  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.
