Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-23T18:45:52.039Z Has data issue: false hasContentIssue false
  • English
  • Français

A method of approximating Markov jump processes

Published online by Cambridge University Press:  01 July 2016

Keith N. Crank  and
Prem S. Puri
Keith N. Crank*
Affiliation:
Vanderbilt university
Prem S. Puri*
Affiliation:
Purdue University
*
∗Postal address: Owen Graduate School of Management, Vanderbilt University, Nashville, TN 37203, USA.
∗∗Postal address: Department of Statistics, Purdue University, Mathematical Sciences Building, West Lafayette, IN 47907, USA.
Get access Rights & Permissions [Opens in a new window]

Abstract

We present a method of approximating Markov jump processes which was used by Fuhrmann [7] in a special case. We generalize the method and prove weak convergence results under mild assumptions. In addition we obtain bounds on the rates of convergence of the probabilities at arbitrary fixed times. The technique is demonstrated using a state-dependent branching process as an example.

Keywords

WEAK CONVERGENCEINTERACTING PROCESSESRATES OF CONVERGENCE
Type
Research Article
Information
Advances in Applied Probability , Volume 20 , Issue 1 , March 1988 , pp. 33 - 58
Copyright
Copyright © Applied Probability Trust 1988 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

Research supported in part by a David Ross fellowship from Purdue University.

Research supported in part by the U.S. National Science Foundation grant No. DMS-8504319.

References

[1] Bartlett, M. S. (1955) An Introduction to Stochastic Processes. Cambridge University Press, Cambridge.Google Scholar
[2] Bartoszynski, R. and Puri, P. S. (1983) On two classes of interacting stochastic processes arising in cancer modeling. Adv. Appl. Prob. 15, 695–712.CrossRefGoogle Scholar
[3] Billingsley, P. (1968) Convergence of Probability Measures. Wiley, New York.Google Scholar
[4] Blumenthal, R. M. and Getoor, R. K. (1968) Markov Processes and Potential Theory. Academic Press, New York.Google Scholar
[5] Chung, K. L. (1967) Markov Chains with Stationary Transition Probabilities. Springer-Verlag, New York.Google Scholar
[6] Crank, Keith N. (1986) Methods of Approximating Markov Jump Processes. , Purdue University.Google Scholar
[7] Fuhrmann, S. (1975) Control of an Epidemic Involving a Multi-Stage Disease. , Purdue University.Google Scholar
[8] Grimvall, A. (1973) On the transition from a Markov chain to a continuous time process. Stoch. Proc. Appl. 1, 335–368.CrossRefGoogle Scholar
[9] Harris, T. E. (1963) The Theory of Branching Processes. Springer-Verlag, Berlin.CrossRefGoogle Scholar
[10] Puri, P. S. (1968) Interconnected birth and death processes. J. Appl. Prob. 5, 334–349.CrossRefGoogle Scholar