A transformation of Markov jump processes and applications in genetic study

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December 2014, 34(12): 5061-5084. doi: 10.3934/dcds.2014.34.5061

A transformation of Markov jump processes and applications in genetic study

Xian Chen ^1, and Zhi-Ming Ma ^1,

1.	Academy of Math and Systems Science, CAS, Zhong-guan-cun East Road 55, Beijing 100190, China, China

Received February 2014 Revised May 2014 Published June 2014

Abstract
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In this paper we provide a verifiable necessary and sufficient condition for a regular q-process to be again a q-process under a transformation of state space. The result as well as some other results on continuous states Markov jump processes is employed to investigate jump processes arising from the study in modeling genetic coalescent with recombination.

Keywords: Markov jump process, $\psi$-transform, q-consistency, q-process, coalescent with recombination..

Mathematics Subject Classification: Primary: 60J75; Secondary: 60J25, 92B0.

Citation: Xian Chen, Zhi-Ming Ma. A transformation of Markov jump processes and applications in genetic study. Discrete and Continuous Dynamical Systems, 2014, 34 (12) : 5061-5084. doi: 10.3934/dcds.2014.34.5061

References:

[1]	F. Ball and G. F. Yeo, Lumpability and marginalisability for continuous-time Markov chains, J. Appl. Probab., 30 (1993), 518-528. doi: 10.2307/3214762.
[2]	R. M. Blumenthal and R. K. Getoor, Markov Processes and Potential Theory, Academic Press, New York-London, 1968.
[3]	C. J. Burke and M. Rosenblatt, A Markovian function of a Markov chain, Ann. Math. Statist., 29 (1958), 1112-1122. doi: 10.1214/aoms/1177706444.
[4]	A. Y. Chen, P. Pollett, H. J. Zhang and B. Cairns, Uniqueness criteria for continuous-time Markov chains with general transition structures, Adv. Appl. Prob., 37 (2005), 1056-1074. doi: 10.1239/aap/1134587753.
[5]	M. F. Chen, From Markov Chains to Non-Equilibrium Particle Systems, 2nd edition, World Scientific, Singapore, 2004. doi: 10.1142/9789812562456.
[6]	M. F. Chen and X. G. Zheng, Uniquness criterion for q-processes, Sci. Sin., 26 (1983), 11-24.
[7]	X. Chen, Z. M. Ma and Y. Wang, Markov jump processes in modeling coalsecent with recombination, Annals of Statistics, to appear.
[8]	D. L. Cohn, Measure Theory, Birkhaeuser, Boston, 1980.
[9]	E. B. Dynkin, Markov processes, Springer, Berlin Heidelberg, 1965.
[10]	L. Gurvits and J. Ledoux, Markov property for a function of a Markov chain: A linear algebra approach, Linear Algebra Appl., 404 (2005), 85-117. doi: 10.1016/j.laa.2005.02.007.
[11]	J. Hachigian, Collapsed Markov chains and the Chapman-Kolmogorov equation, Ann. Math. Statist., 34 (1963), 233-237. doi: 10.1214/aoms/1177704261.
[12]	S. W. He, J. G. Wang and J. A. Yan, Semimartingale Theory and Stochastic Calculus, Science Press, Beijing, 1992.
[13]	Z. T. Hou, The criterion for uniqueness of a Q process, Sci. Sinica, 17 (1974), 141-159.
[14]	Z. T. Hou and G. X. Liu, Markov Skeleton Processes and Their Applications, Science Press, Beijing, 2005.
[15]	O. Kallenberg, Foundations of Modern Probability, Springer, New York, 2002. doi: 10.1007/978-1-4757-4015-8.
[16]	J. G. Kemeny and J. L. Snell, Finite Markov Chains, Springer, New York, 1976.
[17]	J. R. Norris, Markov Chains, Cambridge University Press, Cambridge, 1998.
[18]	M. Rosenblatt, Functions of a Markov process that are Markovian, J. Math. Mech., 8 (1959), 585-596.
[19]	M. Sharpe, General Theory of Markov Processes, Academic Press, Inc., Boston, MA, 1988.
[20]	J. P. Tian and X. S. Lin, Colored coalescent theory, Discrete Contin. Dyn. Syst. suppl., (2005), 833-845. doi: 10.1007/s11538-009-9428-4.
[21]	J. P. Tian and D. Kanna, Lumpability and commutativity of Markov processes, Stoch. Anal. Appl., 24 (2006), 685-702. doi: 10.1080/07362990600632045.
[22]	Y. Wang, Y. Zhou, L. F. Li, X. Chen, Y. T. Liu, Z. M. Ma and S. H. Xu, A new method for modeling coalescent processes with recombination, preprint.
[23]	Z. K. Wang, The Theory of Stochastic Processes, (Chinese) Science Press, Beijing, 1978.

show all references

References:

[1]	F. Ball and G. F. Yeo, Lumpability and marginalisability for continuous-time Markov chains, J. Appl. Probab., 30 (1993), 518-528. doi: 10.2307/3214762.
[2]	R. M. Blumenthal and R. K. Getoor, Markov Processes and Potential Theory, Academic Press, New York-London, 1968.
[3]	C. J. Burke and M. Rosenblatt, A Markovian function of a Markov chain, Ann. Math. Statist., 29 (1958), 1112-1122. doi: 10.1214/aoms/1177706444.
[4]	A. Y. Chen, P. Pollett, H. J. Zhang and B. Cairns, Uniqueness criteria for continuous-time Markov chains with general transition structures, Adv. Appl. Prob., 37 (2005), 1056-1074. doi: 10.1239/aap/1134587753.
[5]	M. F. Chen, From Markov Chains to Non-Equilibrium Particle Systems, 2nd edition, World Scientific, Singapore, 2004. doi: 10.1142/9789812562456.
[6]	M. F. Chen and X. G. Zheng, Uniquness criterion for q-processes, Sci. Sin., 26 (1983), 11-24.
[7]	X. Chen, Z. M. Ma and Y. Wang, Markov jump processes in modeling coalsecent with recombination, Annals of Statistics, to appear.
[8]	D. L. Cohn, Measure Theory, Birkhaeuser, Boston, 1980.
[9]	E. B. Dynkin, Markov processes, Springer, Berlin Heidelberg, 1965.
[10]	L. Gurvits and J. Ledoux, Markov property for a function of a Markov chain: A linear algebra approach, Linear Algebra Appl., 404 (2005), 85-117. doi: 10.1016/j.laa.2005.02.007.
[11]	J. Hachigian, Collapsed Markov chains and the Chapman-Kolmogorov equation, Ann. Math. Statist., 34 (1963), 233-237. doi: 10.1214/aoms/1177704261.
[12]	S. W. He, J. G. Wang and J. A. Yan, Semimartingale Theory and Stochastic Calculus, Science Press, Beijing, 1992.
[13]	Z. T. Hou, The criterion for uniqueness of a Q process, Sci. Sinica, 17 (1974), 141-159.
[14]	Z. T. Hou and G. X. Liu, Markov Skeleton Processes and Their Applications, Science Press, Beijing, 2005.
[15]	O. Kallenberg, Foundations of Modern Probability, Springer, New York, 2002. doi: 10.1007/978-1-4757-4015-8.
[16]	J. G. Kemeny and J. L. Snell, Finite Markov Chains, Springer, New York, 1976.
[17]	J. R. Norris, Markov Chains, Cambridge University Press, Cambridge, 1998.
[18]	M. Rosenblatt, Functions of a Markov process that are Markovian, J. Math. Mech., 8 (1959), 585-596.
[19]	M. Sharpe, General Theory of Markov Processes, Academic Press, Inc., Boston, MA, 1988.
[20]	J. P. Tian and X. S. Lin, Colored coalescent theory, Discrete Contin. Dyn. Syst. suppl., (2005), 833-845. doi: 10.1007/s11538-009-9428-4.
[21]	J. P. Tian and D. Kanna, Lumpability and commutativity of Markov processes, Stoch. Anal. Appl., 24 (2006), 685-702. doi: 10.1080/07362990600632045.
[22]	Y. Wang, Y. Zhou, L. F. Li, X. Chen, Y. T. Liu, Z. M. Ma and S. H. Xu, A new method for modeling coalescent processes with recombination, preprint.
[23]	Z. K. Wang, The Theory of Stochastic Processes, (Chinese) Science Press, Beijing, 1978.

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