Backward Stochastic Differential Equations and Viscosity Solutions of Semilinear Parabolic Deterministic and Stochastic PDE of Second Order

The aim of this article is, in a first step, to present to the reader an introduction to the theory of backward stochastic differential equations (in short, BSDEs) and then to develop the theory of BSDEs in a Markovian framework and its connection with viscosity solutions of semilinear second order partial differential equations of parabolic type. Afterwards, this connection is generalized to that between backward doubly stochastic differential equations (BDSDEs) and stochastic viscosity solutions of semilinear stochastic second order partial differential equations (SPDEs, for short) of parabolic type.