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We prove that in the sense of Baire category, almost all BSDE with bounded continuous coefficients have a unique Solution. Introduction. It is well known that the backward stochastic differential equations (BSDE in short) with globally Lipschitz coefficient have unique Solutions, see e.g. [PP1, EKPQ]. When the coefficient is only continuous, this problem remains open in dimension greater than one. The technics used in dimension one to treat it are specific to R and do not work in R*, d > 1. For this subject see e.g. [PP2, LSM, DHO] and the references therein. Note also that a good lecture in BSDE with applications to control and finance is given in [EKPQ]. Our goal is to prove that most BSDE with boundned and continuous (in both variables y and z) coefficient have unique Solutions. That is in the set of continuous functions endowed with an appropriate metric, the subset of those coefficients, for which the corresponding BSDE has a unique solution, is of the second category of Baire (see [Ox] for the categories of sets). Recall that the subset of Lipschitz functions is of first category, see e.g [AB]. I. Notations. Let (Wt)Q<t<i be a r-dimensional Wiener process defined on a probability space (Ω, ,Τ, P). Let (Ft)Q<t<i denotes the natural filtration of (Wt), where FQ contains all P-null sets of T. Let ξ be an .^-mesurable if-dimensional square integrable random variable and / be a bounded measurable function process defined on 1R+ χ Ω χ JR χ IR with values in JR such that for all (y,z) € H x R, the map (t}u) —> /(£,u;,y,z) is Ji-progressively measurable. We are concerned with the BSDE of the following form Yt = ζ + / /(*, Y„ Z3)ds t Z,dW8 (0 < t < 1) (E') Jt Jt 'Convention CNCPRST Maroc/CNRS France, projet Nf 831

Some generic properties in backward stochastic differential equations with continuous coefficient