Forward-backward stochastic differential equations and quasilinear parabolic PDEs
This paper considers a forward-backward stochastic differential equation (FBSDE) which is a system of the type \[ \begin{aligned} X(t) &= x+\int_0^t f(s,X(s),Y(s),Z(s)) ds +\int_0^t \sigma(s,X(s),Y(s),Z(s)) dB(s),\\ Y(t)&= h(X(T))+\int_t^T g(s,X(s),Y(s),Z(s)) ds -\int_0^t Z(s) dB(s) \end{aligned} \] for \(t\in[0,T]\). Under some technical conditions like Lipschitz, linear growth, or measurability and a simple and very natural monotonicity condition on \(f\) and \(g\), the authors prove results on existence and uniqueness of a solution \(\{(X(s),Y(s),Z(s)):s\in[0,T]\}\). Furthermore, they establish a priori estimates and continuous dependence upon a parameter. Finally, they connect FBSDEs to systems of quasilinear parabolic PDEs of second order. Using their purely probabilistic approach, they prove the existence of viscosity solutions of the PDE.
forward-backward stochastic differential equations
scientific article; zbMATH DE number 1332833