Hostname: page-component-848d4c4894-jbqgn Total loading time: 0 Render date: 2024-06-27T14:10:25.590Z Has data issue: false hasContentIssue false
  • English
  • Français

Variance Optimal Stopping for Geometric Lévy Processes

Part of: Stochastic processes

Published online by Cambridge University Press:  04 January 2016

Kamille Sofie Tågholt Gad  and
Jesper Lund Pedersen
Kamille Sofie Tågholt Gad*
Affiliation:
University of Copenhagen
Jesper Lund Pedersen*
Affiliation:
University of Copenhagen
*
∗ Postal address: Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, 2100 Copenhagen, Denmark.
∗ Postal address: Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, 2100 Copenhagen, Denmark.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The main result of this paper is the solution to the optimal stopping problem of maximizing the variance of a geometric Lévy process. We call this problem the variance problem. We show that, for some geometric Lévy processes, we achieve higher variances by allowing randomized stopping. Furthermore, for some geometric Lévy processes, the problem has a solution only if randomized stopping is allowed. When randomized stopping is allowed, we give a solution to the variance problem. We identify the Lévy processes for which the allowance of randomized stopping times increases the maximum variance. When it does, we also solve the variance problem without randomized stopping.

Keywords

Variance criterionvariance optimal stoppinggeometric Lévy processquadratic optimal stopping

MSC classification

Primary: 60G40: Stopping times; optimal stopping problems; gambling theory
Secondary: 90C20: Quadratic programming
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 47 , Issue 1 , March 2015 , pp. 128 - 145
Copyright
© Applied Probability Trust 

References

Alili, L. and Kyprianou, A. E. (2005). Some remarks on first passage of Lévy processes, the American put and pasting principles. Ann. Appl. Prob. 15, 2062–2080.Google Scholar
Kyprianou, A. E. (2006). Introductory Lectures on Fluctuations of Lévy Processes with Applications. Springer, Berlin.Google Scholar
Markowitz, H. (1952). Portfolio selection. J. Finance 7, 77–91.Google Scholar
Mikosch, T. (2004). Non-Life Insurance Mathematics. Springer, Berlin.Google Scholar
Mordecki, E. (2002). Optimal stopping and perpetual options for Lévy processes. Finance Stoch. 6, 473–493.Google Scholar
Pedersen, J. L. (2011). Explicit solutions to some optimal variance stopping problems. Stochastics 83, 505–518.Google Scholar
Peskir, G. and Shiryaev, A. (2006). Optimal Stopping and Free-Boundary Problems. Birkhäuser, Basel.Google Scholar
Sato, K.-I. (1999). Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press.Google Scholar
Shiryaev, A. N. (1978). Optimal Stopping Rules. Springer, Berlin.Google Scholar