The shape of the value function under Poisson optimal stopping

STOCHASTIC PROCESSES AND THEIR APPLICATIONS

卷:133

Article

Hobson, David¹ 

[1] Univ Warwick, Dept Stat, Coventry CV4 7AL, W Midlands, England

关键词: Poisson optimal stopping;    Diffusion process;    Monotonicity and convexity;    Coupling;    Time-change;   

DOI  :  10.1016/j.spa.2020.12.001

来源: Elsevier

【 摘 要 】

In a classical problem for the stopping of a diffusion process (X-t)(t >= 0), 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)], 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 is an element of T)(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 >= 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 theta(x)g(x)/theta(x)+beta rather than g. Our main e(x)+/- p technique is stochastic coupling. (C) 2020 Elsevier B.V. All rights reserved.

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