Backward stochastic differential equations with subdifferential operator and related variational inequalities
The main goal of the paper is to provide probabilistic representations for the viscosity solutions to parabolic variational inequalities on the whole space and also to the Dirichlet problem for an elliptic variational inequality, by means of the backward stochastic differential equations (BSDE's). To this end, the authors at first study a BSDE of the form \[ dY_t +F(t,Y_t,Z_t)dt\in \partial \varphi (Y_t)dt +Z_tdB_t,\quad 0 \leq t\leq \tau , \] with a terminal condition \(Y_{\tau}=\xi \) where \(B_t\) is a \(d\)-dimensional Brownian motion defined on a probability space \((\Omega , \mathcal F, P)\), \(\tau \) is an a.s. finite stopping time, the functions \(F:\Omega \times [0,\infty)\times R^k\times R^{r\times d} \to R^k\) and \(\varphi :R^k\to (-\infty , \infty ]\) satisfy suitable conditions, and \(\partial \) denotes the subdifferential. Existence, uniqueness and some other useful properties (including a priori estimates for a penalized equation) of solutions to the BSDE are proved in the first part of the paper. In the second part, the results are applied to obtain the probabilistic formulae of viscosity solutions to the variational inequalities.
scientific article; zbMATH DE number 1364493