PRORABILITY .
AND MATHEMATICAL STATISTICS
.
BACKWARD STOCmSTHC NONLINEAR VOILTERRA IiTTEGRBL EQUATIONS WITH LOCAL LTPSCHITZ DRIFT BY
AUGUSTE AM AN
AND
MODESTE M'ZI (ABIDJAN, COTED'IVOIRE)
Abstract. In this paper, we study backward stochastic nonlinear Volterra integral equations. Under a local Lipschitz continuity condition on the drift, we prove the existence and uniqueness result. We also establish a stability property Tor this kind of equations.
AMS Subject Classification: 60H20, 60H05. . Key words and phrases: Backward stochastic differential equation; Volterra integral equation; adapted proms.
1. INTRODUCTION
A linear version of backward stochastic differential equations (BSDE's in short) was first considered by Bismut ([7], 181) in the context of optimal stochastic control. Nonlinear BSDE's have been independently introduced by Pardoux and Peng [23] and DdEe and Epstein [lo]. These equations were intensively investigated in the last years. The main reason for this great interest in these equations is because of their connections with many other fields of research such as: mathematical finance (see El Karoui et al. [13]), stochastic control and stochastic games (see Hamadhe and Lepeltier 1171). These equations also provide probabilistic interpretation for solutions to both elliptic and parabolic nonlinear partial differential equations (see Pardoux and Peng [24], Peng [26]). Indeed, coupled with a forward SDE, such BSDE's give an extension of the celebrate Feynman-Kac formula to the nonlinear case. The classical condition on the drift for proving the existence and uniqueness result is a global Lipschitz one. Many authors have attempted to relax this condition. For instance, several works treat BSDE's with continuous or local Lipschitz drift (see Hamadlne [15], [16], Lepeltier and San Martin [20], N'Zi and Ouknine 1221 and the references therein). In the one-dimensional case, the essential tool is the comparison-theorem technique. In the multidimensional case, the improvements of the Lipschitz condition on the generator concern,
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A. Aman and
M. N'Zi
generally, the variable y only and the conditions considered are global. It seems that the first works treating multidimensional BSDE's with both local conditions on the drift and only square-integrable terminal data are Bahlali 121, 131. This author considered BSDE's with locally Lipschitz coefficients both in y and z. This study has been continued by Bahlali et al. [4], Aman and N'Zi [I] and Essaky et al. [14]. Recently, backward stochastic nonlinear Volterra integral equations (BSNVIE's in short) have been studied by Lin [21] under the global Lipschitz condition qn.the drift. His work is a continuation of a previous one of Hu and Peng rA8-j where backward semilinear stochastic evolution equations with values in a complete separable Hilbert space have been considered. More precisely, Lin [21] gives an existence and uniqueness result for the following nonlinear BSDE of Volterra type:
On the other hand, ordinary stochastic Volterra integral equations have been investigated by Berger and Mizel [S], [dl, Pardoux and Protter [25], Protter [27], Kolodh 1191 and have found applications in mathematical finance (see [9] and [l I]). In this paper, we are concerned with equation (1.1) and our aim is to weaken the global Lipschitz condition on the drift to a Iocal one. The paper is organized as follows. In Section 2, we give essentia1 notions on backward stochastic nonlinear Volterra equations and Section 3 deals with the main result. Finally, Section 4 is devoted to a stability result. 2. ASSUMPTIONS AND FORMULATION OF THE PROBLEM
(~),,,,,, P) be a fiItered probability space satisfying the Let ( 0 , usual conditions and ( W(t),t E LO, T I ) the &dimensional standard Brownian motion defined on it. Define 3 = {(t, s) ER; ; 0 < t < s < T ) and denote by 9 the &algebra of 9, ,,-progressively measurable subsets of 0 x 9. Let M2(t, T; Rk) (resp. M2(9;Rkxd))be the set of Rk-valued (resp. Rk""-valued), &,,-progressively measurable processes which are square-integrable with respect to P@A@L (here R denotes Lebesgue measure over [0, TI). For X E Rk, 1x1 will denote its Euclidean norm. An element Y E Rkx will be considered as a k x d-matrix; its Euclidean norm is given by IYI = JTI (Y Y*) and (Y, Z) = Tr (YZ*). gkstands for the Bore1 G-algebra of Rk. Moreover, we are given the following objects and assumptions: (Al) f : SZ x 9 x Rkx RkX d tion satisfying:
-+ Rk is
a (9@gk@&lkX d/&?k)-measurabIe func-
Volterra integral equations with Lipschitz drift
107
(ii) there exist two constants K > 0 (K sufficiently large) and 0 < a < I such that If ( t ,s, y, z)l < K (1 lyl lzly, P-as., a.e. (t, s)E 9;
+ +
(iii) for every N EN, there exists a constant L,
3
0 such that
for all (yl G N , ly'l < N , for all (t, s) E 9, z E Rk * d, Z' cz Rkxd, where K is' the constant iw(A1)' (ii). -- . b
function which (A2) g : i2 x 9 x Rk + Rk is a ( 9 Q B k / g k d)-mea~~rable satisfies: (i) g -,0) E M Z(9; Rk (ii) 19 (t, s, y)-g (t, s, y')l d K ly -y'l for all y, y' E Rk and for all (t, s)E B7 where K is the constant in (Al) (ii). (m,
(A3) 5 is a square-integrable k-dimensional PT-measurable random vector. R e m a r k 2.1. Note that ( A l ) and (A2) imply
~ ( ' , - , Y ( . ) , Z C . , - ) )and EM~( ~~ ( .;; ,RY~()- ) ) E M ~ ( ~ ; R ~ ~ * ) whenever Y E M2(t, T ; Wk), Z E M 2 (9;R k x
DEFMI~ON 2.2. A solution to BSDE of Volterra type with data ( 5 , f, g) is a pair of &,,-adapted processes {(Y(s), Z (t, s)); (t, s)E g ) with values in M2(t, T; Rk) x M 2 (9;Rk d, which solves (1.1). 3. EXISTENCE AND UNIQUENESS
Before stating the main result, let us give some preliminaries.
LEMMA3.1. Let f denote a process satisfying assumption (Al). Then there exists a sequence of processes (JJnal such that, for euery n > 1, fn is Lipschitzian, satisfies (A1) (i), (A1) (ii) and ( 9 BkQBk d/9?k)-mea~~rable, Q N (fn -f ) -' 0 as a + ao for every fixed N, where
+
Proof. Let $, be a sequence of smooth functions with support in the ball B (0,la + 1) such that $, = 1 in the ball 3 (0, n) and sup $, = 1. One can easily of truncated functions defined by f, = f $, show that the sequence (jJnal satisfies all the properties quoted above. B Let (fn),31 be associated with f by Lemma 3.1. By the results of Lin [21], for every n 2 1, there exists a unique couple of processes ((Y,(s), Z,, (t, s)): (t, s) E 91,an element of M 2 (t , T; Rk)x M2(53; Rk d, solution to the BSDE of
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A. Aman and M. N'Zi
Volterra type with data (5, f,, g). We build the unique solution to equation (2.1) by studying convergence of the sequence {(Ym(s),Z n ( t ,s): (1, s ) E ~ ) . LEMMA 3.2. Assume (AlHA3) hold true. Then there exists a constant C > 0, depending only on T , K and 5 such thal for every n 2 1 T
T
T
E [ I ~ ( s ) 1 ~ d s + ~ j d s ~u)12du l ~ , (< s ,C t
I
for ail ~ E [ T - q TI, ,
S
. .
where --rj < 1/24K2. ._ - Pr@of. Since {(Y, (s),2, ( t , s)): ( t , s) E 3 ) is the unique solution to the BSDE of Volterra type with data ( 0 (to be precised later). For each (t, s) E [T- q , T ] x [t , TI it follows from . . Lemma 2.1 of [21] that
Therefore we obtain
where
.
125
Volterra integral equations with Lipschitz dr$
For each N > 1, let us consider L,, the Lipschitz constant of f in the ball B ( 0 , N) of Rk, and put
The same procedure as in the proof of the existence part in Theorem 3.3 yields
Let us choose
8, = P2 = fi3 = 12K2, 8,
= 8,
and q < 1/24K2. If we define
T
U,,,(t) = E 1IY,(s)-
T
Yo (s)I2ds
and
K , o ( ~=) E
t
then we obtain
. .-
+4 ( 1 2 +~3)~E exp (K,s )
9 -
PAMS 25.1
IZn(t,s)-Zo ( t , s)12 ds, t
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A. A m a n and M. N ' Z i
+
+
+
where K, = 2 24K2 2L, 2L%,K,,K, and C are constants depending only on K, and to.The rest of the proof is identical to that of the existence part of Theorem 3.3.
.
--
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[la
.
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[I81 Y. H u and S. Peng, Adapted solution of backward stochastic evolution equation, Stochastic And. Appl. 9 (1991), pp. 445-459. [I91 A. M. Kolodh, On the existence of solutions of stochastic Volterra integral equations (in Russian), Theory Random Processes 11 (1983), pp. 51-57. [20] J. P. Lepeltier and J. San Martin, Backward stochastic diflerential equations with continuous coejicients, Statist Probab. Lett. 32 (4) (1997), pp. 425430 [21] J. Lin, Adapted solution of backward stochastic nonlinear Volterra integral equation, Stochastic Anal. Appl. 20 (1) (2002), pp. 165-183. [22] M. N'Zi and Y. Ouknine, Backward stochastic diferential equations with continuous coefficients, Random Operators and Stochastic Equations 5 (5) (1997), pp. 5948. [23] E. P a r d o u x and S. Pe-ng, Adapted solution of backward stochastic diflereatial equation, ~ ~ s t q m s ~ o nLett. t r d 4 (19901, pp. 55-61. [24] E. P a r d o Ax and S. P e n & Backward stochastic direrential equation and quasilinear parabolic partial dt@erentiaf equations, in: Stochastic Partial Equations and Their Applications, B. L. Rozovski and R. B. Sowers (Eds.), Lecture Notes in Control Inform. Sci. 176, Springer, Berlin 1992, pp. 2W217. [25] E. Pard o u x and P. P r o t ter, Stochastic Volterra equations with anticipating co&cients, Ann. Probab. 18 (4) (1990), pp. 163551655. 1261 S. Peng, Probabilistic interpretation for systems of quasilinear parabolic purtial diflerential equations, Stochastics 37 (1991), pp. 61-74. [27] P. P r o t t e r , Volterra equations driven by semi-martingales, Ann. Probab. 13 (2) (1985), pp 514-530.
Auguste Aman and Modeste N Z i Laboratoire de Mathimatiques Appliquies UFR de Mathkmatiques et Inforrnatique Universitk de Cocody 22 BP 582 Abidjan 22, Ccite d'Ivoire E-mail:
[email protected];
[email protected] Received on 20.5.2005
