This paper solves a recursive optimal stopping problem with Poisson stopping constraints using the penalized backward stochastic differential equation (PBSDE) with jumps. Stopping in this problem is only allowed at Poisson random intervention times, and jumps play a significant role not only through the stopping times but also in the recursive objective functional and model coefficients. To solve the problem, we propose a decomposition method based on Jacod-Pham that allows us to separate the problem into a series of sub-problems between each pair of consecutive Poisson stopping times. To represent the value function of the recursive optimal stopping problem when the initial time falls between two consecutive Poisson stopping times and the generator is concave/convex, we leverage the comparison theorem of BSDEs with jumps. We then apply the representation result to American option pricing in a nonlinear market with Poisson stopping constraints.

29 pages, 2 figures