The shape of the value function under Poisson optimal stopping

In a classical problem for the stopping of a diffusion process (Xt)t≥0, where the goal is to maximise the expected discounted value of a function of the stopped process Ex[e−βτg(Xτ)], maximisation takes place over all stopping times τ. In a Poisson optimal stopping problem, stopping is restricted to event times of an independent Poisson process. In this article we consider whether the resulting value function Vθ(x)=supτ∈T(Tθ)Ex[e−βτg(Xτ)] (where the supremum is taken over stopping times taking values in the event times of an inhomogeneous Poisson process with rate θ=(θ(Xt))t≥0) inherits monotonicity and convexity properties from g. It turns out that monotonicity (respectively convexity) of Vθ in x depends on the monotonicity (respectively convexity) of the quantity θ(x)g(x)θ(x)+β rather than g. Our main technique is stochastic coupling.