Backward-forward SDE's and stochastic differential games (Q1805788)

The backward-forward stochastic differential equation of the form \[ X_t=x+\int_0^tf(s,X_s,Y_s,Z_s)+\int_0^t\sigma (s,X_s,Y_s,Z_s)dW_s, \] \[ Y_t=g(X_T)-\int_t^Th(s,X_s,Y_s,Z_s)-\int_t^TZ_sdW_s, \] is considered where \(W\) is a standard Brownian motion on \(\mathbb R^m\), \((X,Y,Z)\) is an adapted process with values in \(\mathbb R^m\times \mathbb R^m \times L(\mathbb R^m;\mathbb R^m)\) and \(f,h,\sigma \) and \(g\) are Lipschitz functions. Existence and uniqueness of solutions are shown under certain monotonicity assumptions. The results are applied to study the problem of existence of open-loop Nash equilibrium points of nonzero sum linear-quadratic stochastic differential games with random coefficients.