I am an applied mathematician focusing on stochastic analysis and its applications, specifically in the topic of backward stochastic differential equations and their applications to the homogenization of partial differential equations.

During the last decade, the theory of backward stochastic differential equations took shape as a distinct mathematical discipline. This theory has found a wide field of applications as in stochastic optimal control and stochastic games (see Hamadµene and Lepeltier [40]) and at the same time, in mathematical finance, the theory of hedging and non-linear pricing theory for imperfect markets (see El Karoui and Peng and Quenez [27]). Backward stochastic differential equations also appear to be a powerful tool for constructing Gamma-martingale on manifolds (see Darling [22]) and they provide probabilistic formulae for solutions to partial differential equations (see Pardoux and Peng [67]).
Pardoux and Peng have established the existence and uniqueness of solutions of backward stochastic differential equations under uniform Lipschitz condition on the coefficeint. This asumption is usually not satisfied in many problems, for example in finance. So it is important to find weaker conditions than the Lipschitz one under which the BSDE has a unique solution and the question now is: Are there any weaker conditions than the Lipschitz continuity under which the BSDE has a unique solution?
Since the result of Pardoux and Peng [67], several works have attempted to relax the Lipschitz condition and the growth of the generator function, see Lepeltier and San Martin [49], Hamadène [39], Dermoune et al [24], Barles and Kobylanski [44], N'zi [58] and N'zi-Ouknine [59]. Most of these works deal only with real-valued BSDEs and the terminal condition is bounded because of their dependence on the use of the comparison theorem for BSDEs, the uniqueness does not hold in general. Furthermore, the multidimensional case is also studied even though the comparison theorem does not hold. However, in general, the existence and uniqueness results are obtained only under weaker condition with respect to Y and Lipschitz with respect to Z (see Bahlali et al [6], Briand and Carmona [17], Darling and Pardoux [23], Hamadène [38], Mao [53] and Pardoux [64]). Let us mention nevertheless an exception: in [3], Bahlali has established an existence and uniqueness result for the solution of BSDEs (without reflection) under locally Lipschitz condition with respect to y and z.
In our works, we provide some answers to the question we have raised above. First, when the coefficient is locally monotone with respect to Y and locally Lipschitz with respect to Z. Second, when the coefficient can be neither locally Lipschitz in the variable Y nor in the variable Z (see Puplications in our homepage).
At the same time, we are also intersted to the existence and uniqueness of solutions of some kind of BSDE, for example reflected BSDEs driven by a Brownian motion or reflected BSDEs driven by a Brownian motion and an independant Poisson point process, under some appropriate hypothesis.

Here is a talk on the existence and uniqueness of solutions of BSDE under different conditions (Barcelona, 2006).

 In [12], Bensoussan et al. studied the homogenization of linear second order partial differential operators using a probabilistic approach, based upon the linear Feynman-Kac formula. They left the question of studying the nonlinear case by the probabilistic method as an area open to investigation. Recently, Pardoux and Peng [67, 68] have generalized the Feynman-Kac formula to take into account semi-linear PDE's. This generalization is based upon the theory of backward stochastic differential equations.
It is then by now well known that systems of parabolic semi-linear are closely related to BSDE's. From the knowledge of BSDE's, we can derive some results on systems of semi- linear PDE's (see Pardoux and Peng [67], [68]). This correspondence reduces Bensoussan et al question to a question of stability of BSDEs. This last idea has been used in Pardoux, Veretennikov [71] to give averaging results for semi-linear PDEs where the nonlinear term is a function of the solution and not depend on the gradient, in Pardoux [65] and Ouknine, Pardoux [73] to prove homogenization property for a system of semi-linear PDEs of parabolic type, with rapidly oscillating periodic coeffecients, a singular drift and a singular coeffecient of the zero-th order term. Furthermore, let us recall that other homogenization results have been proved by Buckdahn and al. [18], Gaudron, Pardoux [37], and Lejay [48] where a divergence operators has been involved. On the other hand, from the knowledge of systems of semi-linear PDE's, we can derive some results on BSDE's (see Ma et al. [52] for more details). Now the question is: How to obtain homogenization results for semi-linear variational inequalities and for semi-linear PDE's with singular coeffecient?
In our works, we provide also some answers to this question. We prove some homogenization results for semi-linear PDE's by using an approach based upon the nonlinear Feynman-Kac formula developed in [74] and [68]. This gives a probabilistic formulation for the solutions of systems of semi-linear PDE's via the BSDE's. The problem then reduces to study the stability properties of some kind of BSDE's.

References

[3] K. Bahlali, Backward stochastic differential equations with locally Lipschitz coeffecient, C.R.A.S, Paris, serie I Math. 331, 481-486, (2001).
[6] K.Bahlali, B. Mezerdi, Y. Ouknine, Some generic properties in backward stochastic differential equation. Monte Carlo 2000 conference at Monte Carlo, France, 3-5 jul. 2000. To appear in Monte Carlo Methods and Applications, (2001).
[12] A. Bensoussan, J.L. Lions, G. Papanicolaou : Asymptotic analysis for periodic structure, North Holland, (1978).
[17] P. Briand, R. Carmona, BSDEs with polynomial growth generators, J. Appl. Math. Stochastic Anal. 13, 207-238, (2000).
[18] R. Buckdahn, Y. Hu, S. Peng, Probabilistic approach to homogenization of viscosity solutions of parabolic PDEs, Nonlinear Differential Equations Appl. 6, 395-411, (1999).
[22] R. W. R. Darling, Constructing Gamma-martingales with prescribed limit, using back- wards SDE. Annals of Probability, 23, 1234-1261, (1995).
[23] R. Darling, E. Pardoux, Backward SDE with monotonicity and random terminal time, Ann. of Probab. 25, 1135-1159, (1997).
[24] A. Dermoune, S. Hamadène and Y. Ouknine, Backward stochastic differential equation with local time. Stoc. Stoc. Reports. 66, 103-119, (1999).
[27] N. El Karoui, S. Peng and M.C. Quenez, Backward stochastic differential equations in finance. Mathematical Finance. 7, 1-71, (1997).
[37] G. Gaudron, E. Pardoux, EDSR, convergence en loi et homogènèisation d'EDP paraboliques semi-linèaires, Ann. Inst. Henri Poincaré, Probabilitès-statistiques, 37, 1-40, (2001).
[38] S. Hamadène, Multidimentional Backward SDE's with uniformly continuous coeffecients. Submitted.
[39] S. Hamadène, Equations differentielles stochastiques rétrogrades, le cas localement lips- chitzien. Ann. Inst. Henri Poincaré. 32, 645-660, (1996).
[40] S, Hamadène, J.P, Lepeltier, Zero-sum stochastic differential games and BSDEs, Systems and control letteres, 24, 259-263, (1995).
[44] M. Kobylanski, G. Barles, Existence and uniqueness results of backward stochastic differential equations when the generator has a quadratic growth, Université de Tours, (1996).
[48] A. Lejay, Approche probabiliste de l'homogénéisation des opérateurs sous forme diver- gence en milieu périodique. Thµese de doctorat (Marseille, 2000).
[49] J-P. Lepetier, J. San Martin, Backward stochastic dfferential equations with continuous coeffecients, Statist. Probab. Lett. 32, 4, 425-430, (1997).
[52] J. Ma, P. Protter, J. Yong, Solving forward-backward stochastic differential equations explicitly: a four step scheme. Probab. Theory Rel. Fields, 98, 339-359, (1998).
[58] M. N'zi, Multivalued backward stochastic differential equations with local lipschitz drift. Stochastic and Stoch. Reports 60, 205-218, (1998).
[59] M. N'zi, Y. Ouknine, Multivalued backward stochastic differential equations with con- tinuous drift. Rando. Oper. Stoch. Equations, 5, No. 1, 1-104, (1997).
[65] E. Pardoux, Homogenization of linear and semilinear second order parabolic PDEs with periodic coeffecients: a probabilistic approach, J. Funct. Anal. 167, 498-520, (1999).
[67] E. Pardoux, S. Peng, Adapted solution of a backward stochastic differential equation. System Control Lett. 14, 55-61, (1990).
[68] E. Pardoux, S. Peng, Backward stochastic differential equations and quasilinear parabolic partial differential equations. In: B. L. Rozovski, R. B. Sowers (eds). Stochastics partial equations and their applications, (Lect. Notes control Inf. Sci. 176, 200-217), Springer, Berlin, (1992).
[73] E. Pardoux, Y. Ouknine, Homogenization of PDE's with non linear boundary condition, to appear in Proc. Conf. Ascona, (1999).
[74] S. Peng, Probabilistic interpretation for systems of quasilinear parabolic equations. Stochastics, 37, 61-74, (1991).