Dynamic Programming and a Verification Theorem for the Recursive Stochastic Control Problem of Jump-Diffusion Models with Random Coefficients

In this paper, we consider the stochastic optimal control problem for (forward) stochastic differential equations (SDEs) with jump diffusions and random coefficients under a recursive-type objective functional captured by a backward SDE (BSDE). Due to the jump-diffusion process with random coefficients in both the constraint (forward SDE) and the recursive BSDE objective functional, the associated Hamilton-Jacobi-Bellman equation (HJBE) is an integro-type second-order nonlinear stochastic PDE driven by both Brownian motion and (compensated) Poisson process, which we call the integro-type stochastic HJBE (ISHJBE) with jump diffusions. We first prove the dynamic programming principle for the value function using the backward semigroup associated with the recursive objective functional and the precise estimates of BSDEs, by which the continuity of the value function is also shown. Then we establish a verification theorem, which provides a sufficient condition of optimality and characterizes the value function using the (stochastic) solution of the ISHJBE with jump diffusions. Under suitable assumptions, we show the existence and uniqueness of the weak solution to the ISHJBE via the So