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Hitting Time and Inverse Problems for Markov Chains

Part of: Difference equations Difference and functional equations Stochastic processes Miscellaneous topics - Partial differential equations

Published online by Cambridge University Press:  14 July 2016

Victor de la Peña ,
Henryk Gzyl  and
Patrick McDonald
Victor de la Peña*
Affiliation:
Columbia University
Henryk Gzyl*
Affiliation:
IESA Caracus
Patrick McDonald*
Affiliation:
New College of Florida
*
∗Postal address: Department of Statistics, Columbia University, New York, NY 10027, USA. Email address: vp@stat.columbia.edu
∗∗Postal address: Centro de Finanzas, IESA Caracus, Venezuela. Email address: henryk.gzyl@iesa.edu.ve
∗∗∗Postal address: Division of Natural Science, New College of Florida, Sarasota, FL 34243, USA. Email address: mcdonald@ncf.edu
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Abstract

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Let Wn be a simple Markov chain on the integers. Suppose that Xn is a simple Markov chain on the integers whose transition probabilities coincide with those of Wn off a finite set. We prove that there is an M > 0 such that the Markov chain Wn and the joint distributions of the first hitting time and first hitting place of Xn started at the origin for the sets {-M, M} and {-(M + 1), (M + 1)} algorithmically determine the transition probabilities of Xn.

Keywords

Inverse problemMarkov chainfirst hitting time

MSC classification

Secondary: 60G40: Stopping times; optimal stopping problems; gambling theory 35R30: Inverse problems 39A12: Discrete version of topics in analysis
Type
Research Article
Information
Journal of Applied Probability , Volume 45 , Issue 3 , September 2008 , pp. 640 - 649
Copyright
Copyright © Applied Probability Trust 2008 

References

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