reflected bsde and reflected pdie

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This article was downloaded by: [Florida State University] On: 20 December 2014, At: 08:42 Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK Stochastic Analysis and Applications Publication details, including instructions for authors and subscription information: http://www.tandfonline.com/loi/lsaa20 Reflected BSDE and Reflected PDIE M. Hassani a & Y. Ouknine a b a Faculté des Sciences Semlalia, Département de Mathématiques , Université Cadi Ayyad , Marrakech , Morocco b Faculté des Sciences Semlalia, Département de Mathématiques , Université Cadi Ayyad , B.P.2390, Marrakech , 40000 , Morocco Published online: 15 Feb 2007. To cite this article: M. Hassani & Y. Ouknine (2004) Reflected BSDE and Reflected PDIE, Stochastic Analysis and Applications, 22:3, 559-587, DOI: 10.1081/SAP-120030446 To link to this article: http://dx.doi.org/10.1081/SAP-120030446 PLEASE SCROLL DOWN FOR ARTICLE Taylor & Francis makes every effort to ensure the accuracy of all the information (the “Content”) contained in the publications on our platform. However, Taylor & Francis, our agents, and our licensors make no representations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose of the Content. Any opinions and views expressed in this publication are the opinions and views of the authors, and are not the views of or endorsed by Taylor & Francis. The accuracy of the Content should not be relied upon and should be independently verified with primary sources of information. Taylor and Francis shall not be liable for any losses, actions, claims, proceedings, demands, costs, expenses, damages, and other liabilities whatsoever or howsoever caused arising directly or indirectly in connection with, in relation to or arising out of the use of the Content. This article may be used for research, teaching, and private study purposes. Any substantial or systematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any form to anyone is expressly forbidden. Terms & Conditions of access and use can be found at http:// www.tandfonline.com/page/terms-and-conditions

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Page 1: Reflected BSDE and Reflected PDIE

This article was downloaded by: [Florida State University]On: 20 December 2014, At: 08:42Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954 Registered office: MortimerHouse, 37-41 Mortimer Street, London W1T 3JH, UK

Stochastic Analysis and ApplicationsPublication details, including instructions for authors and subscription information:http://www.tandfonline.com/loi/lsaa20

Reflected BSDE and Reflected PDIEM. Hassani a & Y. Ouknine a ba Faculté des Sciences Semlalia, Département de Mathématiques , Université CadiAyyad , Marrakech , Moroccob Faculté des Sciences Semlalia, Département de Mathématiques , Université CadiAyyad , B.P.2390, Marrakech , 40000 , MoroccoPublished online: 15 Feb 2007.

To cite this article: M. Hassani & Y. Ouknine (2004) Reflected BSDE and Reflected PDIE, Stochastic Analysis andApplications, 22:3, 559-587, DOI: 10.1081/SAP-120030446

To link to this article: http://dx.doi.org/10.1081/SAP-120030446

PLEASE SCROLL DOWN FOR ARTICLE

Taylor & Francis makes every effort to ensure the accuracy of all the information (the “Content”) containedin the publications on our platform. However, Taylor & Francis, our agents, and our licensors make norepresentations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose ofthe Content. Any opinions and views expressed in this publication are the opinions and views of the authors,and are not the views of or endorsed by Taylor & Francis. The accuracy of the Content should not be reliedupon and should be independently verified with primary sources of information. Taylor and Francis shallnot be liable for any losses, actions, claims, proceedings, demands, costs, expenses, damages, and otherliabilities whatsoever or howsoever caused arising directly or indirectly in connection with, in relation to orarising out of the use of the Content.

This article may be used for research, teaching, and private study purposes. Any substantial or systematicreproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in anyform to anyone is expressly forbidden. Terms & Conditions of access and use can be found at http://www.tandfonline.com/page/terms-and-conditions

Page 2: Reflected BSDE and Reflected PDIE

STOCHASTIC ANALYSIS AND APPLICATIONSVol. 22, No. 3, pp. 559–587, 2004

Reflected BSDE and Reflected PDIE

M. Hassani and Y. Ouknine*

Faculté des Sciences Semlalia, Département de Mathématiques,Université Cadi Ayyad, Marrakech, Morocco

ABSTRACT

In this work we consider a large class of multidimensional reflectedbackward stochastic differential equation (RBSDE for short) withgeneral filtration. We establish an existence and uniqueness resultand we derive a probabilistic interpretation for partial differentialand integral equation with reflection (RPDIE for short).

Key Words: Multidimensional reflected backward stochasticdifferential equations; Partial differential and integral equations.

1. INTRODUCTION

In this paper, we study the existence and uniqueness of solution forsome reflected backward stochastic differential equation (RBSDE forshort) and we drive a probabilistic interpretation for PDIE with reflection.

∗Correspondence: Y. Ouknine, Faculté des Sciences Semlalia, Département deMathématiques, Université Cadi Ayyad, B.P. 2390, Marrakech 40000, Morocco;E-mail: [email protected].

559

DOI: 10.1081/SAP-120030446 0736-2994 (Print); 1532-9356 (Online)Copyright © 2004 by Marcel Dekker, Inc. www.dekker.com

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560 Hassani and Ouknine

Such RBSDEs take the form:On a complete stochastic basis ���� � P� ��t�t≥0� (which satisfies the

usual conditions), we seek for a optional process y, a bounded variationprocess K and a martingale M such that

���

�1� A.s. � ∈ �� ∀t ≤ T����

yt = �− ∫ TtF�y��s� �� ds + ∫ T

tK�ds�− ∫ T

tM�ds�

�2� A.s. � ∈ �� ∀t ≤ T���� ∀i = 1 � � � q�

�yt − Si�t� Ui�t�� ≥ 0

�3� A.s. � ∈ �� ∀m ∈ �� �yt −m�t� 10�T��t�K�dt�� ≤ 0�

where T is a bounded stopping time, � is the terminal condition, F is afunction with domain (that we will specify later) and � (the domain ofreflection) is the set of all process m which satisfy

a.s. � ∈ �� ∀t ≤ T���� ∀i = 1 � � � q� �mt − Si�t� Ui�t�� ≥ 0

for some optional processes Ui� Si.Equations without reflection �Ui ≡ 0� were first studied by Pardoux

and Peng, see Refs.6–8�, they considered problems with Brownianstochastic basis and gave a probabilistic interpretation for PDE (theyestablished a new Feyman-Kac formula) in the viscosity sense. The sameproblem was later treated in Ref.3�, but the Feyman-Kac formula isconsidered in Sobolev sense.

The case of problems with reflection Ui = 0 was essentially studiedin dimension one �k = 1� with Brownian basis, see for instance Refs.2,4�.In Ref.4� �Ui= 1� q= 1� the authors gave a probabilistic interpretation inthe viscosity sense while in Ref.2� �U1 = 1� U2 = −1� q = 2� it is done ina weak formulation sense (Sobolev sense).

Here we consider RBSDE in a general setting, namely we study theexistence and uniqueness for problems in a multidimensional case, withgeneral stochastic basis and random domain of reflection. The methodwe use here is in the same spirit of Ref.5� via the so-called penalizationmethod.

A natural consequence is then to give a probabilistic interpretationfor the following PDIE with reflection:

�1� �� +∑di�j=1 Ui�t� x�ki�dt dx�

= f(t� x� ��t� x�� ∗���t� x�� ��t� x+ c�t� x� ���− ��t� x�

)�2� ���t� x�− Si�t� x� Ui�t� x�� ≥ 0

�3� ���t� x�− Si�t� x� Ui�t� x��ki�dt dx� = 0

�4� ��� � x� = ��x� x ∈ ��5� ��t� x� = ��t� x� t ∈ 0�� �� x �

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Reflected BSDE and Reflected PDIE 561

where

���t� x�

�= ��

�t�t� x�+ �b�t� x� ���t� x�� + 1

2

d∑i�j=1

� ∗�ij�t� x��2�

�xi�xj�t� x�

+∫E

[��t� x+ c�t� x� e��− ��t� x�− �c�t� x� e� ���t� x��]��de�

and � is a bounded domain of �d.This paper is organized as follows. In Sec. 2, we give conditions

and approximative scheme to derive existence and uniqueness for theproblem ���. In Sec. 3, we state the main results and their proofs.In Sec. 4, we illustrate these results by an interesting example whichshows the generality of the context presented here. Then by supposing anadditional condition on the data ��� F� Ui� Si��t� (we use the Markovianframework), we devote Sec. 5 to give a probabilistic interpretation forPDIE with reflection.

2. NOTATIONS AND HYPOTHESES

2.1. Notations

Throughout this paper we adopt the following notations anddefinitions:

• � denotes the -algebra of ��t�-optional set.• 2��k� �= �2

���× 0� T�� P�d�� ds�k�, for some k ∈ ∗.• �2��k� the space of �k-valued square integrable martingale M

such that � supt≥0�Mt∧T �2 < .

• c �=y ∈ 2��k� � ∃k ∈ 2��k�� ∃M ∈ �2��k��∃K ∈ �1

���BVc0 �0� T��

k�� such that, a.s.∀t ≥ 0 � yt∧T +

∫ t∧T���0 k�s� �� ds = Mt∧T + Kt∧T

where BVc0 �0� T��

k� is the space of continuous function Kwith bounded variation and K0 = 0, and

• ac �= �y ∈ c such that K = 0�.

For � ∈ �0����T H�� c� (resp. ac

� � denotes the set �y ∈ c such thatyT = �� (resp. c

� ∩ ac�.

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562 Hassani and Ouknine

We also define the following maps

� � c → �2��k�

y �→ M�

and

L � ac → 2��k�

y �→ k�

Let � ∈ �2����T �k�, F � c→2��k� and for i = 1� � � � � q

(for some q ∈ ∗�, let Ui ∈ �0���× 0� T�� P�d��ds�k� and Si ∈

�0���× 0� T�� P�d��ds�k�.

The problem under consideration becomes now: Find y ∈ c� �

K ∈ �1���BV

c0 �0� T��

k�� such that

���

�1� a.s. � ∈ ��∀t ≤ T����

yt = �− ∫ TtF�y��s� ��ds + ∫ T

tK�ds�− ∫ T

t��y��ds��

�2� a.s. � ∈ ��∀t ≤ T����∀i = 1� � � � � q�

�yt − Si�t� Ui�t�� ≥ 0�

�3� a.s. � ∈ ��∀m ∈ �� �yt −m�t� 10�T�����t�K�dt�� ≤ 0�

Our goal is to show that ��� has a unique solution. In order to provethis result, we use the following approximative scheme:

For n ≥ 0

��n�

{yn ∈ ac

L�yn�+ Fn�yn� = 0�

where

Fn � ac−→2��k��

defined by

Fn�y��t� �= F�y��t�− nq∑i=1

�yt − Si�t� Ui�t��−�Ui�t��2

Ui�t��

We will prove under suitable conditions that ��n� has a unique solutionwhich converges to the solution of the initial problem ���.

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Reflected BSDE and Reflected PDIE 563

2.2. Hypotheses

(A)

(i) ∃� ∈ �� � > 0� � < 12 such that ∀y� y′ ∈ c

2�∫ T

0�2�s �F�y�s − F�y′�s + �s�ys − y′s� ys − y′s�ds

≥ ��∫ T

0�ys − y′s�2 ds − ��

∫ T

0�2�s ��y − y′���ds��

where

� �={� ∈ �0

���× 0� T�� P�d��ds���

∫ T

0��s� + ��s�2 ds ∈ �

� ��� P�d����}

and �2�s �= exp�2∫ s∧T0 �r dr��

(ii) ∀R > 0� ∃� > 0 such that ∀y ∈ ac�

�∫ T

0�ys�2 ds ≤ R �⇒ �

∫ T

0�F�y�s�2 ds ≤ ��1+����y�T �2��

and

q∑i=1

�∫ T

0�Si�s��2 ds < �

(iii) ∀y ∈ c� �∀� ∈ ac

0

t ∈ � −→ �∫ T

0�F�y + t��s �s�ds is continuous.

(B) ∃m = M − V ∈ c, � > 0, � < 1, � ∈ �, r > 1, M ∈ �r ���such that

(i) �ms − Si�s� Ui�s�� ≥ �1Ui�s�=0�∀s ≤ T�∀i = 1� � � � � q and a.s.

(ii) ��mT − ��2r +��V + F�m�s ds�2rT < �

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564 Hassani and Ouknine

(iii) ∀y ∈ ac� a.s. � ∈ ��∀t ≤ T���

2∫ T

t�2�s �F�y�s − F�m�s + �s�ys −ms� ys −ms�ds

≥∫ T

tM�ds�− �

∫ T

t�2�s ���y −m���ds��

(C)

(i) s ∈ 0� −→ Ui�s ∧ T� ∈ �r ′��k�+�r ′� ��� P�d��

��0� �k��, where r ′ > 1 such that1r+ 1r ′

= 1.

(ii) ∃m = mT −∫ Ttf ds − ∫ T

tM�ds� such that

� supt≤T

�mt�2 +�∫ T

0�f �2 ds +��MT �2 <

and

�m− Si Ui��· ∧ T� ∈ �r ′���+�r ′� ��� P�d�� ��0� ����

(iii) ��− Si�T� Ui�T�� ≥ 0 a.s.

Remark 2.1. (1) Condition (A) is natural, indeed many authors whosetreat this equation use conditions which imply our one (see Sec. 4). As wewill see, this condition allows us to obtain the existence and uniquenessfor our approximative scheme.

(2) Condition (B) is given to ensure a priori estimations, we notethat this assumption (which appears -perhaps- strong) is natural, indeed:Let us make the following assumption (on F): ∃� < 1

2 � � ∈ �� � > 0 suchthat ∀y� y′ ∈ c

�A′��i�

2∫ T

t�2�s �F�y�s − F�y′�s + �s�ys − y′s� ys − y′s�ds

+ �∫ T

t�2�s ���y − y′���ds� ≥ �

∫ T

t�ys − y′s�2 ds�

then (A)(i) and (B)(iii) follows easily and condition (B) is equivalent

to the following equation: Find r > 1� � > 0� � ∈ �2r ����T �k��

V ∈ �2r� ��BV

c0 �0� T� �

k��, M ∈ �2��k� and m ∈ c such that{�i� mt = �−�− ∫ T

t

[F�m�s ds − V�ds�

]− ∫ TtM�ds�

�ii� �ms − S�i �s� Ui�s�� ≥ 0�∀s ≤ T�∀i = 1� � � � � q and a.s.

where S�i �s� �= Si�s�+ �1Ui�s�=0�Ui�s�/�Ui�s��2�.

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Reflected BSDE and Reflected PDIE 565

Moreover if we make the following additional assumption:∃r > 1� � > 0 such that

• ����2r < • ∀y ∈ c such that �ys − S�i �s� Ui�s�� ≥ 0� ∀s� i and � supt �yt�2r

+����y�T �2r < we have �∫ T0 �F�y�s�ds�2r < .

Then if ∃� ∈ �2r(���T �

k)� V ∈ �2r

(�BVc0 �0� T��

k�)�M ∈ �2��k�

and m ∈ c such that{�i� mt = �+ ∫ T

tV�ds�− ∫ T

tM�ds��

�ii� �ms − S�i �s� Ui�s�� ≥ 0 ∀s ≤ T ∀i = 1� � � � � q and a.s.

then (B) holds.

(3) Condition (C) is used to identified some limits. To be moreprecise, we will need the following fact: If kni converges weakly star to kiin �r

���BV0�0� T��k��

(a) and∑q

i=1

∫ •∧T0 Ui�s�k

ni �ds� converges weakly star to K in

�2����

�0� � ds�k�� then from (C)(i), K• =∑qi=1

∫ •∧T0 Ui�s�ki�ds�.

(b) From (C)(ii), we claim∫ T0 �Si�s� Ui�s���kni �ds�− ki�ds��−→

n→ 0

(see the next section).(c) (C)(ii) allows us to establish condition 2 in ���.

3. MAIN RESULTS

From assumption (A) and the result in Ref.5�, we know that ∃yn ∈ ac� solution of ��n� and the uniqueness follows from Itô formula and

(A)(i). Namely, one can show that Fn satisfies all hypotheses of Ref.5�,to obtain an idea of the proof you may see those of Lemma 8 in Ref.5�.

On the other hand for all y = −�V + �M + ∫ ·0�fs ds ∈ c and �� ∈ �we have

�ynt − yt�2�2��t +∫ T

t�2

��s

[��yn − y�

]�ds�

= ��− yT �2�2��T − 2∫ T

t�2

��s �F�yn�s −�fs +���yns − ys�� y

ns − ys�ds

+ 2∫ T

t�2

��s �yns − ys K

n�ds�−�V�ds��

− 2∫ T

t�2

��s �yns− − ys−��y

n − y��ds��

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566 Hassani and Ouknine

where

Kn�ds� =q∑i=1

Ui�s�kni �ds� and kni �ds� = n

�ynt − Si�t� Ui�t��−�Ui�t��2

ds�

Let y = m��� = ���f = F�m� and �V = V + ∫ �0 f�m�s ds� then�ynt −mt�2�2��t +

∫ T

t�2

��s ��y

n−m���ds�

=��−mT �2�2��T −2∫ T

t�2

��s �F�yn�s−F�m�s+���yns −ms��y

ns −ms�ds

+2∫ T

t�2

��s �yns −msK

n�ds�−�V�ds��

−2∫ T

t�2

��s �yns− −ms−��y

n−m��ds���

We deduce that

�ynt −mt�2�2��t + �1− ��∫ T

t�2

��s ��y

n −m���ds�

+ �∫ T

t�2

��s ���y

n −m��− ���yn −m����ds�

+ 2∫ T

t�2

��s

q∑i=1

n

(�yns − Si�s� Ui�s��−�Ui�s��

)2

ds

+∫ T

tM�ds�+ 2

∫ T

t�2

��s

q∑i=1

�ms − Si�s� Ui�s��kni �ds�

≤ ��−mT �2�2��T − 2∫ T

t�2

��s �yns −ms�V�ds��

− 2∫ T

t�2

��s �yns− −ms−��y

n −m��ds���

Hence, we obtain the following inequality which we denote ∗�

�ynt −mt�2�2��t + �1− ��∫ T

t�2

��s ��y

n −m���ds�

+2∫ T

t�2

��s

q∑i=1

n

(�yns − Si�s� Ui�s��−�Ui�s��

)2

ds

+2∫ T

t�2

��s

q∑i=1

�ms − Si�s� Ui�s��kni �ds�

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Reflected BSDE and Reflected PDIE 567

≤ ��−mT �2�2��T + 2∫ T

t�2

��s �yns −ms���V ��ds�−

∫ T

tM�ds�

−2∫ T

t�2

��s �yns− −ms−��y

n −m��ds��

−�∫ T

t�2

��s ���y

n −m��− ���yn −m����ds��Then

� supn�t

�ynt −mt�2r�2r��t

≤ ��tsupt≤T

(��−mT �2�2��T + 2

∫ T

t�2

��s �yns −ms���V ��ds�

)r�

Thanks to Doob and Holder inequalities, we have ∃C�r� �� ≥ 0� suchthat

�supt�n

�ynt −mt�2r≤C�r����(��−mT �2r+

∣∣∣∣V+∫ ·

0f�m�sds

∣∣∣∣2rT

)(3.1)

On the other hand, from ∗� and Burkholder inequality, there isC�r� �� ≥ 0 such that

�{�1− ��

∫ T

0�2

��s

[��yn −m�

]�ds�

+2∫ T

0�2

��s

q∑i=1

n

(�yns − Si�s� Ui�s��−�Ui�s��

)2

ds

+2∫ T

0�2

��s

q∑i=1

�ms − Si�s� Ui�s��kni �ds�}r

≤ C�r� �����−mT �2r +�∣∣�V ∣∣2r

T+� sup

t≤T�ynt −mt�2r

+�(��yn −m��rT sup

t≤T�ynt −mt�r

)+���yn −m��

r2T +�M�

r2T �

Young’s inequality and (3.1) imply

���yn −m��rT +q∑i=1

nrE

(∫ T

0

(�yns − Si�s� Ui�s��−�Ui�s��

)2

ds

)r

+q∑i=1

�( ∫ T

0�ms − Si�s� Ui�s��kni �ds�

)r≤ C�r� �� ���

(��−mT �2r + ��V �2rT + M�

r2T + 1

)�

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568 Hassani and Ouknine

Therefore

����yn −m��2r +q∑i=1

nr �(∫ T

0

(�yns − Si�s� Ui�s��−�Ui�s��

)2

ds

)r+

q∑i=1

�(∫ T

0�ms − Si�s� Ui�s��kni �ds�

)r≤ C�r� �� ���

(��−mT �2r + ��V �2rT + �M�r + 1)�

(3.2)

In particular, we have the following result.

Proposition 3.1. For a subsequence, there are y ∈ �2����

�0� T�� ds�k���M ∈ �2��k�� ki ∈ �r

� ��BV+0 �0� T����� ˆy0 ∈ �2

�T���k� and

� ∈ 2��k� such that as n→

yn ⇀ y Weakly star in �2����

�0� T�� ds�k���

F�yn� ⇀ � Weakly in 2��k��

��yn� ⇀ M Weakly in �2��k��

yn0 ⇀ˆy0 Weakly in �2

�T���k��

kni ⇀ ki Weakly star in �r���BV

+0 �0� T�����

where BV+0 �0� T��� denotes the space of increasing functions k such that

k0 = 0� And

(i) �yt − Si�t� Ui�t�� ≥ 0 10�T��t� dt P�d�� a.e.

(ii)∑q

i=1

∫ •∧T0 Ui�s�k

ni �ds� converges weakly star to∑q

i=1

∫ •∧T0 Ui�s�ki�ds� in �2

���� �0� � ds�k��.

(iii) yt = �− ∫ Tt�s ds +

∑qi=1

∫ TtUi�s�ki�ds�−

∫ TtM�ds�

10�T��t� dt P�d�� a.s.

(iv) ˆy0 = �− ∫ T0 �sds +∑q

i=1

∫ T0 Ui�s�ki�ds�−

∫ T0 M�ds� P�d�� a.s.

Set

yt �= �−∫ T

t∧T�s ds +

q∑i=1

∫ T

t∧TUi�s�ki�ds�−

∫ T

t∧TM�ds��

Hence

yt = yt 10�T��t�dtP�d�� a.e. and y0 = ˆy0 P�d�� a.s.

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Reflected BSDE and Reflected PDIE 569

Since �yt − Si�t� Ui�t�� ≥ 0 10�T��t� dt P�d�� a.e., �yt − Si�t� Ui�t�� iscàdlàg and ��− Si�T� Ui�T�� ≥ 0, we claim

�yt − Si�t� Ui�t�� ≥ 0 ∀i = 1� � � � � q� ∀t ≤ T��� a.s. � ∈ ��On the other hand, write Itô formula for �yn − y y�� where

yt = yT −∫ T

t∧T��s ds +

∫ T

t∧T�K�ds�−

∫ T

t∧T�M�ds��

�yn0 − y0 y0�= −∫ T

0�2

��s �F�yn�s −�s +���yns − ys�� ys�ds

−∫ T

0�2

��s ���s +��ys yns − ys�ds

+∫ T

0�2

��s

q∑i=1

�ys Ui�s���kni − ki��ds�+∫ T

0�2

��s �yns − ys�K�ds��

−∫ T

0�2

��s ��yns− − ys− �M�ds�� + �ys− ���yn�−M��ds���

−∫ T

0�2

��s ��y

n�−M� �M��ds�−∫ T

0�2

��s �K�K��ds��

If

�{supt≤T

�yt�2 + ��MT �2 +∫ T

0���s�2 ds

}<

then� limn→

{ ∫ T

0�2

��s

q∑i=1

�ys Ui�s���kni − ki��ds�

+∫ T

0�2

��s �yns − ys�K�ds�� −

∫ T

0�2

��s �K�K��ds�

}= 0� (3.3)

If, moreover, �K ≡ 0 then

limn→ �

{ ∫ T

0�2

��s

q∑i=1

�ys Ui�s���kni − ki��ds�

}= 0�

Hence

(1) From (C)(ii), we have

limn→

q∑i=1

�∫ T

0�2

��s �Si�s� Ui�s���kni − ki��ds� = 0 for all �� ∈ ��

(3.4)

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570 Hassani and Ouknine

(2) For all �K ∈ �0��BV c0 �0� T��k�� ∩�r ′∨2

� ����0� T��k��

limn→ �

∫ T

0�yns − ys�K�ds�� = 0� (3.5)

Again, Itô formula implies

��yn0 − y0�2 +�∫ T

0�2

��s ��y

n�−M��ds�

= −2�∫ T

0�2

��s �F�yn�s −�s +���yns − ys�� y

ns − ys�ds

+�(2

q∑i=1

∫ T

0�2

��s �yns − ys Ui �s���kni − ki��ds�+

∫ T

0�2

��s K��ds�

)�

where K�t� =�∑qi=1

∫ t0 Ui�s�ki�ds�.

Then

�∫ T

0�2

��s ��y

n�−M��ds�+ 2�∫ T

0�2

��s �F�yn�s +��yns � yns − ys�ds

+ 2q∑i=1

�∫ T

0�2

��s �yns − Si�s� Ui�s��−kni �ds�+��yn0 − y0�2

+�

{2

q∑i=1

∫ T

0�2

��s �yns − ys Ui�s��ki�ds�−

∫ T

0�2

��s K��ds�

+ 2q∑i=1

∫ T

0�2

��s �ys − Si�s� Ui�s��kni �ds�

}

converges to 0, as n→ .From (3.3) we have

��yn0 − y0�2 +�∫ T

0�2

��s ��y

n�−M��ds�

+ 2�∫ T

0�2

��s �F�yn�s +��yns � yns − ys�ds

+ 2q∑i=1

�∫ T

0�2

��s �yns − Si�s� Ui�s��−kni �ds�

+�{2

q∑i=1

∫ T

0�2

��s �ys Ui�s���ki − kni ��ds�+

∫ T

0�2

��s K��ds�

+ 2q∑i=1

∫ T

0�2

��s �ys − Si�s� Ui�s��kni �ds�

}converges to 0, as n→ .

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Reflected BSDE and Reflected PDIE 571

Hence, from (3.4)

limn→

{�∣∣yn0 − y0

∣∣2 +�∫ T

0�2

��s

[�(yn)−M

]�ds�

+ 2�∫ T

0�2

��s

⟨F�yn�s +��yns � yns − ys

⟩ds

+ 2�q∑i=1

∫ T

0�2

��s

⟨yns − Si�s� Ui�s�

⟩−kni �ds�

+�( ∫ T

0�2

��s

[K]�ds�

+ 2q∑i=1

∫ T

0�2

��s

⟨ys − Si�s� Ui�s�

⟩ki�ds�

)}= 0 (3.6)

Moreover, we have from (A)(i),

lim infn�m→

{2�∫ T

0�2�s⟨F�yn�s − F�ym�s + �s

(yns − yms

) yns − yms

⟩ds

+��∫ T

0�2�s[��yn − ym�

]�ds�− ��

∫ T

0

∣∣yns − yms∣∣2 ds} ≥ 0�

By a few calculation we get

lim infn→

{2�∫ T

0�2�s⟨F�yn�s + �s

(yns) yns − ys

⟩ds

+��∫ T

0�2�s[��yn�−M

]�ds�− ��

∫ T

0

∣∣yns − ys∣∣2 ds} ≥ 0�

Return to (3.6), we have

lim supn→

{�∣∣yn0 − y0

∣∣2 + (1− �)�∫ T

0�2

��s

[�(yn)−M

]�ds�

+ ��∫ T

0

∣∣yns − ys∣∣2 ds

+ 2q∑i=1

�∫ T

0�2

��s

⟨yns − Si�s� Ui�s�

⟩−kni �ds�

}+�( ∫ T

0�2

��s K��ds�

+ 2q∑i=1

∫ T

0�2�s⟨ys − Si�s� Ui�s�

⟩ki�ds�

)≤ 0�

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572 Hassani and Ouknine

Since⟨ys − Si�s� Ui�s�

⟩ ≥ 0, we obtain

(a) limn→

{�∣∣yn0−y0∣∣2+�

∫ T

0�2

��s

[�(yn)−M]�ds�+�

∫ T

0

∣∣yns −ys∣∣2ds+�

q∑i=1

∫ T

0�2

��s

⟨yns −Si�s�Ui�s�

⟩−kni �ds�

}=0�

(b) �ys − Si�s� Ui�s��ki�ds� = 0 and Kt �= Kt − Kt− ≡ 0.

(c) For all �� ∈ �� limn→ ∫ T0 �

2�s �F�yn�s + �s

(yns) yns − ys�ds = 0.

From (b) and the fact that∑q

i=1�ys −msUi�s�� ki�s� = 0 (m of condition�B�), we have 1Ui�s�=0� ki�s� = 0 and y ∈ c

� and ��y� = M� where ki�s�denotes ki�s�− ki�s

−�.Moreover, for all t > 0� � ∈ ac

0 , we have

2�∫ T

0�2�s �F�yn�s − F�y + t��+ �s

(yns − ys − t�s

) yns − ys − t�s�ds

+ ��∫ T

0�2�s[��yn − y − t��

]�ds� ≥ 0�

Passing to the limit, �n→ � we have

2�∫ T

0�2�s �Fs�y + t��−�s �s�ds + 2t

∫ T

0�2�s �s

∣∣�s∣∣2 ds+ �t�

∫ T

0�2�s[����

]�ds� ≥ 0�

Letting, now, t → 0, we have from condition (A)(iii)

�∫ T

0�F�y�−�s ��ds ≥ 0 for all � ∈ ac

0 �

Since ac0 is dense in 2��k�, we claim that � ≡ F�y� (see Ref.5�).

We conclude, the following

Theorem 3.1. Under (A), (B) and (C), equation ��� has a unique solution�y�K� in c

� �1���BV

c0 �0� T��

k�� such that � supt≤T �yt�2 < .Moreover, there are ki ∈ �r

���BV+c0 �0� T���� such that

�i� �ys − Si�s� Ui�s�� ki�ds� = 0 a.s.

�ii� K�t� =q∑i=1

∫ t

0Ui�s� ki�ds��

where BV+c0 �0� T��� denotes the space of continuous and increasing

functions k such that k0 = 0.

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Reflected BSDE and Reflected PDIE 573

Proof. It remains to show the uniqueness:Let �y��K� be another “possible” solution of ���, then from Itô

formula and (A)(i) we have

�∣∣y0 − y0

∣∣2 +�∫ T

0�2

��s

[�(y − y

)]�ds�+ ��

∫ T

0

∣∣ys − ys∣∣2 ds

+ 2q∑i=1

�∫ T

0�2

��s

⟨ys − Si�s� Ui�s�

⟩ki�ds�

− 2�∫ T

0�2

��s

⟨ys − ys�K�ds�

⟩ ≤ 0�

Hence, y = y and �K = K.

4. EXAMPLE

Let �E� !E� be a measurable space and !0 a semiring of setsgenerating the -algebra !E . A martingale measure "�t� ��A� t ≥ 0A ∈ !0 is a random function possessing the following properties:

(i) For fixed A ∈ !0, "��� �� A� is a càdlàg �-valued squareintegrable martingale, and for fixed t� �, "�t� �� �� is acountably additive function on !0.

(ii) If A ∩ B = ∅, then the product "��� �� A�"��� �� B� is amartingale.

(iii) "�0� �� A� = 0.

A random function "�t� �� A� t ≥ 0 A ∈ !0 is called a local martingalemeasure if there is a sequence

( n� n ≥ 0

)of stopping time such that

�i� −"�t ∧ n� �� A� is a martingale measure.

�ii� − n ↗ as n→ �

It is well known (see Ref.10�) that there is a positive measure on the -algebra � × !E (� denotes -algebra of predictable sets) of the form#��� dt� de�P�d�� such that for all t ≥ 0, A ∈ !0

�i� −#��� �0�� de� = 0

�ii� −#��� 0� t�� A� is the predictable process such that∣∣"�t� �� A�∣∣2 − #��� 0� t�� A� is a local martingale

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574 Hassani and Ouknine

Set �2# ��

k� �= �2([0� [×�× E�� × !E� #��� ds� de�P�d���

k).

Let � = (En� !En� !n0� "n� #n)n≥1be a sequence of local martingale

measure such that( ∫ T

0

∫En

Zn�s� �� e�"n�ds� de�∫ T

0

∫Em

Zm�s� �� e�"m�ds� de�

)= 0�

for all n = m�Zn ∈ �2#n��� and Zm ∈ �2

#m��� (i.e., �"n� are stochastically

independent).Consider �2

���k� the space of martingale of the form

∑n=1

∫ •

0

∫En

Zn�s� �� e�"n�ds� de� with �Zn�n≥1 ∈ ⊕n=1

�2#n�

where ⊕n=1

�2#n��k�

�={�Zn�n≥1 ∈

∏n=1

�2#n

(�k)�

∑n=1

0

∫En

∣∣Zn(s� �� e)∣∣20#n(ds� de) < }�

Proposition 4.1. (i) �2���

k� is a closed subspace of �2(�k).

(ii) ∀M ∈ �2��k� there is a unique element �M ′�M ′′� ∈ �2���

k�× �2

���k��⊥ such that M = M ′ +M ′′, where �2

���k��⊥ denotes the

orthogonal subspace of �2���

k�.

Corollary 4.1. For all y ∈ c there is a unique element �Zn�n≥1 ∈⊕ n=1 �

2#n

M ∈ �2���

k��⊥ such that there is f ∈ 2��k� and K ∈�1

���BVc0 �0� T��

k��:

∀T ≥ t ≥ 0� yt +∫ t

0fs ds =

∑n=1

∫ t

0

∫En

Zn�s� �� e�"n� ds� de�+Mt + Kt�

We define the following maps, via the above corollary,

$ � y ∈ c → �Zn�n≥1 ∈ ⊕n=1

�2#n

and % � y ∈ c → M ∈ �2���

k��⊥�

We suppose, in the sequel, that there is an increasing adapted càdlàgprocess K��� t� such that

(i) There is f ∈ �0�0� ×��+� � t =∫ t0 fsK�ds�� ∀0 ≤ t ≤ T

a.s.

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Reflected BSDE and Reflected PDIE 575

(ii) There is a family gn�t� �� de�� t ≥ 0� � ∈ � of positive measure:

#n��� dt� de� = gn�t� �� de�K�dt� ∀0 ≤ t ≤ T a.s.

Let �� Ui� Si as in Sec. 2 and the following map

G � �× 0� T�×�k × ∏n=1

�0�En� !En�k�→ �k

which satisfies:There is � ∈ �1��× 0� T�� K�ds��+�� r ≥ 0� � < 1

4 � and � ∈ �such that ∀y� y′� & ∈ �k�∀Z�Z′ ∈ ∏

n=1�0�En� !En�

k� we have:

(A′1) �t� ��→ G�t��� y� Z� is optional

(A′2) K�dt���P�d�� a.e.

t ∈ � →f�s� ���G�s��� y + ty′� Z + tZ′�� &� is continuous.

(A′3) K�dt���P�d�� a.e.

f�t� ���G�t��� y� Z��2

≤ ��t� ��+ ��t� ��f�t� ���y�2 + r ∑n=1

∫En

�Zn�e��2gn�t� �� de��

(A′4) K�dt���P�d�� a.e.

f�t� ���G�t��� y� Z�−G�t��� y′� Z′�� y − y′�≥ −��t� ��f�t� ���y − y′�2

−� ∑n=1

∫En

�Zn�e�− Z′n�e��2gn�t� �� de��

Theorem 4.1. Under �A′1�–�A

′4�� (Bi)–(Bii), and (C) there is an unique

element y� ki in c� �r

���BV+c0 �0� T���� such that for all T ≥ t ≥ 0

(i) yt = �−∫ T

tG�s� ys� $ys�A�ds�+

q∑i=1

∫ T

tUi�s�ki�ds�

− ∑n=1

∫ T

t

∫En

�$y�n�s� e�"n�ds de�−∫ T

t�% y�d�s��

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576 Hassani and Ouknine

(ii) �ys − Si�s� Ui�s�� ≥ 0� ∀0 ≤ s ≤ T a.s.

(iii) �ys − Si�s� Ui�s�� ki�ds� = 0 a.s.

Proof. Set F the following map

F � c → 2��k�y → G�s� ys� $ys�

By a few calculation, we show that assumptions (A′1)–(A

′4), (Bi)–(Bii),

and (C) on G imply assumptions (A), (B) and (C) (of Sec. 2) on F.Therefore, the previous theorem closes the proof.

5. APPLICATION

In this section, we give a probabilistic interpretation for the solutionof partial differential and integral equation with refection (RPDIE forshort) by means of reflected backward stochastic differential equationwith jumps. For this, we need some additional hypothesis and tools:

• The filtration �t is generated by a M-dimensional Brownianmotion W and a random Poissonian measure N�ds de� on0� T ×E of intensity ds��de�� where �E� !E� ��de�� is a given -finite measured space. Also, we need:

• � be a bounded domain (open) in�d, we denote�c �= �d −�.• � be a positive real number.

Consider the following reflected first boundary (formal) problem

(i) ���t�x�

�= ��

�t�t�x�+�b�t�x����t�x��+ 1

2

d∑i�j=1

� ∗�i�j�t�x��2�

�xi�xj�t�x�

+∫E��t�x+c�t�x�e��−��t�x�−�c�t�x�e����t�x����de�

=f�t�x���t�x�� ∗�t�x����t�x����t�x+c�t�x����−��t�x��

−q∑i=1

Ui�t�x�ki�dtdx��

(ii) ���t� x�− Si�t� x� Ui�t� x�� ≥ 0

a.e. �t� x� ∈ 0�� �×�� ∀i = 1� � � � � q�

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Reflected BSDE and Reflected PDIE 577

(iii) ���t� x�− Si�t� x� Ui�t� x��ki�dt dx� = 0�(iv) ��� � x� = ��x� a.e. x ∈ ��(v) ��t� x� = ��t� x� ∀t ∈ 0�� �∀x ∈ �c�(vi) � ∈ �2�0�� � � 1

2 ��d�k���

Where

• For coefficients b� � c and � we make the assumption (I):

(I1) b� � c are bounded and measurable function and�� ∗�i�j/�xj ∈ � �0�� �×����∀ij�

(I2) For all t ∈ 0�� � and x ∈ �d there is an adapted solution�Xt�x

s �s∈0�� � of

Xt�xs = x +

∫ s∨t

tb�r� Xt�x

r �dr +∫ s∨t

t �r� Xt�x

r �W�dr�

+∫ s∨t

t

∫Ec�r� Xt�x

r− � e�N �dr de��

where N �dr de� �= N�dr de�− dr ��de�.(I3) There is a positive constant ' such that for all " ∈

�0���+�� s� t ∈ 0�� �

'−1∫�"�x�dx ≤

∫�"�Xt�x

s �1�t�xs dx ≤ '∫�"�x�dx�

where

�t�xs �= �� ∈ � � ∀r ∈ 0� s�Xt�x

r ∈ ���

• For f � 0�� �×�×�k ×�M×k ×�2�E� ��de��k�→ �k wemake assumption (J). For a�e� �t� x� ∈ 0�� �×� ∀�y� z� u���y′� z′� u′� in �k ×�M×k ×�2�E� ��de��k�

(J1) �f�t� x� y� z� u�� ≤ ��t� x�+ (��y� + �z� + �u���

(J2) 2�f�t� x� y� z� u�− f�t� x� y′� z′� u′� y − y′�≥ −��t��y − y′�2 − �

{�z− z′�2 + �u− u′�2}�(J3) s ∈ 0� 1�→ �f�t� x� y + sy′� z+ sz′� u+ su′� y′�

is continuous, for some � ∈ �2�0�� �×��+��% ∈ �1�0�� ��+�� ( ≥ 0� � ≤ 1

2 .

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578 Hassani and Ouknine

• For �Ui� Si�i=1�����q � 0�� �×�d → �k we make assumption (K):∀t ∈ 0�� �� ∀i = 1� � � � � q a�e� x ∈ �.

(K1) Ui��∧)t�x� �Xt�x

�∧)t�x��∈�r ′��k�+�r ′

� ���P�d����0� �k��.

(K2) �UiSi���∧)t�x� �Xt�x

�∧)t�x��∈�r ′���+�r ′

� ���P�d����0� ����where )t�x� �= inf�t≤r≤� �Xt�x

r ∈�c�.

(K3) �Ui�� �x���x�−Si�� �x��≥0 ∀x∈� and �Ui�t�x���t�x�−Si�t�x��≥0 ∀�t�x�∈ 0�� �×�c.

• For � b� c��� f� Ui Si� �� � together one make assumption (L):

(L1) There arem∈� 1�2p �0�� �×�d�k� (for some p> 2d+ 4��

� > 0� r > 1 such that

(i) �Ui�t�x�m�t�x�−Si�t�x��≥�1Ui�t�x�=0�.

(ii)∫ �

0

∫��f�t�x�m�t�x�� ∗�t�x��m�t�x��m�t�x+c�t�x����−m�t�x��−�m�t�x��4dtdx< �

(iii)∫��m�� �x�−��� �x��4dx+ supt≤� �x∈�c

�m�t�x�−��t�x��< �

where � 1�2p �0�� ��d�k� denotes the usual Sobolev

space.(L2) For all n ≥ 0 there is �n ∈ � 1�2

p �0�� ��d�k� a weaksolution (in the sense given in Ref.9�) of

(i) ��n�t�x�

=f�t�x��n�t�x�� ∗�t�x���n�t�x���n�t�x+c�t�x����

−�n�t�x��−nq∑i=1

�Ui�t�x��n�t�x�−Si�t�x��−�Ui�t�x��2

×Ui�t�x� a�e� �t�x�∈ 0�� �×��

(ii) �n�� � x� = ��x� a�e� x ∈ �.(iii) �n�t� x� = ��t� x� ∀t ∈ 0�� � ∀x ∈ �c.

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Reflected BSDE and Reflected PDIE 579

Remark 5.1. (1) We know that there is an abundant literatures whichgive sufficient conditions to have (I2), see for example Ref.9� andreference therein. Condition (I3) is adapted (in a more general form) tothis found in Ref.2�. sufficient conditions are given in appendix to have(I3).

(2) Condition (J) and (L), which are natural, are needed to obtainconditions (A) and (B) of previous section.

(3) (K) is linked with assumption (C). We note that:

(a) If Ui� �Ui Si� are in � 1�2p �0�� ��d�k�+��0�� ��k�

then �K� is satisfied (via Itô formula).(b) If c ≡ 0 then Ui� �Ui Si� ∈ ��0�� �× cl����k� suffices.

(4) (L) is given to establish condition (B). Suppose that∫����x��4 dx + sup

t≤� �x∈�c

���t� x�� <

and ∫ �

0

∫��f�t� x� "�t� x�� ∗�t� x��"�t� x�� "�t� x+ c�t� x� ���

−"�t� x��−�"�t� x��4 dt dx < �

for all " ∈ � 1�2p �0�� �×�d�k�, then if ∃m ∈ � 1�2

p �0�� �×�d�k�such that∫

��m�x��4 dx + sup

t≤� �x∈�c

�m�t� x�� +∫ �

0

∫���m�t� x��4 dt dx <

and

�Ui�t� x�m�t� x�− Si�t� x�� ≥ �1Ui�t�x�=0�

then condition (L1) holds.

(L2) is given to derive that the solution of the approximativescheme enjoys with a suitable conditions (especially to obtain thefollowing). To obtain a suitable conditions which give (L2) we refer thereader to Ref.9�.

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580 Hassani and Ouknine

Let, for t ∈ 0�� �, �t the set of x ∈ � such that

(1) A.s. � ∈ �� ∀n ≥ 0� ∀t ≤ s ≤ )t�x�

�n�s�Xt�xs �

=�t�x−∫ )t�x�

s���n�∗ �r�Xt�x

r �W�dr�

−∫ )t�x�

sf�r�Xt�x

r ��n�r�Xt�x

r �� ∗�r�Xt�x

r ���n�r�Xt�x

r ��

�n�r�Xt�xr +c�r�Xt�x

r ����−�n�r�Xt�xr ��dr

+q∑i=1

n∫ )t�x�

s

�Ui�r�Xt�xr ��

n�r�Xt�xr �−Si�r�Xt�x

r ��−�Ui�r�Xt�x

r ��2 Ui�r�Xt�xr �dr

−∫ )t�x�

s

∫E�n�r�Xt�x

r− +c�r�Xt�xr− ����−�n�r�Xt�x

r− ��N �drde��

(2) �F t�x� U t�xi � S

t�xi � �

t�x� satisfy all conditions in Sec. 2, with

Ft�xs �y� �= f�s� Xt�xs � y�$W�y�

∗� $N�y��� U t�xi �s� �= Ui�s� X

t�xs ��

St�xi �s� �= Si�s� Xt�xs �� �

t�x �= ��Xt�x� �1�t�x� + ��)t�x� � X

t�x

)t�x��1��t�x� �c �

mt�xs �= m�s�Xt�x

s �� r = 2 in (B) and mt�x �= 0 in �C��

$W (resp. $N ) denotes the operator defined in Sec. 4 with respect theBrownian motion (resp. the Poisson measure martingale). Note that, frommartingale representation theorem, we have %�y� ≡ y0 for all y ∈ c.

Proposition 5.1. For all t ∈ 0�� �∫�\�t

1dx = 0�

Proof. This is a direct consequence of our hypotheses (I)–(L) (especially(I3)) and Itô formula (see Ref.9�).

Theorem 5.1. There are ��t� x� ∈ �2�0�� �� 12 ��

d�k��� ki�dt dx� ∈�+b �0�� �×�� (the space of finite and positive measure) such that

(i) �� − Si Ui��s� x�1�s �x� ≥ 0� ∀�s� x� ∈ 0�� �×�.(ii) �� − Si Ui��s� x�1�s �x�ki�ds dx� = 0�

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Reflected BSDE and Reflected PDIE 581

(iii) For all h ∈ �1c ����� t ∈ 0�� �� j = 1� � � � � k∫

∫ �

t

{− 1

2� ∗��j�s� x� �h�x�� + �b − a− c ��j��s� x�h�x�

+∫E��j�s� x+ c�s� x� e��− �j�s� x����de�h�x�

}ds dx

=∫�

∫ �

tfj��� �� �� ��

∗ ����� �+ c��� �� e���s� x�−��h�x�ds dx

+∫��j�t� x�h�x�dx−

∫��j�x�h�x�dx

−q∑i=1

∫�

∫ �

th�x��Ui�j�s� x�ki�ds dx��

where

ai �=12

d∑j=1

�� ∗�i�j�xj

for i = 1� � � � � d� and

c �=∫Ec��� �� e���de��

(iv) ��t� x� = ��t� x� ∀t ∈ 0�� �� ∀x ∈ �c�(v) ki��t�×�� ≡ 0� ∀t ∈ 0�� �.

Proof. For t ∈ 0�� �� x ∈ �t let �Y t�x� Zt�x� U t�x� kt�xi � the solution of ���with �F t�x� U t�x

i � St�xi � �

t�x� as a data (with Zt�x = $W�Yt�x� and Ut�x =

$N�Yt�x�). Set ��t� x� �= Y t�xt 1�t �x�+ ��t� x�1�c �x�.

From Sec. 3, we have ∀t ∈ 0�� �� ∀x ∈ �t

*n�t� x� �= ���t� x�− �n�t� x��2r +∫ )t�x�

t�Y t�xs − �n�s� Xt�x

s ��2 ds

+∫ )t�x�

t�Zt�xs − ��n∗ �s� Xt�x

s ��2 ds

+∫ )t�x�

t

∫E�Ut�x

s �e�− ��n�s� Xt�xs− + c�s� Xt�x

s− � e��

−�n�s� Xt�xs− ���2 ds��de�

converges to 0 as n −→ . And, from (3.1) and (3.2), we have

supt

∫�supn

*n�t� x�dx < �

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582 Hassani and Ouknine

Lebesgue dominated theorem gives ∀t ≤ � ,∫�*n�t� x�dx −→

n→ 0 and∫ �

0

∫�*n�t� x�dt dx −→

n→ 0�

Set

kni �dr dx� �= n�Ui�r� x� �n�r� x�− Si�r� x��−

�Ui�r� x��2dr dx

Corollary 5.1. (i) �n�t� x�→ ��t� x� in �2�0�� �� 12 ��

d�k��.

(ii) �n�t� x+ c�t� x� e��− �nt� x→ �n�t� x+ c�t� x� e��− �nt� x in�2�0�� �×�d × E�k�. ∀t ∈ 0�� �� ∀x ∈ �t.

(iii) ��s� Xt�xs = Y t�xs � ��

∗ �s� Xt�xs = Zt�xs � and��s� X

t�xs + c�s� Xt�x

s−� e��−��s� Xt�x

s−� = Ut�xs �e� for a.e. s� e� �.

(iv) f��� �� �n� ∗�#� ���� �+ c��� �e��−��→ f��� �� �n� ∗�#� ���� �+c��� �e��− �� weakly is �2�0�� �×�d�k��

From (I3), we have∫�

∫ �

tkni �dr dx� ≤ '

∫�kt�xi�n�)

t�x� �dx ≤ '2

∫�

∫ �

tkni �dr dx�

where

kt�xi�n�dr� �= n�Ui�r� Xt�x

r � �n�r� Xt�x

r �− Si�r� Xt�xr ��−

�Ui�r� Xt�xr ��2 dr� kt�xi�n�t� = 0�

hence supn∫�

∫ �0 k

ni �dr dx� < , then there is ki�dr dx� a finite positive

measure on 0�� �×� such that, up to a subsequence, ∀� ∈ ��0�� �×���∫

∫ �

0��r� x�kni �dr dx� −→

n→

∫�

∫ �

0��r� x�ki�dr dx��

By approximation, we deduce ∀t ≤ � �∀� ∈ �0�0�� �×��+�∫�

∫ �

t��r� x�ki�dr dx� ≤ '

∫�

∫ )t�x�

t��r� Xt�x

r �kt�xi �dr�dx

≤ '2∫�

∫ �

t��r� x�ki�dr dx�� (5.1)

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Reflected BSDE and Reflected PDIE 583

Set ki�dr dx� �= 1Ui�r�x�=0ki�dr dx�. Then

ki��r�×�� ≡ 0�

Moreover, from (I3) and (3.1) we have ∃C > 0 such that ∀n ≥ 0∫�

∫ �

0��n−m�2�r�x�ki�drdx�

≤'∫�

∫ )0�x�

0��n�r�X0�x

r �−m�r�X0�xr ��2�r�X0�x

r �1U 0�xi �r�=0k

0�xi �dr�dx<C�

Hence there is �i ∈ �2�0�� �×�� ki�dr dx��k�+m such that �n − �i

converges weakly to 0 in �2�0�� ��� ki�dr dx��k�, by consequence

�i�t� x� = ��t� x� = Y t�xt � 1�t �x�ki�dt dx� ∀i a�e�On the other hand, (I3) implies∫

∫ �

t�� − Si Ui��s� x�1�s �x�ki�ds dx�

≤ '∫�t

∫ )t�x�

t�� − Si Ui��s� Xt�x

s �1�s �Xt�xs �k

t�xi �ds�dx

= '∫�t

∫ )t�x�

t�� − �nUi��s� Xt�x

s �1�s �Xt�xs �k

t�xi �ds�dx

+'∫�t

∫ )t�x�

t��n�s� Xt�x

s �− Y t�xs Ui�s� Xt�xs ��1�s �Xt�x

s �kt�xi �ds�dx�

But,∫�t

∫ )t�x�

t�� − �nUi��s� Xt�x

s �1�s �Xt�xs �k

t�xi �ds�dx

=∫�

∫ �

t�� − �nUi��s� x�1�s �x�κ t�s� x�ki�ds dx��

for some bounded and measurable function κt �'−1 ≤ κ

t ≤ '�, then

limn−→

∫�t

∫ )t�x�

t�� − �nUi��s� Xt�x

s �1�s �Xt�xs �k

t�xi �ds�dx = 0�

From (3.5) we have ∀t ∈ 0�� �� ∀i = 1� � � � � q� ∀x ∈ �t�∀+ ∈�2

���× t� )t�x� �� kt�xi ��� ds�P�d���

k�∫ )t�x�

t�Y t�xs − �n�s� Xt�x

s � +�s��kt�xi �ds� −→n→ 0

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584 Hassani and Ouknine

Since∫� supn�Y t�xs − �n�s� Xt�x

s ��2kt�xi �ds�dx < (thanks to (3.1)) we have∀t ∈ 0�� �� ∀i = 1� � � � � q� ∀+ ∈ �2

��� × 0�� �×�� 1t�)t�x� �kt�xi ��� ds�

P�d��dx�k�∫�

∫ )t�x�

t�Y t�xs − �n�s� Xt�x

s � +�s� x��kt�xi �ds�dx −→n→ 0� (5.2)

We deduce that∫�

∫ )t�x�t

�� − Si Ui��s� x�1�s �x�ki�ds dx� = 0� Since�� − Si Ui��s� x�1�s �x� ≥ 0� ∀�s� x� ∈ 0�� �×�, we claim

�� − Si Ui��s� x�1�s �x�ki�ds dx� = 0�

Finally, we show by approximation (using assumption (L2) ) the part (iii)of theorem.

Remark 5.2. Let us point out that by using additional hypotheses onthe regularity of the data, we can obtain the following equality

�t = � for all t ∈ 0�� ��

Hence, we recover the results in Ref.2�.

6. APPENDIX: A SUFFICIENT CONDITIONTO OBTAIN (I3)

Let us make the following assumptions instead (I3) mentioned inSec. 3 �I′3i� For all " ∈ �

0 ���+�� s ∈ 0�� � there is � ∈ � 1�22 �0� s�

�d�k� such that

�i� ���t� x� = 0 a�e� �t� x� ∈ 0� s�×���ii� ��s� x� = "�x� ∀x ∈ ���iii� ��t� x� = 0 ∀�t� x� ∈ 0� s�×�c��iv�

∫� ��t� x�dx = ∫� "�Xt�x

s �1�t�xs ���dx ∀t ∈ 0� s��(I′3ii) There a measure ��dy� on �d such that for all t ∈ 0�� �,

x ∈ � there is a measurable function ,t�x�y� ∈ � �0�� �×�×�d�dt dx��dy��� which satisfy: For all � ⊂�d∫

E1��c�t� x� e�1c�t�x�e�=0��de�

=∫�,t�x�y���dy� and

∫�d

�y���dy� < �

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Reflected BSDE and Reflected PDIE 585

(I′3iii) Div !�t� x� ∈ �1�0�� �� ��� dx��� where

!�t� x� �= b�t� x�− a�t� x�−∫�dy,t�x�y���dy�

+∫ 1

0

∫�dy,t�x−(y�y�1��x − (y���dy��

Div stands for the divergence operator.

Theorem 6.1. Under assumptions (I′3i)–(I′3iii), condition (I3) holds.

Proof. From (I′3i) we have for all t ∈ 0�� �∫�

��

�t�t�x�dx

=−∫�

{�b−a− c����t�x�

+∫E��t�x+c�t�x�e��−��t�x���de�

}dx

=−∫�

{�b−a− c����t�x�+

∫�d��t�x+y�−��t�x���dy�

}dx

=−∫�

{�b−a− c����t�x�

+∫�d

∫ 1

0����t�x+(y�y�×,t�x�y�d(��dy�

}dx

=−∫�

{�b−a− c����t�x�

+∫�d

∫ 1

0����t�x�y�1��x−(y�,t�x−(y�y�d(��dy�

}dx

=−∫��b−a− c+

∫�d

∫ 1

01��x−(y�,t�x−(y�y�yd(��dy�����t�x�dx

=∫�Div!�t�x���t�x�dx�

Then for all k ∈ �1�0�� ����� Itô formula gives∫�"�x�dx exp

∫ s

tkr dr

=∫���t� x�dx+

∫ s

texp∫ �

tkr dr

∫��Div !��� x� + k������ x�dx d��

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586 Hassani and Ouknine

Letting k = ess supx∈��Div !��� x�� we claim

'−1∫�"�x�dx ≤

∫�"�Xt�x

s �1�t�xs ���dx ≤ '∫�"�x�dx�

with ' = exp∫ �0 kr dr� By an approximation argument, we conclude the

proof in same manner as in Ref.3�.

Remark. (1) As in Ref.9�, one can make assumptions on � b� c and �to obtain (i), (ii) and (iii) of (I′3i). Condition (iv) is a direct applicationof Itô formula.

(2) (I′3ii) is similar to the condition given in Ref.9�.

(3) (I′3iii) is more general than the condition in Ref.2� (since theauthors consider the case c = 0 (i.e. ,t�x�y� = 0� ��dy� = 0).

REFERENCES

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2. Bally, V.; Caballero, M.E.; Fernandez, B. Reflected BSDE’s, PDE’sand variational inequalities. Preprint, 1999.

3. Bally, V.; Lesigne. E. SDE, BSDE and PDE. In Backward StochasticDifferential Equation; El Karoui N., Mazliak L., Eds.; PitmanResearch Notes in Math. Series, Longman, 1997; 364.

4. El Karoui, N.; Kapoudijian, C.; Pardoux, E.; Peng, S.; Quenez,M. C. Reflected solutions of backward SDE’s and related obstacleproblems for PDE’s. Preprint, 1995.

5. Hassani, M.; Ouknine Y. On a general result for backward stochasticdifferential equations. Preprint, 2000.

6. Pardoux, E. BSDEs and viscosity solutions of a systems of semilinearparabolic and elliptic PDEs of second order. In Stochastic Analysisand Related Topics VI: The GeiloWorkshop, 1996; Decreusfond, L.,Gjerde, J., Øksendal, B., Üstünel A. S., Eds.; Birkhäuser, 1998;76–127.

7. Pardoux, E.; Peng, S. Adapted solution of a backward stochasticdifferential equation. System Control Lett. 1990, 14, 55–61.

8. Pardoux, E.; Peng, S. Backward SDEs and quasilinear PDEs.In Stochastic Partial Differential Equations and Their Applications;Rozovskii, B. L., Sowers, R., Eds.; LNCIS 176, Springer, 1992;200–217.

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Reflected BSDE and Reflected PDIE 587

9. Rong, S. On solutions of backward stochastic differential equationswith jumps and applications. Stochastic Processes Appl. 1996, 66,209–236.

10. Walsh, J.B. An Introduction to Stochastic Partial DifferentialEquations. In Ecole D’Eté de Saint Flour XIV; Lecture Note in Math.,1180, 265–438.

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