Probabilistic interpretation of a system of semilinear parabolic partial differential equations (Q1366454)

The authors introduce a class of backward stochastic differential equations with respect to both a Brownian motion and a finite sequence of Poisson processes. As in the work by \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)] the authors prove the formula \(Z_t^{t,x,n} =\partial Y_t^{t,x,n} \sigma (x,n)\) relating the components \(Z\) and \(Y\) of the solution of the BSDE. As an application the authors provide a stochastic interpretation for the viscosity solution of a system of nonlinear parabolic partial differential equations. The functions \(u_i(t,x) =Y_t^{t,x,i}\), \(1\leq i\leq k\), satisfy \[ {\partial u_i\over\partial t}+ L^iu_i+f_i \bigl(t,x,u(t,x),(\nabla u_i \sigma_i)(t,x)\bigr)=0, \quad u_i(T,x)= g_i(x). \] The uniqueness of the viscosity solution to this equation is proved.