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Hitting Time and Inverse Problems for Markov Chains

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

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

[1] Arridge, S. R. (1999). Optical tomography in medical imaging. Inverse Problems 15, R41–R93.CrossRefGoogle Scholar
[2] Bal, G. and Chu, T. (2004). On the reconstruction of diffusions from first-exit time distributions. Inverse Problems 20, 1053–1065.CrossRefGoogle Scholar
[3] De la Peña, V., Gzyl, H. and McDonald, P. (2008). Inverse problems for random walks on trees: network tomography. To appear in Statist. Prob. Lett. Google Scholar
[4] Gardiner, C. W. (2004). Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences, 3rd edn. Springer, Berlin.CrossRefGoogle Scholar
[5] Grünbaum, F. A. (1992). Diffuse tomography: the isotropic case. Inverse Problems 8, 409–419.CrossRefGoogle Scholar
[6] Grünbaum, F. A. (2003). Diffuse tomography as a source of challenging nonlinear inverse problems for a general class of networks. Modern Signal Process. 40, 137–146.Google Scholar
[7] Grünbaum, F. A. and Matusevich, L. F. (2002). Explicit inversion formulas for a model in diffuse tomography. Adv. Appl. Math. 29, 172–183.CrossRefGoogle Scholar
[8] Patch, S. (1995). Recursive recovery of a family of Markov transition probabilites from boundary value data. J. Math. Phys. 36, 3395–3412.CrossRefGoogle Scholar