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

A transformation of Markov jump processes and applications in genetic study

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.
Citation: Xian Chen, Zhi-Ming Ma. A transformation of Markov jump processes and applications in genetic study. Discrete & Continuous Dynamical Systems - A, 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.  doi: 10.2307/3214762.  Google Scholar

[2]

R. M. Blumenthal and R. K. Getoor, Markov Processes and Potential Theory,, Academic Press, (1968).   Google Scholar

[3]

C. J. Burke and M. Rosenblatt, A Markovian function of a Markov chain,, Ann. Math. Statist., 29 (1958), 1112.  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.  doi: 10.1239/aap/1134587753.  Google Scholar

[5]

M. F. Chen, From Markov Chains to Non-Equilibrium Particle Systems,, 2nd edition, (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.   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, (1980).   Google Scholar

[9]

E. B. Dynkin, Markov processes,, Springer, (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.  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.  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, (1992).   Google Scholar

[13]

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

[14]

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

[15]

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

[16]

J. G. Kemeny and J. L. Snell, Finite Markov Chains,, Springer, (1976).   Google Scholar

[17]

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

[18]

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

[19]

M. Sharpe, General Theory of Markov Processes,, Academic Press, (1988).   Google Scholar

[20]

J. P. Tian and X. S. Lin, Colored coalescent theory,, Discrete Contin. Dyn. Syst. suppl., (2005), 833.  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.  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, (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.  doi: 10.2307/3214762.  Google Scholar

[2]

R. M. Blumenthal and R. K. Getoor, Markov Processes and Potential Theory,, Academic Press, (1968).   Google Scholar

[3]

C. J. Burke and M. Rosenblatt, A Markovian function of a Markov chain,, Ann. Math. Statist., 29 (1958), 1112.  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.  doi: 10.1239/aap/1134587753.  Google Scholar

[5]

M. F. Chen, From Markov Chains to Non-Equilibrium Particle Systems,, 2nd edition, (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.   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, (1980).   Google Scholar

[9]

E. B. Dynkin, Markov processes,, Springer, (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.  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.  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, (1992).   Google Scholar

[13]

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

[14]

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

[15]

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

[16]

J. G. Kemeny and J. L. Snell, Finite Markov Chains,, Springer, (1976).   Google Scholar

[17]

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

[18]

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

[19]

M. Sharpe, General Theory of Markov Processes,, Academic Press, (1988).   Google Scholar

[20]

J. P. Tian and X. S. Lin, Colored coalescent theory,, Discrete Contin. Dyn. Syst. suppl., (2005), 833.  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.  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, (1978).   Google Scholar

[1]

Benoît Perthame, P. E. Souganidis. Front propagation for a jump process model arising in spacial ecology. Discrete & Continuous Dynamical Systems - A, 2005, 13 (5) : 1235-1246. doi: 10.3934/dcds.2005.13.1235

[2]

Daoyi Xu, Yumei Huang, Zhiguo Yang. Existence theorems for periodic Markov process and stochastic functional differential equations. Discrete & Continuous Dynamical Systems - A, 2009, 24 (3) : 1005-1023. doi: 10.3934/dcds.2009.24.1005

[3]

Sebastian Reich, Seoleun Shin. On the consistency of ensemble transform filter formulations. Journal of Computational Dynamics, 2014, 1 (1) : 177-189. doi: 10.3934/jcd.2014.1.177

[4]

Isabelle Kuhwald, Ilya Pavlyukevich. Bistable behaviour of a jump-diffusion driven by a periodic stable-like additive process. Discrete & Continuous Dynamical Systems - B, 2016, 21 (9) : 3175-3190. doi: 10.3934/dcdsb.2016092

[5]

Donny Citra Lesmana, Song Wang. A numerical scheme for pricing American options with transaction costs under a jump diffusion process. Journal of Industrial & Management Optimization, 2017, 13 (4) : 1793-1813. doi: 10.3934/jimo.2017019

[6]

Wuyuan Jiang. The maximum surplus before ruin in a jump-diffusion insurance risk process with dependence. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 3037-3050. doi: 10.3934/dcdsb.2018298

[7]

Ummugul Bulut, Edward J. Allen. Derivation of SDES for a macroevolutionary process. Discrete & Continuous Dynamical Systems - B, 2013, 18 (7) : 1777-1792. doi: 10.3934/dcdsb.2013.18.1777

[8]

W. Cary Huffman. On the theory of $\mathbb{F}_q$-linear $\mathbb{F}_{q^t}$-codes. Advances in Mathematics of Communications, 2013, 7 (3) : 349-378. doi: 10.3934/amc.2013.7.349

[9]

Gokhan Yener, Ibrahim Emiroglu. A q-analogue of the multiplicative calculus: Q-multiplicative calculus. Discrete & Continuous Dynamical Systems - S, 2015, 8 (6) : 1435-1450. doi: 10.3934/dcdss.2015.8.1435

[10]

Leszek Gasiński, Nikolaos S. Papageorgiou. Dirichlet $(p,q)$-equations at resonance. Discrete & Continuous Dynamical Systems - A, 2014, 34 (5) : 2037-2060. doi: 10.3934/dcds.2014.34.2037

[11]

Volker Rehbock, Iztok Livk. Optimal control of a batch crystallization process. Journal of Industrial & Management Optimization, 2007, 3 (3) : 585-596. doi: 10.3934/jimo.2007.3.585

[12]

Tuvi Etzion, Alexander Vardy. On $q$-analogs of Steiner systems and covering designs. Advances in Mathematics of Communications, 2011, 5 (2) : 161-176. doi: 10.3934/amc.2011.5.161

[13]

Zvi Drezner, Carlton Scott. Approximate and exact formulas for the $(Q,r)$ inventory model. Journal of Industrial & Management Optimization, 2015, 11 (1) : 135-144. doi: 10.3934/jimo.2015.11.135

[14]

Jesús Carrillo-Pacheco, Felipe Zaldivar. On codes over FFN$(1,q)$-projective varieties. Advances in Mathematics of Communications, 2016, 10 (2) : 209-220. doi: 10.3934/amc.2016001

[15]

María Andrea Arias Serna, María Eugenia Puerta Yepes, César Edinson Escalante Coterio, Gerardo Arango Ospina. $ (Q,r) $ Model with $ CVaR_α $ of costs minimization. Journal of Industrial & Management Optimization, 2017, 13 (1) : 135-146. doi: 10.3934/jimo.2016008

[16]

Yaiza Canzani, Dmitry Jakobson, Igor Wigman. Scalar curvature and $Q$-curvature of random metrics. Electronic Research Announcements, 2010, 17: 43-56. doi: 10.3934/era.2010.17.43

[17]

Ali Maalaoui. Prescribing the Q-curvature on the sphere with conical singularities. Discrete & Continuous Dynamical Systems - A, 2016, 36 (11) : 6307-6330. doi: 10.3934/dcds.2016074

[18]

Yun Zhao, Wen-Chiao Cheng, Chih-Chang Ho. Q-entropy for general topological dynamical systems. Discrete & Continuous Dynamical Systems - A, 2019, 39 (4) : 2059-2075. doi: 10.3934/dcds.2019086

[19]

Samer Dweik. $ L^{p, q} $ estimates on the transport density. Communications on Pure & Applied Analysis, 2019, 18 (6) : 3001-3009. doi: 10.3934/cpaa.2019134

[20]

Françoise Pène. Asymptotic of the number of obstacles visited by the planar Lorentz process. Discrete & Continuous Dynamical Systems - A, 2009, 24 (2) : 567-587. doi: 10.3934/dcds.2009.24.567

2018 Impact Factor: 1.143

Metrics

  • PDF downloads (11)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]