Hostname: page-component-8448b6f56d-tj2md Total loading time: 0 Render date: 2024-04-19T07:30:35.576Z Has data issue: false hasContentIssue false
  • English
  • Français

Exit times for a class of random walks exact distribution results

Part of: Stochastic processes Markov processes

Published online by Cambridge University Press:  14 July 2016

Martin Jacobsen
Martin Jacobsen*
Affiliation:
University of Copenhagen, Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, DK-2100 Copenhagen Ø, Denmark. Email address: martin@math.ku.dk
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.

For a random walk with both downward and upward jumps (increments), the joint distribution of the exit time across a given level and the undershoot or overshoot at crossing is determined through its generating function, when assuming that the distribution of the jump in the direction making the exit possible has a Laplace transform which is a rational function. The expected exit time is also determined and the paper concludes with exact distribution results concerning exits from bounded intervals. The proofs use simple martingale techniques together with some classical expansions of polynomials and Rouché's theorem from complex function theory.

Keywords

One-sided exitmean exit timetwo-sided exitpartial eigenfunctionovershoot

MSC classification

Primary: 60G50: Sums of independent random variables; random walks 60G40: Stopping times; optimal stopping problems; gambling theory
Secondary: 60G42: Martingales with discrete parameter 60J05: Discrete-time Markov processes on general state spaces
Type
Part 1. Risk Theory
Information
Journal of Applied Probability , Volume 48 , Issue A: New Frontiers in Applied Probability (Journal of Applied Probability Special Volume 48A) , August 2011 , pp. 51 - 63
Copyright
Copyright © Applied Probability Trust 2011 

References

[1] Asmussen, S., Avram, F. and Pistorius, M. R. J., (2004). Russian and American put options under exponential phase-type Lévy models. Stoch. Process. Appl. 109, 79–111.Google Scholar
[2] Borovkov, A. A. and Borovkov, K. A. J., (2008). Asymptotic Analysis of Random Walks (Encyclopedia Math. Appl. 118). Cambridge University Press.Google Scholar
[3] Ezhov, I. I., Kadankov, V. F. and Kadankova, T. V.}, (2007). Two-boundary problems for a random walk. Ukrainian Math. J. 59, 1668–1692.CrossRefGoogle Scholar
[4] Husak, D. V.}, (2005). On the exit of one class random walks from an interval. Ukrainian Math. J. 57, 1413–1423.Google Scholar
[5] Jacobsen, M.}, (2003). Martingales and the distribution of the time to ruin. Stoch. Process. Appl. 107, 29–51.Google Scholar
[6] Jacobsen, M.}, (2005). The time to ruin for a class of Markov additive process with two-sided jumps. Adv. Appl. Prob. 37, 963–992.Google Scholar
[7] Jacobsen, M.J., (2011). The time to ruin in some additive risk models with random premium rates. Submitted.Google Scholar
[8] Jacobsen, M. and Jensen, A. T.}, (2007). Exit times for a class of piecewise exponential Markov processes with two-sided jumps. Stoch. Process. Appl. 117, 1330–1356.Google Scholar
[9] Kadankova, T. V.}, (2004). Two-boundary problems for a random walk with negative geometric jumps. Theory Prob. Math. Statist. 68, 55–66.Google Scholar
[10] Kadankov, V. F. and Kadankova, T. V.}, (2005). On the distribution of the time of the first exit from an interval and the value of a jump over the boundary for processes with independent increments and random walks. Ukrainian Math. J. 57, 1590–1620.Google Scholar
[11] Kemperman, J. H. B.}, (1961). The Passage Problem for a Stationary Markov Chain (Statist. Res. Monogr. 1). University of Chicago Press.CrossRefGoogle Scholar
[12] Kemperman, J. H. B.}, (1963). A Wiener-Hopf type method for a general random walk with a two-sided boundary. Ann. Math. Statist. 34, 1168–1193.Google Scholar
[13] Kuznetsov, A., Kyprianou, A. E. and Pardo, J. C. J., (2010). Meromorphic Lévy processes and their fluctuation identities. Preprint. Available at http://arxiv.org/abs/1004.4671v1.Google Scholar
[14] Lewis, A. L. and Mordecki, E.}, (2008). Wiener-Hopf factorization for Lévy processes having positive jumps with rational transforms. J. Appl. Prob. 45, 118–134.Google Scholar
[15] Mordecki, E.}, (2003). Ruin probabilities for Lévy processes with mixed-exponential negative jumps. Theory Prob. Appl. 48, 170–176.Google Scholar
[16] Pistorius, M.}, (2006). On maxima and ladder processes for a dense class of Lévy processes. J. Appl. Prob. 43, 208–220.Google Scholar
[17] Spitzer, F.,}, (1976). Principles of Random Walks, 2nd edn. Springer, New York.Google Scholar