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The ruin problem and cover times of asymmetric random walks and Brownian motions

Part of: Stochastic processes Markov processes

Published online by Cambridge University Press:  01 July 2016

K. S. Chong ,
Richard Cowan  and
Lars Holst
K. S. Chong*
Affiliation:
Chinese University of Hong Kong
Richard Cowan*
Affiliation:
University of Sydney
Lars Holst*
Affiliation:
Royal Institute of Technology, Stockholm
*
∗ Postal address: Department of Statistics, Chinese University of Hong Kong, Shatin, Hong Kong. Email address: kschong@hp735.sta.cuhk.edu.hk
∗∗ Postal address: School of Mathematics and Statistics, University of Sydney, NSW 2006, Australia. Email address: rcowan@mail.usyd.edu.au
∗∗∗ Postal address: Department of Mathematics, Royal Institute of Technology, SE 10044, Stockholm, Sweden. Email address: lholst@math.kth.se
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Abstract

A simple asymmetric random walk on the integers is stopped when its range is of a given length. When and where is it stopped? Analogous questions can be stated for a Brownian motion. Such problems are studied using results for the classical ruin problem, yielding results for the cover time and the range, both for asymmetric random walks and Brownian motion with drift.

Keywords

Random walkcover timesBrownian motiongenerating functionsLaplace transformsrangestopping timefirst-passage timeWiener process

MSC classification

Primary: 60J65: Brownian motion
Secondary: 60G40: Stopping times; optimal stopping problems; gambling theory
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 32 , Issue 1 , March 2000 , pp. 177 - 192
Copyright
Copyright © Applied Probability Trust 2000 

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