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Multivalued backward stochastic differential equations with locallipschitz driftModeste N'zi aa University de Cocody , Departement de Mathématiques , 22 BP 582 Abidjan22, COTE d'lVOIREPublished online: 04 Apr 2007.

To cite this article: Modeste N'zi (1997) Multivalued backward stochastic differential equations with local lipschitz drift, Stochastics and StochasticReports, 60:3-4, 205-218

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MULTIVALUED BACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS WITH

LOCAL LIPSCHITZ DRIFT

MODESTE N'ZI* **

Kecencd I I 4pr i l IYY6: In find jorm 10 November 1996)

We deal with a one dimensional multivalued backward stochastic differential equation associated to the subdifferential 13h of a lower semi-continuous convex function h, with a local lipschitz coefficient (drift). When the terminal value is bounded. we prove the existence of a solution by using a suitable approximation of the drift by a double sequence of lipschitz functions. The uniqueness is obtained under the ccndition that the drift is local Lipschitz in y and globally Lipschitz in z. The existence result is an extension to the multivalued setting of the work of Hamadtne [3].

Kejwords: Backward stochastic differential equations; convex function; maximal monotone operator

AMS 1991 Subject Clussificutions: 60H10. 60H20

1. INTRODUCTION

Backward stochastic diffcrcntial cquations (in short, BSDE) have been introduced by Pardoux and Peng [7]. This kind of equation has been thoroughly studied and has found applications in finance (see e.g. El Karoui and Peng and Quenez [2]), in btochastic: control and difI'erentia1 games (see e.g. Hamadkne and Lepeltier [4]). The existence (and uniqueness) result for BSDE was first obtained under the assumption that the drift is globally

*Part of this work was done during a staying of the author at UniversitC d'Evry Val d'Essonne, whose generous support is gratefully acknowledged.

**E-mail: nziy@ci.refer.org. Fax: 225 44 83 97.

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Lipschitz. Later on, many authors have weakened this condition. For example, recently Hamadene 131 considered the case of a iocal Lipschitz coefficient. The aim of this paper is to prove a multivalued analogue of his resiiirs, giving simuitaneotisij: an exieiisiu:: ui' L~:;.x 1:1 G u ~ I ~ ~ K [6j.

The paper is organized as follows: in section 2 we formulate the problem. Sect~on 3 IS devoted to the remits.

2. FORMULATION OF THE PROBLEM

Let [.I7- ( W r : t '> O) be 2 d-dimensional standard Brownian motion ,.i,fi ,,,. 3 .. ,,;; &.-.i-:i:+-. E*- ..-. iii -' . r b , l r . \ l . . i i i i & ,, ..,. i & i i i i i i V jji. 2 . Vj. {F, : t 2 u j stands or ihe naiurai nitration of E-aug~r~er~icd ~ i i h iiie r"-iiiiii sets of F. Let h : M+(-,m. tmj be a iower semicontinuous convex funciion, which is proper, that is, the interior of the sel

is nonempty. We denote by dh the subdifferential of h, which is a multivalued maximal monotone operator defined by its graph:

We consider the muitivalued backward stochastic differentiai equation associated to h, j; < (in short, MBSDE ( h , f , E ) )

where

T > 0 is a terminal time; 0 < is a final value which is a F,-measurable random variable such that

E < ~ < +oo and with values in Dorn (h) ; f : R x [0, TI x iW x W+[W is a coefficient (drift) such that

(i) b'y, z,.f (., y, z ) is Fl-progressively measurable;

(ii) EJ,' (J.(t, 0, 0)12dt < +m;

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MULTlVALUED BSDEs 207

(iii) VN 2 0,3 CN > 0, such that Vw, t , ( y , z ) E [ - N , NId'", (y', 2') E [ - N , yd+'

(iv) 3a E 10.2). 3C > 0, 3 g : R-R, bounded on compact sets such that

{ ( Y t , Z,, Kt) : t E [0, TI) is a Ft-progressively measurable process with values in R x Rd x R satisfying:

(a) ~ ( s u ~ ~ < , g l Y~I ' + ~ ~ / ~ ~ l * d . r ) < +m

(b) { Y , : t E 10, T 1 ) is continuous and takes values in Dom(h)

(c) {Kt : t t~ [O, T J ) is continuous and has bounded variation with -KG = ~ U . S .

(d) Fnr any optional process ( (or,$,) : r f f'3. TI) with va!ues in Gr(ah), the measure (Y, - a,)(dK, + ,dtdt) is as . negative.

( Y , Z , K) is called a solution of the MBSDE (h , f , J ) .

3. RESULTS

The fvllowing comparison lemma will be useful in the construction of a solution.

LEMMA 1 k t f l , f 2 be two functions in (y ,z ) uniformly Lipschitz with respect to (w, t ) , h' , h2 be two lower semicontinuous convex functions which are proper, and J 1 , J2 be two square integrable random variables with values in Dom(hl) and Dom(h2) respectively. Assume that

(i) V(x , y) E Dom(hl) x ~ o r n ( h ~ ) with x < y

1z1(y) - h1 (x) 2 h2(y) - h2(x);

(ii) 5 J2 a.s.;

(iii) f l ( t , Y:,z,') - f 2 ( t , Y:,z:) 5 0 a.s.

Let ( Y i , Z', Ki) , i = 1,2, be a solution of the MBSDE (hi, f', c). Then we have

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Proof Recalling that the domain of h' is Dom (h') = {x E R : hi(x) < tx), we easily see that there is nothing to prove if the interior of ~ o m ( h ' ) n ~ o m ( l t ~ ) is empty. Hence, suppose the contrary and pick a point ,3 in this set. Let us put

and let (h'): be the right derivative of hi. We have (a,, (hi):(a,)) E Gr(dh). Therefore

which implies that

Now, since f'(s, Y : , z ~ ) < f 2 ( s , Y ,~ , z , ' ) as., we have

where k is a Lipschitz constant for f'.

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It follows that

Y-. I u+, thc end ~ ? f thc proof follows l ic i~i i Grcinwilil's Icrnrna. 0

We will also need thc following result of Saisho [9].

L E M M A 2 Let { k n : n > 0) and { y n : n > 0) be t w o sequences in C([O, TI, R') which convrrge untf2jrrnly to k und .v respectivel-y. Assume rhut !cn has hounded variu:ion, und :ha;

whcrc j j . standk$ fur the total ariati ti on on [0, TI. Thcn k !OF hounded vcrrbrhn, ~ l n d

In the sequel, C > 0 is a generic constant not necessarily the same at each occurrenee, We begi!-: i.iiiii: I!::: cxi:,!r:;ct:

Proof k t !QN . n 2 0) ( r ep . (9"' : .E 2 0)) he m increasing l r p c n r - -..Y.

decreasing) sequence of Lipschitz functions with pointwise limit f ' Q { f 2 0 )

(resp. f Q c~<o}). Let us set @" = an + !Pm. For any n, nz 2 0, q,"' is Lipschitz and we have

We also have

Now, let (Y.", zn.", ZPn') be the unique solution of the MBSDE (h , qPm, () given by the results of Oukninc 161. The rest of the proof IS divided into three steps.

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210 M. N'ZI

Step 1 Let us prove that

Let (cr,p) E Gr(dh). By virtue of Itb's formula, we have

,;llL,L 6@."' is bounded, by virtue of Girszmv's themem (see [5. p.!9!j) there exists a prohahility measure P n i m on (R; 3); equivalent to P: with Radon- Nikodym derivative

under which the process Wn,m = W - ~ , 6 4 " ~ " ( s , Y;,", Z;>")ds is a 3,- Brownian motion. Now, we have

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MULTIVALUED BSDEs 21 1

It is easy to see that three exists C > O (independent of n, m) such that

in view of the inequality (Y:," - a:)(dk-?," + Oris) < O a.s. and the boundness of J, we deduce that there exists C > O (independent of n, nz: s-uch that

Let us denote by En,, the expectation operator under Pn>". Since Jd Z;~"(Y:~" - a)dW:lm is a 3,-martingale under Pn3", we have

Now, let u < t . Then

By virtue of Gronwall's lemma, we deduce that

En.,(/ Y:'" - a12/3,) < C exp(C(T - t ) )

For u = t, we obtain

(Yylm - a12 < C exp(CT),

which ends the proof of (i) since C is independent of n, m.

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212 M. N'ZI

By virtue of (2) and since Jd Z~." ' (Y ,~~" ' - a)dW, is a 3,-martingale under P, u-e have

From t h ~ negativity of the measure (Y:" - a)(rlK,"?" + 0d.v). we obtain

Now, the boundness of the double sequence { Y","' : n , m > 0 ) and the properties of 4"3" imply that there exists C > 0 (indepdendent of n, m) such that

By virtue of Young's inequality

we deduce that

Therefore. (ii) follows from the fact a E [0,2j.

Step I1 Let us prove that (Yn>", Znsm, Kn9") converges in some sense to a process ( Y, Z, K). By virtue of the comparison Lemma 1, {Ynvm : n 2 0) is an increasing sequence. Since it is bounded, this sequence admits a poinwise limit Y m . Again, the comparison Lemma 1 implies that { Y m : m > 0) is a decreasing

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MULTIVALUED BSDEs 213

sequence. Since it is also bounded, there exists a process Y which is a pointwise limit of this sequence. Now, let n, m, k , p > 0. E y virtue of 116's formula, we have

Proposition 4.3 in Ckpa 111 implies that

On the other hand, the expectation of the stochastic integral is zero. Hence, by using the boundness of the double sequence {Yn>" : n, m 2 0) and ( I ) , it is not difficult to see that there exists C > 0 (independent of n, m , p , k) such that

Let

Holder's inequality implies that

where r = 2/(2 - a) .

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214 M. N'ZI

Now, let {Ykm," : m 2 0) be a subsequence of {Yn." : n, m > 0) converging pointwise to Y. By virtue of the Lcbesgue dominated convergence theorem, Ykm,m converges to Y in LS([O, T ] x fl:dt @ dP). It follows that {Zkm." : m > 0) is a Cauchy sequence in ~ '(10, TI x 0. dt @ d F ) . Therefore there exists Z such that z k m 3 " converges to Z in L*([o. T ] x f l ,

dt '8 dP). So we have

By the same arguments as in Hamadene [3]: one car, prove that

-

+ 2 ~ ( ,I,.,7- sup / /'(Z?," - z > . P ) ( Y ~ ~ ~ - y2g )dWy i ) \ .q ,> .> , ! ., :: 4,

In view of the Burkhoider-Davis-Gundy inequality and the boundness of {Y"," : n, m > 0), there exists C > 0 (independent of n , m) such that

It follows that { Y k ~ ~ " ' : m > 0) is a Cauchy sequence in ~ ' ( 0 ; C([O, TI, R ) ) . Hence

lim E sup / Y ? " - Y , ' ) = O "-tea ( O<t<T

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MULTIVALUED BSDES

Now, let us put

(2)-(5) imply that

sup / K ? , ~ - & 1 2 ) = 0 .

Step I11 Let us prove that (Y, 2, K ) is a solution of the MBSDE (h , f , [). It is clear that (a ) is satisfied. Since for any tn 2 0; Y : ' " * ~ E Dorn(h) o.s., we deduce that

The continuity of I' foiiows from the uniform convergence of ykrn," to Y, .-~1.1.1. t .. 1 wrlicr~ ledus io (bj . <-;.,ice jy k"." -A& ccnvergez u n i f ~ m j y :o K, it foiiows that K is c~fiiiiiiioii~. Now, let us prove that K has bounded variation. To this cnd, we adopt a simi!ar argument as in the proof ofProposition 4.2 in Pardoux and Rascanu [8]. We put

We may supijosc =ithorn lost of generality i'riai ili beiongs to the interior of Dom(8h). There exists r > 0 such that for any / z / 5 1, rz is an element of the interior of Dom(3h) and

where

A" ( x ) = Projahix) ( 0 ) .

For any process {U, : t > 0 ) such that Vt E [0, TI, I U,( 5 1 we have

We deduce that

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216

where

M. N'ZI

Since

by virtue of (I), the boundness of the sequence {r" : m 1 01, and

we have

which implies that K has bounded variation. Therefore, (c) is verified. Since for any fixed m 2 0, (r",Z",I?) is the solution of the MBSDE(h, Jm,[), for any optional process (U, V) with values in Gr(ah), we have

By letting m--, + m, using the above estimates and Lemma 2, we deduce that for any 0 j a j b 5 T

in probability, which leads to (d).

Now we deal with the uniqueness of the solution.

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MULTIVALUED BSDES

then the MBSDE(h, f , 6) has at most one solution.

Proof Let ( Y, Z, K) and ( Y ' , Z ' , K ' ) be two solutions and put

in vizw of 1~6 ' s formula, we have

The conditions on f and the fact that ( Y i - Y:) (dK, - dK;! < O a.s.. show that there exists C, > 0 such that

By using classical arguments based on the inequality

and Gronwall's iemma, we obtaln

Now, Fatou's lemma leads to Y! = 1'; as . . It follou~s that Y = Y ' : Z = Z', and K = K' a s . 0

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M. N'ZI

References

[l] Cepa, E. (1996). Equation different~elles stochastiques multivoques. Lecture Notes Math., 1613, 86-10?,

iij Ei -carou:, x 7 - c. < :u., reng, 2. a l x QLICI~CL, :V;.C. Rifiii;iA ~~:ii:i;~ ~f ha~k: ;~: ! SEE':: Backward stochastic differential equations, Finance and Optimization (submitted).

[3] Hamadene, S. (1996) Equations differentielles stochastiques retrogrades: le cas localement lipschitzien. Ann. Inst, Henri Poincare, 32(5), 6455659.

[4] Hamadtne, S. and Lepe!!ie:, J.P. Z e r ~ sgm stochastic differentia! games and backward equations. To appear in System and Control Letters.

[5] Karatzas, I. and Shreve, S. (1988). Brownian Motion and Srochastic Calculus. Springer. [6] Ouknine, Y. Multivalued backward stochastic differential equations (submitted). [7] Pardoux, E. and Peng, S. Adapted solution of backward stochastic differential equation.

System and Control Letters, 14, 55-61. [8] Pardoux. E. and Rascanu, A. Backward stochastic differential equations with maximal

monotone operators (submitted). 19j Saisho, Y. (1987). Stnchatx differential eqiiationl for mulridimensional domains with

reflectin2 boundary. Probab. Theorv Rel. Fields, 74, 455477.

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