American Institute of Mathematical Sciences

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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

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 & 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.  Google Scholar

[2]

R. M. Blumenthal and R. K. Getoor, Markov Processes and Potential Theory, Academic Press, New York-London, 1968.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

M. F. Chen, From Markov Chains to Non-Equilibrium Particle Systems, 2nd edition, World Scientific, Singapore, 2004. doi: 10.1142/9789812562456.  Google Scholar

[6]

M. F. Chen and X. G. Zheng, Uniquness criterion for q-processes, Sci. Sin., 26 (1983), 11-24.  Google Scholar

[7]

X. Chen, Z. M. Ma and Y. Wang, Markov jump processes in modeling coalsecent with recombination,, Annals of Statistics, ().   Google Scholar

[8]

D. L. Cohn, Measure Theory, Birkhaeuser, Boston, 1980.  Google Scholar

[9]

E. B. Dynkin, Markov processes, Springer, Berlin Heidelberg, 1965.  Google Scholar

[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.  Google Scholar

[11]

J. Hachigian, Collapsed Markov chains and the Chapman-Kolmogorov equation, Ann. Math. Statist., 34 (1963), 233-237. doi: 10.1214/aoms/1177704261.  Google Scholar

[12]

S. W. He, J. G. Wang and J. A. Yan, Semimartingale Theory and Stochastic Calculus, Science Press, Beijing, 1992.  Google Scholar

[13]

Z. T. Hou, The criterion for uniqueness of a Q process, Sci. Sinica, 17 (1974), 141-159.  Google Scholar

[14]

Z. T. Hou and G. X. Liu, Markov Skeleton Processes and Their Applications, Science Press, Beijing, 2005. Google Scholar

[15]

O. Kallenberg, Foundations of Modern Probability, Springer, New York, 2002. doi: 10.1007/978-1-4757-4015-8.  Google Scholar

[16]

J. G. Kemeny and J. L. Snell, Finite Markov Chains, Springer, New York, 1976.  Google Scholar

[17]

J. R. Norris, Markov Chains, Cambridge University Press, Cambridge, 1998.  Google Scholar

[18]

M. Rosenblatt, Functions of a Markov process that are Markovian, J. Math. Mech., 8 (1959), 585-596.  Google Scholar

[19]

M. Sharpe, General Theory of Markov Processes, Academic Press, Inc., Boston, MA, 1988.  Google Scholar

[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.  Google Scholar

[21]

J. P. Tian and D. Kanna, Lumpability and commutativity of Markov processes, Stoch. Anal. Appl., 24 (2006), 685-702. doi: 10.1080/07362990600632045.  Google Scholar

[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., ().   Google Scholar

[23]

Z. K. Wang, The Theory of Stochastic Processes, (Chinese) Science Press, Beijing, 1978. Google Scholar

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.  Google Scholar

[2]

R. M. Blumenthal and R. K. Getoor, Markov Processes and Potential Theory, Academic Press, New York-London, 1968.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

M. F. Chen, From Markov Chains to Non-Equilibrium Particle Systems, 2nd edition, World Scientific, Singapore, 2004. doi: 10.1142/9789812562456.  Google Scholar

[6]

M. F. Chen and X. G. Zheng, Uniquness criterion for q-processes, Sci. Sin., 26 (1983), 11-24.  Google Scholar

[7]

X. Chen, Z. M. Ma and Y. Wang, Markov jump processes in modeling coalsecent with recombination,, Annals of Statistics, ().   Google Scholar

[8]

D. L. Cohn, Measure Theory, Birkhaeuser, Boston, 1980.  Google Scholar

[9]

E. B. Dynkin, Markov processes, Springer, Berlin Heidelberg, 1965.  Google Scholar

[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.  Google Scholar

[11]

J. Hachigian, Collapsed Markov chains and the Chapman-Kolmogorov equation, Ann. Math. Statist., 34 (1963), 233-237. doi: 10.1214/aoms/1177704261.  Google Scholar

[12]

S. W. He, J. G. Wang and J. A. Yan, Semimartingale Theory and Stochastic Calculus, Science Press, Beijing, 1992.  Google Scholar

[13]

Z. T. Hou, The criterion for uniqueness of a Q process, Sci. Sinica, 17 (1974), 141-159.  Google Scholar

[14]

Z. T. Hou and G. X. Liu, Markov Skeleton Processes and Their Applications, Science Press, Beijing, 2005. Google Scholar

[15]

O. Kallenberg, Foundations of Modern Probability, Springer, New York, 2002. doi: 10.1007/978-1-4757-4015-8.  Google Scholar

[16]

J. G. Kemeny and J. L. Snell, Finite Markov Chains, Springer, New York, 1976.  Google Scholar

[17]

J. R. Norris, Markov Chains, Cambridge University Press, Cambridge, 1998.  Google Scholar

[18]

M. Rosenblatt, Functions of a Markov process that are Markovian, J. Math. Mech., 8 (1959), 585-596.  Google Scholar

[19]

M. Sharpe, General Theory of Markov Processes, Academic Press, Inc., Boston, MA, 1988.  Google Scholar

[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.  Google Scholar

[21]

J. P. Tian and D. Kanna, Lumpability and commutativity of Markov processes, Stoch. Anal. Appl., 24 (2006), 685-702. doi: 10.1080/07362990600632045.  Google Scholar

[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., ().   Google Scholar

[23]

Z. K. Wang, The Theory of Stochastic Processes, (Chinese) Science Press, Beijing, 1978. Google Scholar

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