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On two classes of interacting stochastic processes arising in cancer modeling

Published online by Cambridge University Press:  01 July 2016

Robert Bartoszyński  and
Prem S. Puri
Robert Bartoszyński*
Affiliation:
Polish Academy of Sciences
Prem S. Puri*
Affiliation:
Purdue University
*
∗Present address: Department of Statistics, Ohio State University, 1958 Neil Avenue, Columbus, OH 43210, U.S.A.
∗∗Postal address: Department of Statistics, Purdue University, Mathematical Sciences Building, West Lafayette, IN 47907, U.S.A.
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Abstract

The processes X and Y are said to interact if the laws governing the changes of either of them at time t depend on the values of the other process at times up to t. For bivariate interacting Markov processes, their limiting behavior is analysed by means of an approximation suggested by Fuhrmann, consisting of discretizing time, and assuming that in each time interval the processes develop independently, according to the laws obtained by fixing the value of the other process at its level attained at the beginning of the interval.

In this way the conditions for almost sure extinction, escape to ∞ with positive probability, etc., are obtained (by using the martingale convergence theorem) for state-dependent branching processes studied by Roi, and for bivariate processes with one component piecewise determined.

Keywords

INTERACTING MARKOV PROCESSESSTATE-DEPENDENT BRANCHING PROCESSESMARTINGALESLIMIT THEOREMS
Type
Research Article
Information
Advances in Applied Probability , Volume 15 , Issue 4 , December 1983 , pp. 695 - 712
Copyright
Copyright © Applied Probability Trust 1983 

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Footnotes

Research carried out while this author was visiting the Departments of Statistics and Mathematics at Purdue University. The support of these departments is gratefully acknowledged.

Research supported in part by U.S. National Science Foundation Grant No. MCS-8102733.

References

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