Multidimensional backward stochastic differential equations with uniformly continuous coefficients

This type of equation, at least in the nonlinear case, was first introduced by Pardoux and Peng (1990a), who proved the existence and uniqueness of a solution under suitable assumptions on f and ?, the most important of which are the Lipschitz continuity of f and the square integrability of ?. Their aim was to give a probabilistic interpretation of a solution to a second-order quasilinear partial differential equation. Since then, these equations have gradually become an important mathematical tool in many fields such as financial mathematics (see, for example, El-Karoui et al. 1997a; 1997b; Buckdahn and Hu 1998; Cvitanic and Karatzas 1996), stochastic games and optimal control (Hamadene and Lepeltier 1995a; 1995b; Hamadene et al. 1997; 1999; Cvitanic and Karatzas 1996; Dermoune et al. 1999), partial differential equations and homogenization (Pardoux and Peng 1990b; 1992; Pardoux 1999; Peng 1991; Darling and Pardoux 1997; Buckdahn and Peng 1999) and construction of F-martingales (Darling 1995). A further problem under widespread discussion is how to improve the existence and uniqueness result of Pardoux and Peng (1990a) by weakening the Lipschitz continuity condition on f. Hamadene (1996), Kobylanski (2000) and Lepeltier and San Martin (1997; 1998) have dealt with the situation where Y is a unidimensional process. They obtained an existence result without assuming f Lipschitz continuous. However, the solution is not necessarily unique. Hamadene (1996) takes f to be just locally Lipschitz, while Kobylanski

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