A REMARK ON GENERALIZED FEYNMAN-KAC FORMULA

In this Note, we are interested in non linear PDE's of kind (1), A being the infinitesimal generator of a diffusion and f the generator of a backward stochastic differential equation. Such problems have been studied by Peng [5] and Pardoux-Peng [4] who give a probabilistic interpretation of solutions, assuming that the coefficients of A are Lipschitz, by studying the diffusion of generator A and rite backward stochastic differential equation of generator f. We extend this result to the case of locally Lipschitz diffusion's coefficients, assuming nevertheless the existence of a Lyapunov function, h, to obtain non-explosion of the diffusion process [3]. We obtain a generalization of Feynman-Kac formula and a result concerning existence and uniqueness of viscosity solutions of h-polynomial growth order.