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Optimal Stopping for Processes with Independent Increments, and Applications

Part of: Stochastic processes Sequential methods

Published online by Cambridge University Press:  14 July 2016

G. Deligiannidis ,
H. Le  and
S. Utev
G. Deligiannidis*
Affiliation:
University of Nottingham
H. Le*
Affiliation:
University of Nottingham
S. Utev*
Affiliation:
University of Nottingham
*
∗Current address: Department of Mathematics, University of Leicester, University Road, Leicester, LE1 7RH, UK.
∗∗Postal address: School of Mathematical Sciences, University of Nottingham, University Park, Nottingham, NG7 2RD, UK.
∗∗Postal address: School of Mathematical Sciences, University of Nottingham, University Park, Nottingham, NG7 2RD, UK.
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Abstract

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In this paper we present an explicit solution to the infinite-horizon optimal stopping problem for processes with stationary independent increments, where reward functions admit a certain representation in terms of the process at a random time. It is shown that it is optimal to stop at the first time the process crosses a level defined as the root of an equation obtained from the representation of the reward function. We obtain an explicit formula for the value function in terms of the infimum and supremum of the process, by making use of the Wiener–Hopf factorization. The main results are applied to several problems considered in the literature, to give a unified approach, and to new optimization problems from the finance industry.

Keywords

Optimal stoppingrandom walkLévy processoption pricing

MSC classification

Secondary: 62L15: Optimal stopping 60G50: Sums of independent random variables; random walks
Type
Research Article
Information
Journal of Applied Probability , Volume 46 , Issue 4 , December 2009 , pp. 1130 - 1145
Copyright
Copyright © Applied Probability Trust 2009 

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