Stochastic optimal control for nonlinear markov jump diffusion processes

We consider the problem finite horizon stochastic optimal control for nonlinear markov jump diffusion processes. In particular, by using stochastic calculus for markov jump diffusions processes and the logarithmic transformation of the value function we demonstrate the transformation of the corresponding Hamilton-Jacobi-Bellman (HJB) Partial Differential Equation (PDE) to the backward Chapman Kolmogorov PDE for jump diffusions. Furthermore we derive the Feynman-Kac lemma for nonlinear markov jump diffusions processes and apply it to the transformed HJB equation. Application of the Feynman-Kac lemma yields the solution of the transformed HJB equation. The path integral interpretation is derived. Finally, conclusions and future directions are discussed.