  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.
