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March  2021, 26(3): 1499-1529. doi: 10.3934/dcdsb.2020170

On discrete-time semi-Markov processes

1. 

Faculty of Computing, Engineering and Science, University of South Wales, UK

2. 

Department of Mathematics "G. Peano", University of Torino, Italy

*Corresponding author: Costantino Ricciuti

Received  September 2019 Revised  February 2020 Published  March 2021 Early access  May 2020

In the last years, several authors studied a class of continuous-time semi-Markov processes obtained by time-changing Markov processes by hitting times of independent subordinators. Such processes are governed by integro-differential convolution equations of generalized fractional type. The aim of this paper is to develop a discrete-time counterpart of such a theory and to show relationships and differences with respect to the continuous time case. We present a class of discrete-time semi-Markov chains which can be constructed as time-changed Markov chains and we obtain the related governing convolution type equations. Such processes converge weakly to those in continuous time under suitable scaling limits.

Citation: Angelica Pachon, Federico Polito, Costantino Ricciuti. On discrete-time semi-Markov processes. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1499-1529. doi: 10.3934/dcdsb.2020170
References:
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D. Applebaum, Lévy Processes and Stochastic Calculus, Cambridge Studies in Advanced Mathematics, 116, Cambridge University Press, Cambridge, 2009. doi: 10.1017/CBO9780511809781.  Google Scholar

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V. S. Barbu and N. Limnios, Semi-Markov Chains and Hidden Semi-Markov Models Toward Applications, Lecture Notes in Statistics, 191, Springer, New York, 2008. doi: 10.1007/978-0-387-73173-5.  Google Scholar

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P. Becker-KernM. M. Meerschaert and H.-P. Scheffler, Limit theorems for coupled continuous-time random walks, Ann. Probab., 32 (2004), 730-756.  doi: 10.1214/aop/1079021462.  Google Scholar

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L. Beghin, Fractional relaxation equations and Brownian crossing probabilities of a random boundary, Adv. in Appl. Probab., 44 (2012), 479-505.  doi: 10.1017/S0001867800005693.  Google Scholar

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L. Beghin and E. Orsingher, Fractional Poisson processes and related planar random motions, Electron. J. Probab., 14 (2009), 1790-1827.  doi: 10.1214/EJP.v14-675.  Google Scholar

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L. Beghin and E. Orsingher, Poisson-type processes governed by fractional and higher-order recursive differential equations, Electron. J. Probab., 15 (2010), 684-709.  doi: 10.1214/EJP.v15-762.  Google Scholar

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L. Beghin and C. Ricciuti, Time-inhomogeneous fractional Poisson processes defined by the multistable subordinator, Stoch. Anal. Appl., 37 (2019), 171-188.  doi: 10.1080/07362994.2018.1548970.  Google Scholar

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J. Bertoin, Subordinators: Examples and Applications, in Lectures on Probability Theory and Statistics, Lectures Notes in Math., 1717, Springer, Berlin, 1999, 1–91. doi: 10.1007/978-3-540-48115-7_1.  Google Scholar

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N. H. Bingham, Limit theorems for occupation times of Markov processes, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 17 (1971), 1-22.  doi: 10.1007/BF00538470.  Google Scholar

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R. GarraE. Orsingher and F. Polito, State-dependent fractional point processes, J. Appl. Probab., 52 (2015), 18-36.  doi: 10.1239/jap/1429282604.  Google Scholar

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R. GarraA. GiustiF. Mainardi and G. Pagnini, Fractional relaxation with time-varying coefficient, Fract. Calc. Appl. Anal., 17 (2014), 424-439.  doi: 10.2478/s13540-014-0178-0.  Google Scholar

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M. E. Hernández-HernándezV. N. Kolokoltsov and L. Toniazzi, Generalized fractional evolutions equations of Caputo type, Chaos Solitons Fractals, 102 (2017), 184-196.  doi: 10.1016/j.chaos.2017.05.005.  Google Scholar

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N. N. Leonenko, M. M. Meerschaert, R. L. Shilling and A. Sikorskii, Correlation structure of time-changed Lévy processes, Commun. Appl. Ind. Math., 6 (2014), 22pp. doi: 10.1685/journal.caim.483.  Google Scholar

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show all references

References:
[1]

D. Applebaum, Lévy Processes and Stochastic Calculus, Cambridge Studies in Advanced Mathematics, 116, Cambridge University Press, Cambridge, 2009. doi: 10.1017/CBO9780511809781.  Google Scholar

[2]

V. S. Barbu and N. Limnios, Semi-Markov Chains and Hidden Semi-Markov Models Toward Applications, Lecture Notes in Statistics, 191, Springer, New York, 2008. doi: 10.1007/978-0-387-73173-5.  Google Scholar

[3]

P. Becker-KernM. M. Meerschaert and H.-P. Scheffler, Limit theorems for coupled continuous-time random walks, Ann. Probab., 32 (2004), 730-756.  doi: 10.1214/aop/1079021462.  Google Scholar

[4]

L. Beghin, Fractional relaxation equations and Brownian crossing probabilities of a random boundary, Adv. in Appl. Probab., 44 (2012), 479-505.  doi: 10.1017/S0001867800005693.  Google Scholar

[5]

L. Beghin and E. Orsingher, Fractional Poisson processes and related planar random motions, Electron. J. Probab., 14 (2009), 1790-1827.  doi: 10.1214/EJP.v14-675.  Google Scholar

[6]

L. Beghin and E. Orsingher, Poisson-type processes governed by fractional and higher-order recursive differential equations, Electron. J. Probab., 15 (2010), 684-709.  doi: 10.1214/EJP.v15-762.  Google Scholar

[7]

L. Beghin and C. Ricciuti, Time-inhomogeneous fractional Poisson processes defined by the multistable subordinator, Stoch. Anal. Appl., 37 (2019), 171-188.  doi: 10.1080/07362994.2018.1548970.  Google Scholar

[8]

J. Bertoin, Subordinators: Examples and Applications, in Lectures on Probability Theory and Statistics, Lectures Notes in Math., 1717, Springer, Berlin, 1999, 1–91. doi: 10.1007/978-3-540-48115-7_1.  Google Scholar

[9]

P. Billingsley, Convergence of Probability Measures, John Wiley & Sons, Inc., New York-London-Sydney, 1968.  Google Scholar

[10]

N. H. Bingham, Limit theorems for occupation times of Markov processes, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 17 (1971), 1-22.  doi: 10.1007/BF00538470.  Google Scholar

[11]

N. H. Bingham, C. M. Goldie and J. L. Teugels, Regular Variation, Encyclopedia of Mathematics and its Applications, 27, Cambridge University Press, Cambridge, 1987. doi: 10.1017/CBO9780511721434.  Google Scholar

[12]

D. Buraczewski, E. Damek and T. Mikosch, Stochastic Models with Power-Law Tails, Springer Series in Operations Research and Financial Engineering, Springer, Cham, 2016. doi: 10.1007/978-3-319-29679-1.  Google Scholar

[13]

E. Çinlar, Markov additive processes and semi-regeneration, in Proceedings of the Fifth Conference on Probability Theory, Editura Acad. R. S. R., Bucharest, 1977, 33-49.  Google Scholar

[14]

D. R. Cox, Renewal Theory, Methuen & Co. Ltd., London; John Wiley & Sons, Inc., New York, 1962.  Google Scholar

[15]

L. Devroye, A triptych of discrete distributions related to the stable law, Statist. Probab. Lett., 18 (1993), 349-351.  doi: 10.1016/0167-7152(93)90027-G.  Google Scholar

[16]

M. D'Ovidio, Wright functions governed by fractional directional derivatives and fractional advection diffusion equations, Methods Appl. Anal., 22 (2015), 1-36.  doi: 10.4310/MAA.2015.v22.n1.a1.  Google Scholar

[17]

W. Feller, On semi-Markov processes, Proc. Nat. Acad. Sci. U.S.A., 51 (1964), 653-659.  doi: 10.1073/pnas.51.4.653.  Google Scholar

[18]

R. GarraE. Orsingher and F. Polito, State-dependent fractional point processes, J. Appl. Probab., 52 (2015), 18-36.  doi: 10.1239/jap/1429282604.  Google Scholar

[19]

R. GarraA. GiustiF. Mainardi and G. Pagnini, Fractional relaxation with time-varying coefficient, Fract. Calc. Appl. Anal., 17 (2014), 424-439.  doi: 10.2478/s13540-014-0178-0.  Google Scholar

[20]

N. Georgiou, I. Z. Kiss and E. Scalas, Solvable non-Markovian dynamic network, Phys. Rev. E (3), 92 (2015), 9pp. doi: 10.1103/PhysRevE.92.042801.  Google Scholar

[21]

B. V. Gnedenko and A. N. Kolmogorov, Limit Distributions for Sums of Independent Random Variables, Addison-Wesley Educational Publishers Inc., Cambridge, Mass., 1954.  Google Scholar

[22]

I. I. Gihman and A. V. Skorohod, The Theory of Stochastic Processes. II, der Mathematischen Wissenschaften, Band 218, Springer-Verlag, New York-Heidelberg, 1975.  Google Scholar

[23]

C. W. J. Granger and R. Joyeux, An introduction to long-memory time series models and fractional differencing, J. Time Ser. Anal., 1 (1980), 15-29.  doi: 10.1111/j.1467-9892.1980.tb00297.x.  Google Scholar

[24]

A. Gut, Probability: A Graduate Course, Springer Texts in Statistics, 75, Springer, New York, 2013. doi: 10.1007/978-1-4614-4708-5.  Google Scholar

[25]

M. E. Hernández-Hernández and V. N. Kolokoltsov, Probabilistic solutions to nonlinear fractional differential equations of generalized Caputo and Riemann–Liouville type, Stochastics, 90 (2018), 224-255.  doi: 10.1080/17442508.2017.1334059.  Google Scholar

[26]

M. E. Hernández-HernándezV. N. Kolokoltsov and L. Toniazzi, Generalized fractional evolutions equations of Caputo type, Chaos Solitons Fractals, 102 (2017), 184-196.  doi: 10.1016/j.chaos.2017.05.005.  Google Scholar

[27]

J. Jacod, Systèmes régenératifs and processus semi-markoviens, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 31 (1974/75), 1-23.  doi: 10.1007/BF00538712.  Google Scholar

[28]

V. N. Kolokol'tsov, Generalized continuous-time random walks, subordination by hitting times, and fractional dynamics, Theory Probab. Appl., 53 (2009), 594-609.  doi: 10.1137/S0040585X97983857.  Google Scholar

[29]

V. N. Kolokoltsov, Markov Processes, Semigroups and Generators, De Gruyter Studies in Mathematics, 38, Walter de Gruyter & Co., Berlin, 2011. doi: 10.1515/9783110250114.  Google Scholar

[30]

V. Korolyuk and A. Swishchuk, Semi-Markov Random Evolutions, Mathematics and its Applications, 308, Kluwer Academic Publishers, Dordrecht, 1995. doi: 10.1007/978-94-011-1010-5.  Google Scholar

[31]

A. KumarE. Nane and P. Vellaisamy, Time-changed Poisson processes, Statist. Probab. Lett., 81 (2011), 1899-1910.  doi: 10.1016/j.spl.2011.08.002.  Google Scholar

[32]

T. G. Kurtz, Comparison of semi-Markov and Markov processes, Ann. Math. Statist., 42 (1971), 991-1002.  doi: 10.1214/aoms/1177693327.  Google Scholar

[33]

N. N. Leonenko, M. M. Meerschaert, R. L. Shilling and A. Sikorskii, Correlation structure of time-changed Lévy processes, Commun. Appl. Ind. Math., 6 (2014), 22pp. doi: 10.1685/journal.caim.483.  Google Scholar

[34]

N. N. LeonenkoE. Scalas and M. Trinh, The fractional non-homogeneous Poisson process, Statist. Probab. Lett., 120 (2017), 147-156.  doi: 10.1016/j.spl.2016.09.024.  Google Scholar

[35]

P. Levy, Processus semi-markoviens, Proceedings of the International Congress of Mathematicians, 1954, Amsterdam, Erven P. Noordhoff N. V., Groningen; North-Holland Publishing Co., Amsterdam, 1956,416–426.  Google Scholar

[36]

F. MainardiR. Gorenflo and E. Scalas, A fractional generalization of the Poisson process, Vietnam J. Math., 32 (2007), 53-64.   Google Scholar

[37]

M. M. Meerschaert and H.-P. Scheffler, Limit theorems for continuous-time random walks with infinite mean waiting times, J. Appl. Probab., 41 (2004), 623-638.  doi: 10.1239/jap/1091543414.  Google Scholar

[38]

M. M. Meerschaert and H.-P. Scheffler, Triangular array limits for continuous time random walks, Stochastic Process. Appl., 118 (2008), 1606-1633.  doi: 10.1016/j.spa.2007.10.005.  Google Scholar

[39]

M. M. MeerschaertE. Nane and P. Vellaisamy, The fractional Poisson process and the inverse stable subordinator, Electron. J. Probab., 16 (2011), 1600-1620.  doi: 10.1214/EJP.v16-920.  Google Scholar

[40]

M. M. Meerschaert and A. Sikorskii, Stochastic Models for Fractional Calculus, De Gruyter Studies in Mathematics, 43, Walter de Gruyter & Co., Berlin, 2012. doi: 10.1515/9783110258165.  Google Scholar

[41]

M. M. Meerschaert and P. Straka, Semi-Markov approach to continuous time random walk limit processes, Ann. Probab., 42 (2014), 1699-1723.  doi: 10.1214/13-AOP905.  Google Scholar

[42]

M. M. Meerschaert and B. Toaldo, Relaxation patterns and semi-Markov dynamics, Stochastic Process. Appl., 129 (2019), 2850-2879.  doi: 10.1016/j.spa.2018.08.004.  Google Scholar

[43]

T. Michelitsch, G. Maugin, S. Derogar and A. Nowakowski, Sur une généralisation de l'opérateur fractionnaire, preprint, arXiv: 1111.1898v1. Google Scholar

[44]

J. R. Norris, Markov Chains, Cambridge Series in Statistical and Probabilistic Mathematics, 2, Cambridge University Press, Cambridge, 1998. doi: 10.1017/CBO9780511810633.  Google Scholar

[45]

E. Orsingher and F. Polito, Fractional pure birth processes, Bernoulli, 16 (2010), 858-881.  doi: 10.3150/09-BEJ235.  Google Scholar

[46]

E. Orsingher and F. Polito, On a fractional linear birth-death process, Bernoulli, 17 (2011), 114-137.  doi: 10.3150/10-BEJ263.  Google Scholar

[47]

E. OrsingherF. Polito and L. Sakhno, Fractional non-linear, linear and sub-linear death processes, J. Stat. Phys., 141 (2010), 68-93.  doi: 10.1007/s10955-010-0045-2.  Google Scholar

[48]

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