthal

0

^-

1

^law ^that

u(t, x) =

^I^E

l(X ₀ ^t,x ) + Z t

0 f(r, X _r ^t,x , Y _r ^t,x , Z _r ^t,x ) ds +

Z t 0

g (r, X _r ^t,x , Y _r ^t,x ) dB r | F _t ^B

.

^(1.5)

Itiswellknowthat

u

^is ^a

F ^B

-measurableeld. However, tothebest ofour knowledge,todate there has been no disussion inthe literature onerning the path regularity of the proess

Z

when

f

^and

l

âre ônly ^Lipshitz ôntinuous, êven ⁱⁿ^the ^speial âses ^where^the ôeientsâre

enoughregular.

Our goal in this paper is twofold. First we show that if the oeients

l

^and

f

^are ^on-

tinuously dierentiable, then the visosity solution

u

^of^the ^SPDE

(1.4)

^will^have^a ^ontinuous

spatial gradient

∂ x u

^and, ^more ^important,^the ^followingprobabilisti representation holds:

∂ x u(t, x) =

^I^E

l(X ₀ ^t,x ) N ₀ ^t + Z s

0 f (r, X _r ^t,x , Y _r ^t,x , Z _r ^t,x )N _r ^s dr +

Z t 0

g(r, X _r ^t,x , Y _r ^t,x )N _r ^s dB r | F _t ^B

(1.6)

where

N _. ^s

^is^some^proess ^dened ^on

[0, s]

^,^dependingônlyôn^the^solutionsôf^the ^forward ^SDE

(1.3)

^and ^itsvariationalequationrespetively. This representation an be thought ofas a new typeofnonlinear Feynman-Kaformulaforthederivativeof

u

^,^whih ^does^not ^seem^to^existⁱⁿ

theliterature. Themainsignianeofthe formula, however, liesinthatitdoesnotdependon

the derivativesof the oeients of the bakward doubly BSDE

(1.2)

^. ^Beause ^of ^this ^speial

feature, we an then derive arepresentation

Z _s ^t,x = ∂ x u(t, X _s ^t,x )σ(t, X _s ^t,x ), s ∈ [0, t],

^(1.7)

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underonlyaLipshitzonditionon

l

^and

f

^. ^This^latterrepresentationthenenablesustoprove the path regularity of the proess

Z

^, ^the ^seond ^goal ôf ^this ^paper, êven ⁱⁿ ^the âse ^where

the terminal value of

Y ^t,x

^is ^of ^the ^form

l(X t 0 , ..., X t n )

^, ^where

π : 0 = t 0 < .. < t n = t

^is ^any

partitionof

[0, t]

^, â^result ^that ^does ^not ^seem ^to^beâmendable ^by âny êxisting ^method.

Letus reall that this two representations have already be given by Maand Zhang [11℄ in

the ase of a probabilistirepresentation for solutionsof PDEs via BSDEs. Consequently our

approah isbeinspired by their works. However, there are partiularities: rst, the derivative

notionistakeinthe owsense (independentof

ω 1

⁾^beause

u

^,^the ^stohasti ^visosity ^solution

oftheSPDE

(1.4)

^,îsâ^randomêld. ^Seondly^,^the ^proof^to^theôntinuityôf^therepresentation of the proess

Z

^need, ^sine

F ^t

s = (F _s ^B ⊗ F _s,t ^W ) _0≤s≤t

^is ^not ^a ^ltration,

G ^t

s = (F _s ^B ⊗ F _0,t ^W ) _0≤s≤t

whihis altration.

The rest of this paper is organized as follows. In setion 2 we give all the neessary pre-

liminaries. In setion 3 we establish the new Feynman-Ka formula between oupled forward

bakward doubly SDE

(1.1)

^-

(1.3)

^and ^the ^SPDE

(1.4)

^, ^under ^the

C ¹

-assumption of the oef- ients. The setion

4

^is ^devoted ^to ^give ^the ^main representation theorem assuming only the Lipshitz ondition of the oeients

l

^and

f

^. ^In ^setion

5

^we ^study ^the ^path ^regularity ^of^the

proess

Z

^.

2 Preliminaries

Let

T > 0

^a ^xed ^time ^horizon. ^Throughout ^this ^paper

{W _t , 0 ≤ t ≤ T }

^and

{B _t , 0 ≤ t ≤ T }

^will^denote ^two independent

d

-dimensionalBrownian motionsdened onthe omplete probabilityspaes

(Ω 1 , F ₁ ,

^I^P

1 )

^and

(Ω 2 , F ₂ ,

^I^P

2 )

respetively. Forany proess

{U _s , 0 ≤ s ≤ T }

dened on

(Ω i , F _i ,

^I^P

i ) (i = 1, 2)

^,^we ^write

F _s,t ^U = σ(U r − U s , s ≤ r ≤ t)

^and

F _t ^U = F _0,t ^U

^. ^Unless

otherwise speied we onsider

Ω = Ω 1 × Ω 2 , F = F ₁ ⊗ F ₂

^and ^I^P

=

^I^P

1 ⊗

^I^P

₂ .

In addition,we put for eah

t ∈ [0, T ]

^,

F = {F s = F _s ^B ⊗ F _s,T ^W ∨ N , 0 ≤ s ≤ T }

where

N

îs ^the ôlletion ôf Î^{P -null} ^sets. În ôther ^words, ^the ôlletion

F

is IP -omplete but is neither inreasing nor dereasing so that, it is not a ltration. Let us tellalso that random

variables

ξ(ω 1 ), ω 1 ∈ Ω 1

^and

ζ(ω 2 ), ω 2 ∈ Ω 2

âre ônsidered âs ^random ^variables ôn

Ω

^via ^the

followingidentiation:

ξ(ω 1 , ω 2 ) = ξ(ω 1 ); ζ(ω 1 , ω 2 ) = ζ(ω 2 ).

Let

E

^denote â ^generi Êulidean ^spae; ând ^regardless ôf îts ^dimension ^we ^denote

h; i

^to ^be

the innerprodut and

|.|

^the^normⁱⁿ

E

^. îf ânôtherÊulidean ^spaesâre^needed, ^we^shall^label

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them as

E 1 ; E 2 , ., .,

^et. ^Furthermore, we use the notation

∂ t = _∂t ^∂ , ∂ x = ( _∂x ^∂ ₁ , _∂x ^∂ ₂ , .., _∂x ^∂

d )

^and

∂ ² = ∂ _xx = (∂ _x ² _i _x _j ) ^d _i,j=1

^, ^for

(t, x) ∈ [0, T ] ×

^I^R

^d

^. ^Note ^that ^if

ψ = (ψ ¹ , .., ψ ^d ) :

^I^R

^d →

^I^R

^d

^, ^then

∂ _x ψ , (∂x _j ψ ⁱ ) ^d _i,j=1

îs â^matrix. ^The ^meaningôf

∂ _xy , ∂ _yy

^,^et. ^should ^be^lear ^from^the ^ontext.

The following spaes willbe used frequently in the sequel (let

X

^denote^a ^generi ^Banah

spae):

1. For

t ∈ [0, T ], L ⁰ ([0, t]; X )

îs ^the ^spae ôf âll^measurable ^funtions

ϕ : [0, t] 7→ X

^.

2. For

0 ≤ t ≤ T, C([0, t]; X )

îs ^the ^spae ôf âllôntinuous ^funtions

ϕ : [0, t] 7→ X

^; ^further,

for any

p > 0

^we ^denote

|ϕ| ^∗,p _0,t = sup

0≤s≤t

kϕ(s)k ^p _X

^when ^the ^ontext^is ^lear.

3. Forany

k, n ≥ 0, C ^k,n ([0, T ]×E; E ₁ )

îs^the^spaeôfâll

E ₁

^-valued^funtions

ϕ(t, e), (t, e) ∈ [0, T ] × E

^, ^suh^that ^they ^are

k

^-times ontinuously dierentiablein

t

^and

n

^-times ^ontin-

uously dierentiablein

e

^.

4.

C _b ¹ ([0, T ] × E; E 1 )

^is ^the ^spae ^of ^those

ϕ ∈ C ¹ ([0, T ] × E; E 1 )

^suh ^that ^all ^the ^partial

derivativesare uniformlybounded.

5.

W ^1,∞ (E, E 1 )

îs ^the ^spae ôf âll ^measurable ^funtions

ψ : E 7→ E 1

^, ^suh ^that ^for ^some

onstant

K > 0

^it ^holds ^that

|ψ(x) − ψ(y)| _E ₁ ≤ K |x − y| _E , ∀x, y ∈ E

^.

6. Forany sub-

σ

^-eld

G ⊆ F _T ^B

^and

0 ≤ p < ∞, L ^p (G; E)

^denote ^all

E

^-valued

G

-measurable random variable

ξ

^suh ^that ^I^E

|ξ| ^p < ∞

^. ^Moreover,

ξ ∈ L ^∞ (G; E)

^means ^it ^is

G

^-

measurable and bounded.

7. For

0 ≤ p < ∞, L ^p ( F , [0, T ]; X )

îs ^the ^spae ôf âll

X

^-valued,

F

-adapted proesses

ξ

satisfying IE

Z T 0

kξ _t k ^p _X dt

< ∞

^;^and^also,

ξ ∈ L ^∞ ( F , [0, T ];

^I^R

^d )

^means^that^the^proess

ξ

^is ^uniformlyessentially bounded in

(t, ω)

^.

8.

C( F , [0, T ] × E; E ₁ )

^is^the^spae ^of

E ₁

^-valued,^ontinuous^random^eld

ϕ : Ω × [0, T ] × E

^,

suh thatfor xed

e ∈ E

^,

ϕ(., ., e)

^is ^an

F

-adapted proess.

Tosimplifynotationweoftenwrite

C([0, T ]×E ; E 1 ) = C ^0,0 ([0, T ]×E ; E 1 )

^;^and^if

E 1 =

^I^R,^then

weoftensuppress

E 1

^for^simpliity^(e.g.,

C ^k,n ([0, T ] ×E;

^I^R

) = C ^k,n ([0, T ]× E), C ^k,n ( F , [0, T ]×

E;

^I^R

) = C ^k,n ( F , [0, T ] × E), ...,

êt.). ^Finally, ûnless ôtherwise ^speied ^(suh âs ^proess

Z

mentionedin Setion1), allvetors inthe paperwill be regarded asolumn vetors.

Throughoutthis paper we shall make use of the followingstanding assumptions:

( A1 )

^The ^funtions

σ ∈ C _b ^0,1 ([0, T ] ×

^I^R

^d ;

^I^R

^d×d ), b ∈ C _b ^0,1 ([0, T ] ×

^I^R

^d ;

^I^R

^d )

^;^and ^all^the ^partial

derivativesof

b

^and

σ

^(with ^respet ^to

x

⁾ âre ûniformly^bounded ^by â ômmonônstant

K > 0

^. ^Fûrther, ^there êxists ônstant

c > 0

^, ^suh ^that

ξ ^T σ(t, x)σ(t, x) ^T ξ ≥ c|ξ| ² , ∀x, ξ ∈

^I^R

^d , t ∈ [0, T ].

^(2.1)

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( A2 )

^The ^funtion

f ∈ C(F ^B , [0, T ] ×

^I^R

^d ×

^I^R

×

^I^R

^d ) ∩ W ^1,∞ ([0, T ] ×

^I^R

^d ×

^I^R

×

^I^R

^d )

^and

l ∈ W ^1,∞ (

^I^R

^d )

^. ^Furthermore,we denote the Lipshitz onstants of

f

^and

l

^by ^a^ommon

one

K > 0

^asⁱⁿ

( A1 )

^;^and ^we^assume ^that

sup

0≤t≤T

{|b(t, 0)| + |σ(t, 0)| + |f (t, 0, 0, 0)| + |g(0)|} ≤ K.

^(2.2)

( A3 )

^The ^funtion

g ∈ C _b ^0,2,3 ([0, T ] ×

^I^R

^d ×

^I^R

;

^I^R

^d )

The following results are either standard or slight variations of the well-know results in

SDE and bakward doubly SDE literature;wegiveonly the statement forready referene.

Lemma 2.1 Suppose that

b ∈ C( F , [0, T ] ×

^I^R

^d ;

^I^R

^d ) ∩ L ⁰ ( F , [0, T ]; W ^1,∞ (R ^d ;

^I^R

^d )),

σ ∈ C( F , [0, T ] ×

^I^R

^d ;

^I^R

^d×d ) ∩ L ⁰ ( F , [0, T ]; W ^1,∞ (R ^d ;

^I^R

^d×d ))

^, ^withâômmon^Lipshitzônstant

K > 0

^. ^Suppose ^also ^that

b(t, 0) ∈ L ² ( F , [0, T ];

^I^R

^d )

^and

σ(t, 0) ∈ L ² ( F , [0, T ];

^I^R

^d×d )

^. ^Let

X

^be

the unique solution of the followingforward SDE

X s = x + Z t

s

b(r, X r ) dr + Z t

s

σ(r, X r ) dW r .

^(2.3)

Then for any

p ≥ 2

^, ^there êxists â ônstant

C > 0

^depending ^only ^on

p, T

^and

K

^, ^suh ^that

E(|X| ^∗,p _0,t ) ≤ C

|x| ^p +

^I^E

Z T

0 [|b(s, 0)| ^p + |σ(s, 0)| ^p ] ds

(2.4)

Lemma 2.2 Assume

f ∈ C( F , [0, T ] ×

^I^R

×

^I^R

^d ) ∩ L ⁰ ( F , [0, T ]; W ^1,∞ (

^I^R

× R ^d ))

^, ^with^a ^uniform

Lipshitz onstant

K > 0

^, ^suh ^that

f (s, 0, 0) ∈ L ² ( F , [0, T ])

^and

g ∈ C( F , [0, T ] ×

^I^R

×

IR

d ;

^I^R

^d ) ∩ L ⁰ ( F , [0, T ]; W ^1,∞ (

^I^R

× R ^d ;

^I^R

^l ))

^with â ômmon ûniform ^Lipshitz ônstant

K > 0

with respet the rst variable and the Lipshitz onstant

0 < α < 1

^whih ^respet ^the ^seond

variableandsuhthat

g(s, 0, 0) ∈ L ² ( F , [0, T ])

^. ^For^any

ξ ∈ L ² (F ₀ ;

^I^R

)

^,^let

(Y, Z )

^be^the^adapted

solution to the BDSDE:

Y s = ξ + Z s

0 f(r, Y r , Z r ) dr + Z s

0 g(r, Y r , Z r ) dB r − Z s

0 Z r ↓ dW r .

^(2.5)

Thenthere existsa onstant

C > 0

T

ând ôn ^the^Lipshitz ônstants

K

^and

α

^, ^suh ^that

IE

Z T 0

|Z s | ² ds ≤ C

^I^E

|ξ| ² + Z T

0 [|f(s, 0, 0)| ² + |g(s, 0, 0)| ² ] ds

.

^(2.6)

Moreover, for all

p ≥ 2

^, ^there êxistsâ ônstant

C p > 0

^, ^suh ^that

IE

(|Y | ^∗,p _0,t ) ≤ C p

^I^E

|ξ| ^p + Z T

0 [|f(s, 0, 0)| ^p + |g(s, 0, 0)| ^p ] ds

(2.7)

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tothebakward doublySDEs(see Pardoux-Peng

[

¹⁵

]

^). ^F^or^any^random^variables

ξ

^of^the^form

ξ = F

Z T 0

ϕ 1 dW t , .., Z T

0 ϕ n dW s ; Z T

0 ψ 1 dB s , ..., Z T

0 ψ p dB s

with

F ∈ C _b ^∞ (

^I^R

^n+p ), ϕ 1 , ..., ϕ n ∈ L ² ([0, T ],

^I^R

^d ), ψ 1 , ..., ψ n ∈ L ² ([0, T ],

^I^R

^d ),

^we ^let

D _t ξ =

n

X

i=

∂F

∂x i

Z T 0

ϕ ₁ dW _t , .., Z T

0 ϕ _n dW _s ; Z T

0 ψ ₁ dB _s , ..., Z T

0 ψ _p dB _s

ϕ _i (t).

For suh a

ξ

^, ^we ^dene^its

1, 2

^-norm ^as:

kξk ² _1,2 =

^I^E

|ξ| ² +

^I^E

Z T

0 |D r ξ| ² dr

.

S

^denoting ^the ^set ôf ^random^variable ôf ^the âbove ^form,^we ^dene ^the ^Sobolev^spae

ID

1,2 , S ^k.k ^1,2 .

The "derivation operator"

D .

êxtends âs ânôperator^from Î^D

1,2

into

L ² (Ω, L ² ([0, T ],

^I^R

^d ))

^.

We shall apply the previous antiipativealulusto the oupledforward bakward doubly

SDEs

(1.3)

^-

(1.2)

^. În ^this ^fat, ^letûs ônsider ^the ^following variational equation that will play a important role inthis paper: for

i = 1, .., d

^,

∇ i X _s ^t,x = e i + Z t

s

∂ x b(r, X _r ^t,x )∇ i X _r ^t,x dr +

d

X

j=1

Z t s

∂ x σ ^j (r, X _r ^t,x )∇ i X _r ^t,x ↓ dW _r ^j ,

∇ _i Y _s ^t,x = ∂ x l(X ₀ ^t,x )∇ _i X ₀ ^t,x +

Z s 0

[∂ x f(r, Ξ ^t,x (r))∇ i X _r ^t,x + ∂ y f (r, Ξ ^t,x (r))∇ i Y _r ^t,x + h∂ z f (r, Ξ ^t,x (r)), ∇ i Z _r ^t,x i]dr +

Z s 0

[∂ x g(r, Θ ^t,x (r))∇ i X _r ^t,x + ∂ y g(r, Θ ^t,x (r))∇ i Y _r ^t,x ]dB r − Z s

0 ∇ i Z _r ^t,x ↓ dW r ,

^(2.8)

where

e i = (0, ..., 1, ..., ⁱ 0) ^T ∈

^I^R

^d , Ξ ^t,x = (Θ ^t,x , Z ^t,x ), Θ ^t,x = (X ^t,x , Y ^t,x )

^and

σ ^j (.)

^is ^the

j

^-th

olumnofthe matrix

σ(.)

^. ^We^reall^again^that ^the supersription

t,x

indiatesthe dependene

of thesolutiononthe initialdate

(t, x)

^,ând ^will^beômitted^when^the ôntextîs^lear. ^Wê âlso

remark that under the above assumptions,

∇X ^t,x , ∇Y ^t,x , ∇Z ^t,x

∈ L ² ( F ; C([0, T ];

^I^R

^d×d ) × C([0, T ];

^I^R

^d ) × L ² ([0, T ];

^I^R

^d×d )).

Further the

d × d

-matrix-valued proess

∇X ^t,x

^satises ^a ^linear ^SDE ^and

∇X _t ^t,x = I

^, ^so ^that

[∇X _s ^t,x ] ⁻¹

^exists ^for

s ∈ [0, t],

^I^P-a.s. ^and ^we^have ^the ^following:

(9)

Lemma 2.3 Assume that

( A1 )

^holds; ^and ^supp^ose ^that

f ∈ C _b ^0,1 ([0, T ] ×

^I^R

^2d+1 )

^and

g ∈ C _b ^0,2,3 ([0, T ] ×

^I^R

^d+1 ;

^I^R

^d )

^. ^Then

(X, Y, Z) ∈ L ² ([0, T ];

^I^D

^1,2 (

^I^R

^2d+1 ))

^, ând ^there êxists â

version of

(D s X r , D s Y r , D s Z r )

^that ^satises







D s X r = ∇X _r (∇X _s ) ⁻¹ σ(s, X s ) 1 _{s≤r} , D _s Y _r = ∇Y _r (∇X _s ) ⁻¹ σ(s, X _s ) 1 _{s≤r} , D s Z r = ∇Z r (∇X s ) ⁻¹ σ(s, X s ) 1 _{s≤r} ,

0 ≤ s, r ≤ t.

^(2.9)

Lemma 2.4 Suppose that

F ∈

^I^D

^1,2

^. ^Then

(i)

(Integration byparts formula): for any

u ∈ Dom(δ)

^suh ^that

F u ∈ L ² ([0, T ] × Ω;

^I^R

^d )

^, ^one

has

F u ∈ Dom(δ)

^, ^and ^it ^holds ^that

Z T

0 hF u t , dW t i = δ(F u) = F Z T

0 hu t , dW t i − Z T

0 D t F u t dt;

(ii)

(Clark-Hausman-Oone formula):

F =

^I^E

(F ) + Z T

0

IE

{D _t F | F _t }dW _t .

3 Relations to stohasti PDE revisited

Inthissetionweprovetherelation(1.7)betweentheforwardbakward doublySDE

(1.2)

^-

(1.3)

andthequasi-linearSPDE

(1.4)

^,ûnder^theôndition^that^theôeientsâreônlyontinuously dierentiable. Indeed,sine Bukdahn andMa

[

³

,

⁴

]

^provide^that, ^if

f

^and

l

^are ^only^Lipshitz

ontinuous, the quantity

u(t, x) = Y _t ^t,x

îs â ^stohasti ^visosity ^solution ôf ^the quasi-linear SPDE

(1.4)

^, ^relation ⁱⁿ ^(1.7) ^beomes questionable. Our objetive is to ll this gap in the literature and toextend the results ofMa and Zhang

[

¹¹

]

^given ⁱⁿ^the ^ase ^of ^the probabilisti interpretation of PDEs viathe BSDEs.

Theorem 3.1 Assume

( A1 )

^and

( A3 )

^and ^supp^ose ^that

f ∈ C _b ^0,1 ([0, T ] ×

^I^R

^d ×

^I^R

×

^I^R

^d )

^and

l ∈ C _b ¹ (

^I^R

^d )

^. ^Let

(X ^t,x , Y ^t,x , Z ^t,x )

^be ^the ^adapted ^solution ^to ^the ^FBDSDE

(1.2)

^-

(1.3)

^, ^and ^set

u(t, x) = Y _t ^t,x

^the ^stohasti ^visosity ^of ^SPDE

(1.4)

^. ^Then,

(i) ∂ _x u(t, x)

^exists ^for ^all

(t, x) ∈ [0, T ] ×

^I^R

^d

^; ^and ^for ^eah

(t, x)

^and ^i=1,...,d, ^the ^following

representation holds:

∂ x i u(t, x) =

^I^E

∂ x l(X ₀ ^t,x )∇ i X ₀ ^t,x +

Z t 0

[∂ x f (r, Ξ ^t,x (r))∇ _i X _r ^t,x + ∂ y f (r, Ξ ^t,x (r))∇ _i Y _r ^t,x + ∂ z f(r, Ξ ^t,x (r))∇ _i Z _r ^t,x ]dr +

Z t 0

[∂ x g (r, Θ ^t,x (r))∇ _i X _r ^t,x + ∂ y g(r, Θ ^t,x (r))∇ _i Y _r ^t,x ]dB r | F _t ^B

(3.1)

(10)

where

Θ ^t,x = (X ^t,x , Y ^t,x ), Ξ ^t,x = (Θ ^t,x , Z ^t,x )

^, ^and

(∇X ^t,x , ∇Y ^t,x , ∇Z ^t,x )

^the ^unique ^solution ^of

equation

(2.8)

^;

(ii) ∂ x u(t, x)

îs ôntinuous ôn

[0, T ] ×

^I^R

^d

^;

(iii) Z _s ^t,x = ∂ x u(s, X _s ^t,x )σ(s, X _s ^t,x ), ∀ s ∈ [0, t],

^I^P-a.s.

Proof. Forthesimple presentationwetake

d = 1

^. ^The^higherdimensionalasean betreated inthesamewaywithoutsubstantialdiulty. Weusethesimplernotations

l _x , (f _x , f _y , f _z ), (g _x , g _y , g _z )

respetively for the partialderivatives of

l, f

^and

g

^.

The proof is inspired by the approah of Ma and Zhang

[

¹¹

]

^(see ^Theorem

3.1

^). Nevertheless, thereexists slightdierenedue inthefatthatthe solutionofSPDE'sisarandomeld; more

preisely willshow thatit is aonditional expetation with respet the ltration

(F _t ^B ) 0≤t≤T

^.

We rst prove

(i)

^. ^Let

(t, x) ∈ [0, T ] ×

^I^R ^be ^xed. ^F^or

h 6= 0

^,^we^dene:

∇X _s ^h = X _s ^t,x+h − X _s ^t,x

h ; ∇Y _s ^h = Y _s ^t,x+h − Y _s ^t,x

h ; ∇Z _s ^h = Z _s ^t,x+h − Z _s ^t,x

h s ∈ [0, t].

It follows analogouslyof the proof of Theorem 2.1in [15℄) that

IE

{|∆Y ^h | ^∗,2 _0,t =

^I^E

{|∇Y ^h − ∇Y ^t,x | ^∗,2 _0,t } → 0 as h → 0.

^(3.2)

We know also that proesses

Y ^t,x , Y ^t,x+h , ∇Y ^h

^and

∆Y ^h

âre âll âdapted ^to ^the

σ

^-algebra

F ^t = (F _s ^t ) _0≤s≤t

^, ^where

F _s ^t = F _s ^B ⊗ F _s,t ^W

^. ^In ^partiular, ^sine

W

îs â ^Brownian ^motion ôn

(Ω ₂ , F ₂ ,

^I^P

₂ )

^, ^applying ^the ^Blumenthal

0

^-

1

^law ^(see, ^e.g, ^[9℄),

Y _t ^t,x = u(t, x), Y _t ^t,x+h = u(t, x + h), ∇Y _t ^h = _h ¹ [u(t, x + h) − u(t, x)]

^and

∆Y _t ^h

^are ^allindependent of (ora onstant with respet to)

ω 2 ∈ Ω 2

^. ^Therefore ^we ^onlude ^from ^the ^above ^that

∂ x u

êxist, âs ^the ^random êld ând

∂ _x u(t, x) = ∇Y _t ^t,x

^, ^for ^all

(t, x)

^. ^Finally, ^taking ^the ônditional êxpetation ôn ^the ^both ^sides

of (2.8) at

s = t

^,^the representation (3.1) hold and nish the prove of

(i)

^.

We nowprove(ii). Let

(t _i , x _i ) ∈ [0, T ] ×

^I^R

, i = 1, 2

^. ^Knowing ^that

t ₁

^and

t ₂

^played ^inverse

rolesoneanother, weassumewithoutlosingageneralitythat

t 1 < t 2

^. ^Sine

∂ x u

îsâônditional

expetation with respet the ltration

(F _s ^B ) 0≤s≤t

^,^we ^have

|∂ x u(t 1 , x 1 ) − ∂ x u(t 2 , x 2 )| ≤

^I^E

{A(t 1 , x 1 ) − A(t 2 , x 2 ) | F _t ^B ₁ } +

^I^E

{A(t 2 , x 2 ) | F _t ^B ₁ } −

^I^E

{A(t 2 , x 2 ) | F _t ^B ₂ }

,

^(3.3)

where

A(t, x) = l x (X ₀ ^t,x )∇X ₀ ^t,x +

Z t 0

[f _x (r, Ξ ^t,x (r))∇X _r ^t,x + f _y (r, Ξ ^t,x (r))∇Y _r ^t,x + f _z (r, Ξ ^t,x (r))∇Z _r ^t,x ] dr +

Z t 0

[g x (r, Θ ^t,x (r))∇X _r ^t,x + g y (r, Θ ^t,x (r))∇Y _r ^t,x ]dB r .

^(3.4)

(11)

Thanks tothe quasi-left-ontinuity of

(F _s ^B ) 0≤s≤t

^, ^we ^see ^that

t lim 1 ↓t 2

^I^E

(A(t 2 , x 2 ) | F _t ^B ₁ ) −

^I^E

(A(t 2 , x 2 ) | F _t ^B ₂ )

= 0,

^(3.5)

independently of

x ₂

^. ^In ^virtue ^of

(3.3)

^and

(3.5)

^),^to ^prove

(ii)

^it^remain^to ^show^that

t 1 ↓t lim 2 x 1 →x 2

IE

{A(t 1 , x 1 ) − A(t 2 , x 2 ) | F _t ^B ₁ } = 0.

^(3.6)

Tothis end,sine

A(t, x)

îsâ^stohasti ^proess ândⁱⁿ^virtue ôfKolmogorov-CentsovTheorem (see [9℄),it sues toshowthat

IE

|A(t ₁ , x 1 ) − A(t 2 , x 2 )| ²

≤ C(|t ₁ − t 2 | ² + |x ₁ − x 2 | ² ),

what we donow. Reallingthe denition of

A(t i , x i ), i = 1, 2

^and ^denoting

G ^t,x (r) = f x (r, Ξ ^t,x (r))∇X _r ^t,x + f y (r, Ξ ^t,x (r))∇Y _r ^t,x + f z (r, Ξ ^t,x (r))∇Z _r ^t,x

and

H ^t,x (r) = g x (r, Θ ^t,x (r))∇X _r ^t,x + g y (r, Θ ^t,x (r))∇Y _r ^t,x ,

we get

|A(t ₁ , x 1 ) − A(t 2 , x 2 )| ≤ |l _x (X ₀ ^t ¹ ^,x ¹ )∇X ₀ ^t ¹ ^,x ¹ − l x (X ₀ ^t ² ^,x ² )∇X ₀ ^t ² ^,x ² | +

Z t 1

0 |G ^t ¹ ^,x ¹ (r) − G ^t ² ^,x ² (r)|dr +

Z t 1

0 (H ^t ¹ ^,x ¹ (r) − H ^t ² ^,x ² (r))dB r

+

Z t 2

t 1

|G ^t ² ^,x ² (r)|dr +

Z t 2

t 1

H ^t ² ^,x ² (r)dB r

.

Taking the expetation, it follows by Hölder's and Burkölder-Gundy Davis inequalitiesthat

IE

|A(t ₁ , x ₁ ) − A(t ₂ , x ₂ )| ²

≤ C

^I^E

|l _x (X ₀ ^t ¹ ^,x ¹ )∇X ₀ ^t ¹ ^,x ¹ − l _x (X ₀ ^t ² ^,x ² )∇X ₀ ^t ² ^,x ² | ² +

Z t 1

0 |G ^t ¹ ^,x ¹ (r) − G ^t ² ^,x ² (r)| ² dr +

Z t 1

0 (H ^t ¹ ^,x ¹ (r) − H ^t ² ^,x ² (r))dB r

2 + Z t 2

t 1

|G ^t ² ^,x ² (r)| ² dr +

Z t 2

t 1

H ^t ² ^,x ² (r)dB r

2 )

≤ C

^I^E

|l _x (X ₀ ^t ¹ ^,x ¹ )∇X ₀ ^t ¹ ^,x ¹ − l _x (X ₀ ^t ² ^,x ² )∇X ₀ ^t ² ^,x ² | ² +

Z t 1

0 |G ^t ¹ ^,x ¹ (r) − G ^t ² ^,x ² (r)| ² dr + Z t 1

0 |H ^t ¹ ^,x ¹ (r) − H ^t ² ^,x ² (r)| ² dr + (t 2 − t 1 )

Z t 2

t 1

|G ^t ² ^,x ² (r)| ² dr + (t 2 − t 1 ) Z t 2

t 1

|H ^t ² ^,x ² (r)| ² dr

.

Bysimilarstandard omputationsinMaand Zhang[11℄ (see proof ofTheorem 3.1), weobtain

IE

|A(t ₁ , x 1 ) − A(t 2 , x 2 )| ²

≤ C(|t ₂ − t 1 | ² + |x ₂ − x 1 | ² )

(12)

that providethe proof of

(ii)

^.

It remainstoprove

(iii)

^. ^Fôrâôntinuous ^funtion

ϕ

^,^let^us^onsider

{ϕ ^ε } _ε>0

^a^family^of

C ^0,∞

funtionsthatonvergesto

ϕ

^uniformly^. ^Sine

b, σ, l, f

âreâllûniformly^Lipshitzôntinuous,

wemayassumethattherstorderpartialderivativesof

b ^ε , σ ^ε , l ^ε , f ^ε

âreâllûniformly^bounded,

by the orresponding Lipshitz onstants of

b, σ, l, f

^uniformlyⁱⁿ

ε > 0

^. ^Now ^we ^onsider ^the

family of FBDSDEs parameterized by

ε > 0

^:







X _s ^t,x = x + R t

s b ^ε (r, X _r ^t,x )dr + R t

s σ ^ε (r, X _r ^t,x ) ↓ dW _r ; Y _s ^t,x = l ^ε (X ₀ ^t,x ) + R s

0 f ^ε (r, X _r ^t,x , Y _r ^t,x , Z _r ^t,x )dr + R s

0 g(r, X _r ^t,x , Y _r ^t,x )dB _r − R s

0 Z _r ^t,x ↓ dW _r

(3.7)

and denote it solution by

(X ^t,x (ε), Y ^t,x (ε), Z ^t,x (ε))

^. ^W^e ^dene

u ^ε (t, x) = Y _t ^t,x (ε)

^. ^Theorem

3.2

of

[

¹⁵

]

^provide ^that

u ^ε

^is ^the ^lassial^solution ^of ^stohasti ^PDE

du ^ε (t, x) = [L ^ε u(t, x) + f ^ε (t, x, u ^ε (t, x), (∇u ^ε σ ^ε )(t, x))] dt +g(t, x, u ^ε (t, x)) dB t , (t, x) ∈ (0, T ) ×

^I^R

^d , u ^ε (0, x) = l ^ε (x), x ∈

^I^R

^d .

(3.8)

For any

{x ^ε } ⊂

^I^R

ⁿ

^suh ^that

x ^ε → x

^as

ε → 0

^, ^dene

(X ^ε , Y ^ε , Z ^ε ) = (X ^t,x ^ε (ε), Y ^t,x ^ε (ε), Z ^t,x ^ε (ε))

^. ^Thenîtîs^well^knowâording^the^workôf^Pardoux

and Peng [15℄ that

Y _s ^ε = u ^ε (s, X _s ^ε ); Z _s ^ε = ∂ x u ^ε (s, X _s ^ε )σ ^ε (s, X _s ^ε ), ∀ s ∈ [0, t],

^I^P

− a.s.

^(3.9)

Now by Lemma

2.1

^and ^Lemma

2.2

^,^for ^all

p ≥ 2

^it^hold ^that

IE

|X ^ε − X| ^∗,p _0,t + |Y ^ε − Y | ^∗,p _0,t + Z t

0 |Z _s ^ε − Z s | ² ds

→ 0

^(3.10)

as

ε → 0

^. ^Moreover ^let ^us ^reall

(∇X ^ε , ∇Y ^ε , ∇Z ^ε )

^the ^unique ^solution ^of ^the variational equation of

(3.7)

^. ^Using ^again ^Lemma

2.1

^and ^Lemma

2.2

^we^get

IE

|∇X ^ε − ∇X| ^∗,p _0,t + |∇Y ^ε − ∇Y | ^∗,p _0,t + Z t

0 |∇Z _s ^ε − ∇Z _s | ² ds

→ 0,

^(3.11)

as

ε → 0

^. ^Thus ^it^is ^readily ^seen ^that

IE

{l ^ε _x (X ₀ ^ε )∇X ₀ ^ε |F _t ^B } →

^I^E

{l _x (X 0 )∇X ₀ |F _t ^B },

IP -a.s., as

ε → 0

^. ^Furthermore,by the analoguestep used in [11℄, one an show that IE

Z t 0

[f _x ^ε (r)∇X _r ^ε + f _y ^ε (r)∇Y _r ^ε + f _z ^ε (r)∇Z _r ^ε ]dr + Z t

0 [g x (r)∇X _r ^ε + g y (r)∇Y _r ^ε ]dB r |F _t ^B

(13)

IE

Z t 0

[f x (r)∇X r + f y (r)∇Y r + f z (r)∇Z r ]dr + Z t

0 [g x (r)∇X r + g y (r)∇Y r ]dB r |F _t ^B

IP -a.s., as

ε → 0

^. ^Therefore, ^we ^get

∂ x u ^ε (t, x ^ε ) → ∂ x u(t, x), as ε → 0

^I^P

− a.s.,

for eahxed

(t, x) ∈ [0, T ] ×

^I^{R .} Consequently, possibly alonga subsequene, we obtain

Z _s ^ε = lim

ε→0 ∂u ^ε (s, X _s ^ε )σ ^ε (s, X ^ε ) = ∂u(s, X s )σ(s, X s ), ds × d

^I^P

− a.e.

Sine forIP

− a.e. ω, ∂ x u(., .)

^and

X

âre ^both ôntinuous, ^theâbove êqualitiesâtually ^holds

for all

s ∈ [0, t]

^, ^I^{P -a.s.,} ^proving

(iii)

^and ^end ^the ^proof.

The following orollary is the diret onsequene of the Theorem

3.1

^. ^The ^onvention ^on

the generi onstant

C > 0

^still^true.

Corollary 3.2 AssumethatthesameonditionsasinTheorem

3.1

^hold,^and^let

(X ^t,x , Y ^t,x , Z ^t,x )

be the solution of FBDSDE

(1.2)

^-

(1.3)

^. ^Then, ^there êxistsâ ônstant

C > 0

K, T,

^and ^for ^any

p ≥ 1,

^a ^positive

L ^p (Ω, (F _s ^t ) 0≤s≤t ,

^I^P

)

^-proess

Γ ^t,x

^, ^suh ^that

|∂ _x u(t, x)| ≤ CΓ ^t,x _t , ∀ (t, x) ∈ [0, T ] ×

^I^R

^d ,

^I^P

− a.s.

^(3.12)

Consequently, one has

|Z _s ^t,x | ≤ CΓ ^t,x _s (1 + |X _s ^t,x |), ∀s ∈ [0, t],

^I^P

− a.s.

^(3.13)

Furthermore,

∀ p > 1

C p > 0

^, ^depending ^on

K, T

^, ^and

p

^suh ^that

IE

|X ^t,x | ^∗,p _0,t + |Y ^t,x | ^∗,p _0,t + |Z ^t,x | ^∗,p _0,t ≤ C p (1 + |x| ^p ).

^(3.14)

Proof. We assume rst that

p ≥ 2

^. ^The ^ase

1 < p < 2

^then ^follows ^easily^from ^Hölder

inequality. ByLemma

2.1

^and ^Lemma

2.2

^,^we ^an ^nd ^onstant

C > 0

^suh ^that

IE

(

|∇X ^t,x | ^∗,p _0,t + |∇Y ^t,x | ^∗,p _0,t + Z T

0 |∇Z _r ^t,x | ² dr ^p/2 )

≤ C.

Then, from the identity

(3.1)

^, ^we ^dedue immediatelythat

|∂ _x u(t, x)| ≤ CΓ ^t,x _t

^, ^for ^all

(t, x) ∈ [0, T ] ×

^I^R,^where

Γ ^t,x _s =

^I^E

|∇X ₀ ^t,x | + Z s

0 [|∇X _r ^t,x | + |∇Y _r ^t,x | + |∇Z _r ^t,x |]dr +

Z s 0

[∇X _r ^t,x + ∇Y _r ^t,x ]dB r

| F _s ^t

.

(14)

Moreover we get for

s ∈ [0, t],

^I^E

(|Γ _s ^t,x | ^p ) ≤ C

^. ^Then^Theorem

3.1 (iii)

^implies ^that

|Z _s ^t,x | ≤ CΓ ^t,x _s (1 + |X _s ^t,x |), ∀s ∈ [0, t],

^I^{P -a.s.}

Now, applyingagain Lemma

2.1

^and

2.2

^and ^realling

(3.13)

^we ^get

(3.14)

^,^for

p ≥ 2

^.

To onlude this setion, we would like to point out that in Theorem

3.1

^, ^the ^funtions

f

^and

l

^are ^assumed ^to ^be ontinuously dierentiable in all spatial variables with uniformly bounded partial derivatives, whih is muh stronger than standing assumption

( A2 )

^. ^The

followingtheorem redues the requirementon

f

^and

l

^to ônly ûniformly^Lipshitz ôntinuous,

whihwillbeimportantin our future disussion.

Theorem 3.3 Assume

( A1 )

^-

( A4 )

^, ^and ^let

(X, Y, Z )

^be ^the ^solution ^to ^the ^FBDSDE

(1.2)

^-

(1.3)

^. ^Then ^for ^all

p > 0

C _p > 0

^suh^that

IE

|X| ^∗,p _0,t + |Y | ^∗,p _0,t + ess sup

0≤s≤t

|Z _s | ^p

≤ C _p (1 + |x| ^p ).

^(3.15)

Proof. In the light of the orollary

3.2

^, ^we ^need ^only ^onsider

p ≥ 2

^. ^By ^Lemma

2.1

^and

Lemma

2.2

^it ^follows ^that ^for ^any

p > 0

^there ^exists

C p > 0

^suh ^that

IE

{|X| ^∗,p _0,t + |Y | _0,t ^∗,p } ≤ C _p (1 + |x| ^p ).

^(3.16)

Next,bysimilarargumentof Theorem

3.1 (iii)

^, ^we^onsider ^two ^sequenes ^of^smooth ^funtions

{f ^ε } ε

^and

{l ^ε } ε

^with ^their^rst ^order derivativesin

(x, y, z)

^uniformly^bounded ⁱⁿ

t

^and

ε

^suh

that

lim ε→0

( sup

(t,x,y,z)

|f ^ε (t, x, y, z ) − f (t, x, y, z)| + sup

x

|l ^ε (x) − l(x)|

)

= 0.

Denoting

(X ^ε , Y ^ε , Z ^ε )

^the ^unique^solution^of^the orrespondingFBDSDEsand applyingCorol- lary

3.2

^, ^we ^an ^nd ^for ^any

p ≥ 2

^a ^onstant

C p > 0

^,independent of

ε

^, ^suh^that

IE

|Z ^ε | ^∗,p _0,t

≤ C p (1 + |x| ^p ).

^(3.17)

Furthermore, by

(3.10)

^we ^know ^that ^I^E

Z t

0 |Z _s ^ε − Z s | ² ds → 0

^as

ε → 0

^. ^Thus, ^possibly ^along

a sequene say

(ε n ) n≥1

^we ^have

lim n→∞ Z ^ε ⁿ = Z ds × d

Î^{P -a.s.} Âpplying ^Fâtou's ^lemma ând

realling

(3.17)

^we ^the ^obtain

IE

ess sup

0≤s≤t

|Z _s | ^p

≤ C _p (1 + |x| ^p )

whihleads to

(3.15)

^,^as ^desired.

(15)

In this setion we shall prove the rst main theorem of the paper. This theorem an be

regarded as an extension of the nonlinear Feynman-Ka formula obtained by Pardoux-Peng

[

¹⁵

]

^. ^It ^gives ^a probabilisti representation of the gradient (rather than the solution itself) of the stohasti visosity solution, whenever it exists, to a quasi-linear paraboli stohasti

PDE. Unlike the ases studied in

(3.1)

^, ⁱⁿ ^this ^setion, ^our representation does not depend on the partial derivatives of the funtions

f, l

^and

g

^. ^In ^this ^ontext ^suh representation is the best tool for us to study the path regularity of the proess

Z

ⁱⁿ ^the ^BDSDE ^with ^non-

smoothoeients. Fornotationalsimpliity,weshalldropthesupersript

t,x

fromthesolution

(X, Y, Z)

^of ^FBDSDE

(1.2)

^-

(1.3)

^.

To begin with, letusintroduethe two important stohasti integralsthat willplay a key

role in the representation:

M _r ^s = Z s

r

[σ ⁻¹ (τ, X τ )∇X _τ ] ^T ↓ dW τ

and

N _r ^s = 1

s − r (M _r ^s ) ^T [∇X _r ] ⁻¹ , 0 ≤ r < s ≤ t.

Let usreall that

IE

|M _r ^s | ^2p ≤ C _p

^I^E

Z s

r

|σ ⁻¹ (τ, X _τ )∇X _τ | ² dτ p

(4.1)

≤ C p (s − r) ^p

^I^E

|∇X _τ | ^∗,2p _s,r

≤ C p (s − r) ^p ,

where

C p > 0

îsâ ^generi ônstan^t.

An other hand, let us dene the ltration

G ^t =

F _s ^B ⊗ F _t ^W , 0 ≤ s ≤ t

^whih ^will ^play ^a

importantrole in the proof of the ontinuity of the proess

Z

ⁱⁿ ^the ^BDSDE.

Lemma 4.1 For any xed

t ∈ [0, T ]

^and ^any

H ∈ L ^∞ ( F ^t , [0, T ];

^I^R

)

^we ^have

(i)

^I^E

| R s

0 1

s−r H r M _s ^r dB r | < +∞

(ii)

^for ^I^P

.a.e., ω ∈ Ω

^, ^the ^mapping

s 7→ R s 0

1 s−r H r (ω)M _s ^r (ω)dB r (ω)

îs ôntinuous ôn

[0, t]

(iii)

^for ^I^P

.a.e. , ω ∈ Ω

^, ^the^mapping

s 7→

^I^E

{ R s 0

1 s−r H r M _s ^r dB r /G _s ^t }(ω)

îsôntinuous ôn

[0, t]

Proof. First, for any

0 ≤ τ < s ≤ t

^we ^denote

A ^s _τ =





 R s

τ 1

s−r H r M _r ^s dr, 0 ≤ τ < s

0, if s = τ.

(4.2)

(16)

Tosimplifynotation,when

τ = 0

^we^denote

A ^s ₀ = A s

^. ^F^urther,^let

β

^be^suh^that

α = 1−2β < ¹ ₂

and

β < 1

^. ^Consider ^the ^random^variable

M ^∗ = sup

0≤t 1 <t 2 ≤t

|M _t ^t ₁ ² |

(t 2 − t 1 ) ^α ;

^(4.3)

then by

(4.2)

^and ^Theorem

2.1

^of ^Revuz-Yor

[

¹⁶

]

^,^we ^see ^that ^I^E

[M ^∗ ] ² < +∞

^.

To prove (i) we note that for any

0 ≤ τ ≤ s ≤ t

^by Burkhölder-Gundy- Davis's inequality one has

IE

|A ^s _τ | ≤ C

^I^E

Z s

τ

H r M _r ^s s − r

2 dr

! 1/2

≤ C

^I^E

Z s

τ

|H r | ²

(s − r) ^2β . |M _r ^s | ² (s − r) ^2α dr

1/2

≤ C

^I^E

Z s

τ

|H _r (s − r) ^β

2 dr

! 1/2

M ^∗

≤ C

^I^E

Z s

τ

dr (s − r) ^2β

1/2

kHk _∞ M ^∗ = C(s − τ) ^(1/2)−β

^I^E

(kHk _∞ M ^∗ ),

^(4.4)

where

k.k

^denotes ^the^norm^of

L ^∞ ([0, T ])

^. ^Again^letting

C > 0

^be^a^generi^onstant^depending

only on

β

^and

T

^,^we ^have

IE

|A ^s _τ | ≤ C{

^I^E

kHk _∞ ² } ^1/2 {

^I^E

(M ^∗ ) ² } ^1/2

≤ CkHk L ^∞ ([0,T ]×Ω) kM ^∗ k _L ² _(Ω) < ∞.

^(4.5)

Setting

τ = 0

ⁱⁿ

(4.5)

^we ^proved

(i)

^.

To prove

(ii)

^let

τ = 0

ând ôbserve ^that, ⁱⁿ ^view ôf

(i), A s

îs â ^stohasti întegral ^for

0 < s ≤ t

^. Consequently, the mapping

s 7→ A s

îs ôntinuous ôn^[0,t℄. Ît ^remain ^to ^prove

(iii)

^.

In this fat, we remark that the right-hand side of the inequality

(4.4)

^(with

τ = 0

⁾^is ^learly

in

L ¹

^; ^thus ^we ^hek ^easily ^that ^the ^proess

A

^is ^uniformly integrable. Therefore, by similar step in Ma and Zhang [11℄ (see proof for

(iii)

^of ^Theorem ^4.1) ^it^follows ^that ^the

G ^t

-optional projetion of

A

^, ^denoting

^o A _s =

^I^E

(A _s |G _s ^t ), s ∈ [0, t]

^, ^has ^ontinuous ^path. ^This ^prove

(iii)

^,

whene the lemma.

Theorem 4.2 Assume that the assumptions

( A1 )

^-

( A4 )

^hold, ^and ^let

(X, Y, Z)

^be ^the ^adapted

solution to FBDSDE

(1.3)

^-

(1.2)

^. ^Then

(i)

^the ^following^identity ^holds ^I^P-almost ^surely:

Z s =

^I^E

l(X 0 )N ₀ ^s + Z s

0 f (r, X r , Y r , Z r )N _r ^s dr + Z s

0 g(r, X r , Y r )N _r ^s dB r |F _s ^t

σ(s, X s )

∀ 0 ≤ s ≤ t;

^(4.6)

Representation theorems for backward doubly stochastic differential equations

HAL Id: hal-00196841

https://hal.archives-ouvertes.fr/hal-00196841v4

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