Infinite time horizon stochastic recursive control problems with jumps: dynamic programming and stochastic verification theorems

Abstract: This paper is devoted to studying an infinite time horizon stochastic recursive control problem with jumps, where infinite time horizon stochastic differential equation and backward stochastic differential equation with jumps describe the state process and cost functional, respectively. For this, the first is to explore the wellposedness and regularity of these two equations in $L^p$-sense ($p\geq2$). By establishing the dynamic programming principle, we relate the value function of the control problem with integral-partial differential equation of HJB type in the sense of viscosity solutions. On the other hand, stochastic verification theorems are also studied to provide sufficient conditions to verify the optimality of the given admissible controls. Such a study is carried out in the framework of classical solutions but also in that of viscosity solutions. Our work emphasizes important differences from the approach for finite time horizon problems. In particular, we have to work in an $L^p$-setting for $p>4$ in order to study the verification theorem in viscosity sense.