Consider an agent with an asset to sell. We might model the asset as a diffusion process $(X_t)_{t \geq 0}$, and the selling problem as a classical optimal stopping problem where the goal is to maximise the expected discounted value of a function of the stopped process $\E^x[e^{-\beta \tau}g(X_\tau)]$, and maximisation takes place over all stopping times $\tau$.

Now suppose there are liquidity constraints. We might model these by saying that it is not possible for the agent to sell the asset at a moment of their choosing, but rather they must wait for a selling opportunity. The selling opportunities might be modelled as the event times of an independent Poisson process. This motivates study of a constrained optimal stopping problem in which stopping is restricted to event times of an independent Poisson process.

In this talk we consider whether the resulting value function $V_\theta(x) = \sup_{\tau \in \sT(\T^\theta)}\E^x[e^{-\beta \tau}g(X_\tau)]$ (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 $g$.

The main technique is stochastic coupling.

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