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Exit times for a class of random walks exact distribution results

Part of:Markov processesStochastic 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
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Abstract

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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 walks60G40: Stopping times; optimal stopping problems; gambling theory
Secondary:60G42: Martingales with discrete parameter60J05: Discrete-time Markov processes on general state spaces

Information

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 

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