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Optimal Stopping for Processes with Independent Increments, and Applications

Part of: Sequential methods Stochastic processes

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 

References

[1] Asmussen, S., Avram, F. and Pistorius, M. R. (2004). Russian and American put options under exponential phase-type Lévy models. Stoch. Process. Appl. 109, 79–111.CrossRefGoogle Scholar
[2] Bingham, N. H. (1975). Fluctuation theory in continuous time. Adv. Appl. Prob. 7, 705–766.CrossRefGoogle Scholar
[3] Boyarchenko, S. and Levendorskii, S. Z. (2006). General option exercise rules, with applications to embedded options and monopolistic expansion. Contrib. Theoret. Econom. 6, 53 pp.Google Scholar
[4] Boyarchenko, S. and Levendorskii, S. (2007). Practical guide to real options in discrete time. Internat. Econom. Rev. 48, 311–342.CrossRefGoogle Scholar
[5] Darling, D. A., Liggett, T. and Taylor, H. M. (1972). Optimal stopping for partial sums. Ann. Math. Statist. 43, 1363–1368.CrossRefGoogle Scholar
[6] Feller, W. (1971). An Introduction to Probability Theory and Its Applications, Vol. II, 2nd edn. John Wiley, New York.Google Scholar
[7] Greenwood, P. and Pitman, J. (1980). Fluctuation identities for random walk by path decomposition at the maximum. Adv. Appl. Prob. 12, 291–293.CrossRefGoogle Scholar
[8] Greenwood, P. and Pitman, J. (1980). Fluctuation identities for Lévy processes and splitting at the maximum. Adv. Appl. Prob. 12, 893–902.CrossRefGoogle Scholar
[9] Kyprianou, A. E. (2006). Introductory Lectures on Fluctuations of Lévy Processes with Applications. Springer, Berlin.Google Scholar
[10] Kyprianou, A. E. and Surya, B. A. (2005). On the Novikov-Shiryaev optimal stopping problems in continuous time. Electron. Commun. Prob. 10, 146–154.CrossRefGoogle Scholar
[11] Mordecki, E. (2002). Optimal stopping and perpetual options for Lévy processes. Finance Stoch. 6, 473–493.CrossRefGoogle Scholar
[12] Mordecki, E. (2002). The distribution of the maximum of a Lévy process with positive Jumps of phase-type. Theory Stoch. Process. 8, 309–316.Google Scholar
[13] Novikov, A. A. and Shiryaev, A. N. (2005). On an effective case of the solution of the optimal stopping problem for random walks. Theory Prob. Appl. 49, 344–354.CrossRefGoogle Scholar
[14] Novikov, A. and Shiryaev, A. (2007). On a solution of the optimal stopping problem for processes with independent increments. Stochastics 79, 393–406.CrossRefGoogle Scholar
[15] Rogers, L. C. G. and Williams, D. (2000). Diffusions, Markov Processes, and Martingales, Vol. 1. Cambridge University Press.Google Scholar
[16] Surya, B. A. (2007). An approach for solving perpetual optimal stopping problems driven by Lévy processes. Stochastics 79, 337–361.CrossRefGoogle Scholar