  This paper is devoted to studying an infinite horizon stochastic recursive
control problem with jumps, where infinite 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
$L^p$-frameworks ($p\geq2$). By establishing the dynamic programming principle
of the control problem, we relate the value function of the control problem
with a kind of 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 the sufficient conditions to verify the optimality of
given admissible controls. Such a study is carried out in the framework of
classical solutions but also in that of viscosity solutions. By these research,
we reveal several essential differences from the finite horizon problem.
