Journal of Applied Probability

Exit times for a class of random walks exact distribution results

Martin Jacobsen


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

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

First available in Project Euclid: 18 October 2011

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

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


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.

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