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A method of approximating Markov jump processes

  • Keith N. Crank (a1) and Prem S. Puri (a2)

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.

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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.

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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.

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References

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[1] Bartlett, M. S. (1955) An Introduction to Stochastic Processes. Cambridge University Press, Cambridge.
[2] Bartoszynski, R. and Puri, P. S. (1983) On two classes of interacting stochastic processes arising in cancer modeling. Adv. Appl. Prob. 15, 695712.
[3] Billingsley, P. (1968) Convergence of Probability Measures. Wiley, New York.
[4] Blumenthal, R. M. and Getoor, R. K. (1968) Markov Processes and Potential Theory. Academic Press, New York.
[5] Chung, K. L. (1967) Markov Chains with Stationary Transition Probabilities. Springer-Verlag, New York.
[6] Crank, Keith N. (1986) Methods of Approximating Markov Jump Processes. , Purdue University.
[7] Fuhrmann, S. (1975) Control of an Epidemic Involving a Multi-Stage Disease. , Purdue University.
[8] Grimvall, A. (1973) On the transition from a Markov chain to a continuous time process. Stoch. Proc. Appl. 1, 335368.
[9] Harris, T. E. (1963) The Theory of Branching Processes. Springer-Verlag, Berlin.
[10] Puri, P. S. (1968) Interconnected birth and death processes. J. Appl. Prob. 5, 334349.

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A method of approximating Markov jump processes

  • Keith N. Crank (a1) and Prem S. Puri (a2)

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