The shape of the value function under Poisson optimal stopping (Q1994915)

The author studies a Poisson optimal stopping of a diffusion process, i.e., the maximization takes place over all event times of an independent, inhomogeneous Poisson process with rate \(\theta=\theta(X_t)\), \(t\geq 0\). The aim is to check if the value function \(V_\theta(x)=\sup_{\tau\in \mathcal T}\textbf{E}_x[e^{-\beta\tau}g(X_\tau)]\), where \(\mathcal{T}\) is the set of all stopping times taking values at an event time of prescribed Poisson process, inherits monotonicity and convexity properties from \(g\). It is shown that monotonicity (respectively convexity) of \(V_\theta(x)\) in \(x\) depends on the monotonicity (respectively convexity) of the quantity \(\frac{\theta(x)g(x)}{\theta(x)+\beta}\) rather than g. The proof uses stochastic coupling technique. It is worth noting that a closely related formulation of the optimal stopping task considered in this paper is the Eflving's problem (see [\textit{G. Elfving}, J. Appl. Probab. 4, 77--89 (1967; Zbl 0183.20705); \textit{D. O. Siegmund}, Ann. Math. Stat. 38, 1627--1640 (1967; Zbl 0183.20706); \textit{R. Righter}, Probab. Eng. Inf. Sci. 1, 189--202 (1987; Zbl 1133.90415)]).