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One-sided solutions for optimal stopping problems with logconcave reward functions

Part of: Markov processes Stochastic processes Sequential methods

Published online by Cambridge University Press:  22 July 2019

Yi-Shen Lin  and
Yi-Ching Yao
Yi-Shen Lin*
Affiliation:
Academia Sinica
Yi-Ching Yao*
Affiliation:
Academia Sinica
*
*Postal address: Institute of Statistical Science, Academia Sinica, 128 Academia Road, Section 2, Nankang, Taipei 11529, Taiwan, ROC.
*Postal address: Institute of Statistical Science, Academia Sinica, 128 Academia Road, Section 2, Nankang, Taipei 11529, Taiwan, ROC.
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Abstract

In the literature on optimal stopping, the problem of maximizing the expected discounted reward over all stopping times has been explicitly solved for some special reward functions (including (x+)ν, (ex − K)+, (K − e− x)+, x ∈ ℝ, ν ∈ (0, ∞), and K > 0) under general random walks in discrete time and Lévy processes in continuous time (subject to mild integrability conditions). All such reward functions are continuous, increasing, and logconcave while the corresponding optimal stopping times are of threshold type (i.e. the solutions are one-sided). In this paper we show that all optimal stopping problems with increasing, logconcave, and right-continuous reward functions admit one-sided solutions for general random walks and Lévy processes, thereby generalizing the aforementioned results. We also investigate in detail the principle of smooth fit for Lévy processes when the reward function is increasing and logconcave.

Keywords

American option pricingthreshold typerandom walkLévy processsmooth fit

MSC classification

Primary: 60G40: Stopping times; optimal stopping problems; gambling theory
Secondary: 62L15: Optimal stopping 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 60J65: Brownian motion
Type
Original Article
Information
Advances in Applied Probability , Volume 51 , Issue 1 , March 2019 , pp. 87 - 115
Copyright
© Applied Probability Trust 2019 

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