Generalized BSDEs and nonlinear Neumann boundary value problems (Q1390765)

The authors provide probabilistic formulas for viscosity solutions of systems of semilinear partial differential equations (of parabolic or elliptic type), with nonlinear Neumann boundary condition [for similar programmes, see also \textit{S. Peng}, Stochastics Stochastics Rep. 37, No. 1/2, 61-74 (1991; Zbl 0739.60060), \textit{E. Pardoux} and \textit{S. Peng}, in: Stochastic partial differential equations and their applications. Lect. Notes Control Inf. Sci. 176, 200-217 (1992; Zbl 0766.60079), \textit{Y. Hu}, Stochastic Processes Appl. 48, No. 1, 107-121 (1993; Zbl 0789.60047), \textit{E. Pardoux}, \textit{F. Pradeilles} and \textit{Z. Rao}, Ann. Inst. Henri Poincaré, Probab. Stat. 33, No. 4, 467-490 (1997; Zbl 0891.60054), \textit{R. W. R. Darling} and \textit{E. Pardoux}, Ann. Probab. 25, No. 3, 1135-1159 (1997; Zbl 0895.60067)]. For this, one needs to study a backward stochastic differential equation with an additional term, which is an integral with respect to a continuous increasing process (local time). Also, in the probabilistic formula for the solution of the partial differential equation, one uses a reflected diffusion process, the properties of which are studied [see also \textit{P. L. Lions} and \textit{A. S. Sznitman}, Commun. Pure Appl. Math. 37, 511-537 (1984; Zbl 0598.60060)]. An infinite horizon backward stochastic differential equation, of same type, is also solved.