Journal of Applied Probability

Exit times for a class of random walks exact distribution results

Martin Jacobsen

Abstract

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.

Article information

Source
J. Appl. Probab., Volume 48A (2011), 51-63.

Dates
First available in Project Euclid: 18 October 2011

Permanent link to this document
https://projecteuclid.org/euclid.jap/1318940455

Digital Object Identifier
doi:10.1239/jap/1318940455

Mathematical Reviews number (MathSciNet)
MR2865616

Zentralblatt MATH identifier
1235.60046

Subjects
Primary: 60G50: Sums of independent random variables; random walks 60G40: Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60]
Secondary: 60G42: Martingales with discrete parameter 60J05: Discrete-time Markov processes on general state spaces

Keywords
One-sided exit mean exit time two-sided exit partial eigenfunction overshoot

Citation

Jacobsen, Martin. Exit times for a class of random walks exact distribution results. J. Appl. Probab. 48A (2011), 51--63. doi:10.1239/jap/1318940455. https://projecteuclid.org/euclid.jap/1318940455


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