The shape of the value function under Poisson optimal stopping

In a classical problem for the stopping of a diffusion process $(X_t)_{t \geq 0}$, where the goal is to maximise the expected discounted value of a function of the stopped process ${\mathbb E}^x[e^{-\beta \tau}g(X_\tau)]$, maximisation takes place over all stopping times $\tau$. 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 $V_\theta(x) = \sup_{\tau \in {\mathcal T}({\mathbb T}^\theta)}{\mathbb E}^x[e^{-\beta \tau}g(X_\tau)]$ (where the supremum is taken over stopping times taking values in the event times of an inhomogeneous Poisson process with rate $\theta = (\theta(X_t))_{t \geq 0}$) inherits monotonicity and convexity properties from $g$. It turns out that monotonicity (respectively convexity) of $V_\theta$ in $x$ depends on the monotonicity (respectively convexity) of the quantity $\frac{\theta(x) g(x)}{\theta(x) + \beta}$ rather than $g$. Our main technique is stochastic coupling.