A Stochastic HJB Equation for Optimal Control of Forward-Backward SDEs.

Summary We study optimal stochastic control problems of general coupled systems of forward-backward stochastic differential equations with jumps. By means of the Ito-Ventzell formula, the system is transformed into a controlled backward stochastic partial differential equation. Using a comparison principle for such equations we obtain a general stochastic Hamilton-Jacobi-Bellman (HJB) equation for the value function of the control problem. In the case of Markovian optimal control of jump diffusions, this equation reduces to the classical HJB equation. The results are applied to study risk minimization in financial markets.