Constrained optimal stopping, liquidity and effort

Abstract

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 E x [e −βτ g(Xτ )], maximisation takes place over all stopping times τ . In a constrained 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 θ (x) = sup τ ∈T (T θ ) E x [e −βτ g(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 V θ 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.

Constrained optimal stopping under a regime-switching model

“…Solving it, they showed by a probabilistic argument that the solution to the VI coincides with the value function. There are other many works dealing with this issue such as [12], [13], [15], [16], [18] and so forth.…”

Two-sided Poisson control of linear diffusions

“…Nowadays the literature on Poisson type control problems is quite extensive. Some examples include optimal stopping [9,17], stopping games [21,22], ergodic control [31,19], optimal switching [20], extensions to inhomogeneous Poisson processes [12,13] and more general signal processes [27,26,28].…”

The shape of the value function under Poisson optimal stopping

“…Hobson and Zeng [13] consider an extension of (2) in which the agent can choose the rate of the Poisson process (dynamically) subject to a cost which depends on the chosen rate. Motivated by this example, in this paper we consider the extension of (2) to a state-dependent, inhomogeneous Poisson process and the problem of finding…”