Pardoux, Etienne; Zhang, Shuguang Generalized BSDEs and nonlinear Neumann boundary value problems. (English) Zbl 0909.60046 Probab. Theory Relat. Fields 110, No. 4, 535-558 (1998). 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 S. Peng, Stochastics Stochastics Rep. 37, No. 1/2, 61-74 (1991; Zbl 0739.60060), E. Pardoux and S. Peng, in: Stochastic partial differential equations and their applications. Lect. Notes Control Inf. Sci. 176, 200-217 (1992; Zbl 0766.60079), Y. Hu, Stochastic Processes Appl. 48, No. 1, 107-121 (1993; Zbl 0789.60047), E. Pardoux, F. Pradeilles and Z. Rao, Ann. Inst. Henri Poincaré, Probab. Stat. 33, No. 4, 467-490 (1997; Zbl 0891.60054), R. W. R. Darling and 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 P. L. Lions and 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. Reviewer: Mihai Gradinaru (Nancy) Cited in 6 ReviewsCited in 75 Documents MSC: 60H30 Applications of stochastic analysis (to PDEs, etc.) 60H99 Stochastic analysis 35J60 Nonlinear elliptic equations Keywords:backward stochastic differential equation; semilinear partial differential equation; nonlinear Neumann boundary condition; viscosity solution; reflected diffusion process Citations:Zbl 0739.60060; Zbl 0766.60079; Zbl 0789.60047; Zbl 0895.60067; Zbl 0891.60054; Zbl 0598.60060 PDFBibTeX XMLCite \textit{E. Pardoux} and \textit{S. Zhang}, Probab. Theory Relat. Fields 110, No. 4, 535--558 (1998; Zbl 0909.60046) Full Text: DOI