• Previous Article
    Stabilization of stochastic differential equations driven by G-Lévy process with discrete-time feedback control
  • DCDS-B Home
  • This Issue
  • Next Article
    A game-theoretic framework for autonomous vehicles velocity control: Bridging microscopic differential games and macroscopic mean field games
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  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, doi: 10.3934/dcdsb.2020170
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]

E. OrsingherC. Ricciuti and B. Toaldo, Time-inhomogeneous jump processes and variable order operators, Potential Anal., 45 (2016), 435-461.  doi: 10.1007/s11118-016-9551-4.  Google Scholar

[49]

E. OrsingherC. Ricciuti and B. Toaldo, On semi-Markov processes and their Kolmogorov's integro-differential equations, J. Funct. Anal., 275 (2018), 830-868.  doi: 10.1016/j.jfa.2018.02.011.  Google Scholar

[50]

A. PachonF. Polito and L. Sacerdote, Random graphs associated to some discrete and continuous time preferential attachment models, J. Stat. Phys., 162 (2016), 1608-1638.  doi: 10.1007/s10955-016-1462-7.  Google Scholar

[51]

A. PachonL. Sacerdote and S. Yang, Scale-free behaviour of networks with the copresence of preferential and uniform attachment rules, Phys. D, 371 (2018), 1-12.  doi: 10.1016/j.physd.2018.01.005.  Google Scholar

[52]

R. N. Pillai and K. Jayakumar, Discrete Mittag–Leffler distributions, Statist. Probab. Lett., 23 (1995), 271-274.  doi: 10.1016/0167-7152(94)00124-Q.  Google Scholar

[53]

R. Pyke, Markov renewal processes with finitely many states, Ann. Math. Statist., 32 (1961), 1243-1259.  doi: 10.1214/aoms/1177704864.  Google Scholar

[54]

R. Pyke, Markov renewal processes with infinitely many states, Ann. Math. Statist., 35 (1964), 1746-1764.   Google Scholar

[55]

M. Raberto, F. Rapallo and E. Scalas, Semi-Markov graph dynamics, PLoS ONE, 6 (2011). doi: 10.1371/journal.pone.0023370.  Google Scholar

[56]

C. Ricciuti and B. Toaldo, Semi-Markov models and motion in heterogeneous media, J. Stat. Phys., 169 (2017), 340-361.  doi: 10.1007/s10955-017-1871-2.  Google Scholar

[57]

S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives. Theory and Applications, Gordon and Breach Science Publishers, Yverdon, 1993.  Google Scholar

[58]

K. Sato, Lèvy Processes and Infinitely Divisible Distributions, Cambridge Studies in Advanced Mathematics, 68, Cambridge University Press, Cambridge, 1999.  Google Scholar

[59]

R. L. Schilling, R. Song and Z. Vondraček, Bernstein Functions. Theory and Applications, De Gruyter Studies in Mathematics, 37, Walter de Gruyter & Co., Berlin, 2010. doi: 10.1515/9783110215311.  Google Scholar

[60]

A. V. Skorohod, Limit theorems for stochastic processes, Teor. Veroyatnost. i Primenen., 1 (1956), 289-319.  doi: 10.1137/1101022.  Google Scholar

[61]

A. V. Skorohod, Limit theorems for stochastic processes with independent increments, Teor. Veroyatnost. i Primenen., 2 (1957), 145-177.   Google Scholar

[62]

W. L. Smith, Regenerative stochastic processes, Proc. Roy. Soc. London Ser. A, 232 (1955), 6-31.  doi: 10.1098/rspa.1955.0198.  Google Scholar

[63]

F. W. Steutel and K. van Harn, Infinite Divisibility of Probability Distributions on the Real Line, Monographs and Textbooks in Pure and Applied Mathematics, 259, Marcel Dekker, Inc., New York, 2004.  Google Scholar

[64]

P. Straka and B. I. Henry, Lagging and leading coupled continuous time random walks, renewal times and their joint limits, Stochastic Process. Appl., 121 (2011), 324-336.  doi: 10.1016/j.spa.2010.10.003.  Google Scholar

[65]

B. Toaldo, Lévy mixing related to distributed order calculus, subordinators and slow diffusions, J. Math. Anal. Appl., 430 (2015), 1009-1036.  doi: 10.1016/j.jmaa.2015.05.024.  Google Scholar

[66]

W. Whitt, Some useful functions for functional limit theorems, Math. Oper. Res., 5 (1980), 67-85.  doi: 10.1287/moor.5.1.67.  Google Scholar

[67]

W. Whitt, Stochastic-Process Limits, Springer Series in Operations Research, Springer-Verlag, New York, 2002. doi: 10.1007/b97479.  Google Scholar

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]

E. OrsingherC. Ricciuti and B. Toaldo, Time-inhomogeneous jump processes and variable order operators, Potential Anal., 45 (2016), 435-461.  doi: 10.1007/s11118-016-9551-4.  Google Scholar

[49]

E. OrsingherC. Ricciuti and B. Toaldo, On semi-Markov processes and their Kolmogorov's integro-differential equations, J. Funct. Anal., 275 (2018), 830-868.  doi: 10.1016/j.jfa.2018.02.011.  Google Scholar

[50]

A. PachonF. Polito and L. Sacerdote, Random graphs associated to some discrete and continuous time preferential attachment models, J. Stat. Phys., 162 (2016), 1608-1638.  doi: 10.1007/s10955-016-1462-7.  Google Scholar

[51]

A. PachonL. Sacerdote and S. Yang, Scale-free behaviour of networks with the copresence of preferential and uniform attachment rules, Phys. D, 371 (2018), 1-12.  doi: 10.1016/j.physd.2018.01.005.  Google Scholar

[52]

R. N. Pillai and K. Jayakumar, Discrete Mittag–Leffler distributions, Statist. Probab. Lett., 23 (1995), 271-274.  doi: 10.1016/0167-7152(94)00124-Q.  Google Scholar

[53]

R. Pyke, Markov renewal processes with finitely many states, Ann. Math. Statist., 32 (1961), 1243-1259.  doi: 10.1214/aoms/1177704864.  Google Scholar

[54]

R. Pyke, Markov renewal processes with infinitely many states, Ann. Math. Statist., 35 (1964), 1746-1764.   Google Scholar

[55]

M. Raberto, F. Rapallo and E. Scalas, Semi-Markov graph dynamics, PLoS ONE, 6 (2011). doi: 10.1371/journal.pone.0023370.  Google Scholar

[56]

C. Ricciuti and B. Toaldo, Semi-Markov models and motion in heterogeneous media, J. Stat. Phys., 169 (2017), 340-361.  doi: 10.1007/s10955-017-1871-2.  Google Scholar

[57]

S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives. Theory and Applications, Gordon and Breach Science Publishers, Yverdon, 1993.  Google Scholar

[58]

K. Sato, Lèvy Processes and Infinitely Divisible Distributions, Cambridge Studies in Advanced Mathematics, 68, Cambridge University Press, Cambridge, 1999.  Google Scholar

[59]

R. L. Schilling, R. Song and Z. Vondraček, Bernstein Functions. Theory and Applications, De Gruyter Studies in Mathematics, 37, Walter de Gruyter & Co., Berlin, 2010. doi: 10.1515/9783110215311.  Google Scholar

[60]

A. V. Skorohod, Limit theorems for stochastic processes, Teor. Veroyatnost. i Primenen., 1 (1956), 289-319.  doi: 10.1137/1101022.  Google Scholar

[61]

A. V. Skorohod, Limit theorems for stochastic processes with independent increments, Teor. Veroyatnost. i Primenen., 2 (1957), 145-177.   Google Scholar

[62]

W. L. Smith, Regenerative stochastic processes, Proc. Roy. Soc. London Ser. A, 232 (1955), 6-31.  doi: 10.1098/rspa.1955.0198.  Google Scholar

[63]

F. W. Steutel and K. van Harn, Infinite Divisibility of Probability Distributions on the Real Line, Monographs and Textbooks in Pure and Applied Mathematics, 259, Marcel Dekker, Inc., New York, 2004.  Google Scholar

[64]

P. Straka and B. I. Henry, Lagging and leading coupled continuous time random walks, renewal times and their joint limits, Stochastic Process. Appl., 121 (2011), 324-336.  doi: 10.1016/j.spa.2010.10.003.  Google Scholar

[65]

B. Toaldo, Lévy mixing related to distributed order calculus, subordinators and slow diffusions, J. Math. Anal. Appl., 430 (2015), 1009-1036.  doi: 10.1016/j.jmaa.2015.05.024.  Google Scholar

[66]

W. Whitt, Some useful functions for functional limit theorems, Math. Oper. Res., 5 (1980), 67-85.  doi: 10.1287/moor.5.1.67.  Google Scholar

[67]

W. Whitt, Stochastic-Process Limits, Springer Series in Operations Research, Springer-Verlag, New York, 2002. doi: 10.1007/b97479.  Google Scholar

[1]

Yueyuan Zhang, Yanyan Yin, Fei Liu. Robust observer-based control for discrete-time semi-Markov jump systems with actuator saturation. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020105

[2]

Guangjun Shen, Xueying Wu, Xiuwei Yin. Stabilization of stochastic differential equations driven by G-Lévy process with discrete-time feedback control. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020133

[3]

Yuefen Chen, Yuanguo Zhu. Indefinite LQ optimal control with process state inequality constraints for discrete-time uncertain systems. Journal of Industrial & Management Optimization, 2018, 14 (3) : 913-930. doi: 10.3934/jimo.2017082

[4]

Agnieszka B. Malinowska, Tatiana Odzijewicz. Optimal control of the discrete-time fractional-order Cucker-Smale model. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 347-357. doi: 10.3934/dcdsb.2018023

[5]

Xi Zhu, Meixia Li, Chunfa Li. Consensus in discrete-time multi-agent systems with uncertain topologies and random delays governed by a Markov chain. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020111

[6]

Samuel N. Cohen. Uncertainty and filtering of hidden Markov models in discrete time. Probability, Uncertainty and Quantitative Risk, 2020, 5 (0) : 4-. doi: 10.1186/s41546-020-00046-x

[7]

Qiuli Liu, Xiaolong Zou. A risk minimization problem for finite horizon semi-Markov decision processes with loss rates. Journal of Dynamics & Games, 2018, 5 (2) : 143-163. doi: 10.3934/jdg.2018009

[8]

Eduardo Liz. A new flexible discrete-time model for stable populations. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2487-2498. doi: 10.3934/dcdsb.2018066

[9]

Lih-Ing W. Roeger, Razvan Gelca. Dynamically consistent discrete-time Lotka-Volterra competition models. Conference Publications, 2009, 2009 (Special) : 650-658. doi: 10.3934/proc.2009.2009.650

[10]

Bara Kim, Jeongsim Kim. Explicit solution for the stationary distribution of a discrete-time finite buffer queue. Journal of Industrial & Management Optimization, 2016, 12 (3) : 1121-1133. doi: 10.3934/jimo.2016.12.1121

[11]

Ciprian Preda. Discrete-time theorems for the dichotomy of one-parameter semigroups. Communications on Pure & Applied Analysis, 2008, 7 (2) : 457-463. doi: 10.3934/cpaa.2008.7.457

[12]

Lih-Ing W. Roeger. Dynamically consistent discrete-time SI and SIS epidemic models. Conference Publications, 2013, 2013 (special) : 653-662. doi: 10.3934/proc.2013.2013.653

[13]

H. L. Smith, X. Q. Zhao. Competitive exclusion in a discrete-time, size-structured chemostat model. Discrete & Continuous Dynamical Systems - B, 2001, 1 (2) : 183-191. doi: 10.3934/dcdsb.2001.1.183

[14]

Alexander J. Zaslavski. The turnpike property of discrete-time control problems arising in economic dynamics. Discrete & Continuous Dynamical Systems - B, 2005, 5 (3) : 861-880. doi: 10.3934/dcdsb.2005.5.861

[15]

Veena Goswami, Gopinath Panda. Optimal information policy in discrete-time queues with strategic customers. Journal of Industrial & Management Optimization, 2019, 15 (2) : 689-703. doi: 10.3934/jimo.2018065

[16]

Veena Goswami, Gopinath Panda. Optimal customer behavior in observable and unobservable discrete-time queues. Journal of Industrial & Management Optimization, 2019  doi: 10.3934/jimo.2019112

[17]

Vladimir Răsvan. On the central stability zone for linear discrete-time Hamiltonian systems. Conference Publications, 2003, 2003 (Special) : 734-741. doi: 10.3934/proc.2003.2003.734

[18]

Yung Chung Wang, Jenn Shing Wang, Fu Hsiang Tsai. Analysis of discrete-time space priority queue with fuzzy threshold. Journal of Industrial & Management Optimization, 2009, 5 (3) : 467-479. doi: 10.3934/jimo.2009.5.467

[19]

Jianquan Li, Zhien Ma, Fred Brauer. Global analysis of discrete-time SI and SIS epidemic models. Mathematical Biosciences & Engineering, 2007, 4 (4) : 699-710. doi: 10.3934/mbe.2007.4.699

[20]

Sofian De Clercq, Koen De Turck, Bart Steyaert, Herwig Bruneel. Frame-bound priority scheduling in discrete-time queueing systems. Journal of Industrial & Management Optimization, 2011, 7 (3) : 767-788. doi: 10.3934/jimo.2011.7.767

2019 Impact Factor: 1.27

Article outline

[Back to Top]