• Previous Article
    Cauchy problem for stochastic non-autonomous evolution equations governed by noncompact evolution families
  • DCDS-B Home
  • This Issue
  • Next Article
    Diffusive limit to a selection-mutation equation with small mutation formulated on the space of measures
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  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:
[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]

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, 2021, 26 (2) : 755-774. doi: 10.3934/dcdsb.2020133

[2]

Cuicui Li, Lin Zhou, Zhidong Teng, Buyu Wen. The threshold dynamics of a discrete-time echinococcosis transmission model. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020339

[3]

Veena Goswami, Gopinath Panda. Optimal customer behavior in observable and unobservable discrete-time queues. Journal of Industrial & Management Optimization, 2021, 17 (1) : 299-316. doi: 10.3934/jimo.2019112

[4]

Ming Chen, Hao Wang. Dynamics of a discrete-time stoichiometric optimal foraging model. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 107-120. doi: 10.3934/dcdsb.2020264

[5]

Peter Giesl, Zachary Langhorne, Carlos Argáez, Sigurdur Hafstein. Computing complete Lyapunov functions for discrete-time dynamical systems. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 299-336. doi: 10.3934/dcdsb.2020331

[6]

Haixiang Yao, Ping Chen, Miao Zhang, Xun Li. Dynamic discrete-time portfolio selection for defined contribution pension funds with inflation risk. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020166

[7]

Xu Zhang, Chuang Zheng, Enrique Zuazua. Time discrete wave equations: Boundary observability and control. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 571-604. doi: 10.3934/dcds.2009.23.571

[8]

Xin Zhao, Tao Feng, Liang Wang, Zhipeng Qiu. Threshold dynamics and sensitivity analysis of a stochastic semi-Markov switched SIRS epidemic model with nonlinear incidence and vaccination. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2021010

[9]

Lars Grüne, Matthias A. Müller, Christopher M. Kellett, Steven R. Weller. Strict dissipativity for discrete time discounted optimal control problems. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020046

[10]

Stefan Siegmund, Petr Stehlík. Time scale-induced asynchronous discrete dynamical systems. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 1011-1029. doi: 10.3934/dcdsb.2020151

[11]

Ting Liu, Guo-Bao Zhang. Global stability of traveling waves for a spatially discrete diffusion system with time delay. Electronic Research Archive, , () : -. doi: 10.3934/era.2021003

[12]

Theresa Lange, Wilhelm Stannat. Mean field limit of ensemble square root filters - discrete and continuous time. Foundations of Data Science, 2021  doi: 10.3934/fods.2021003

[13]

Philipp Harms. Strong convergence rates for markovian representations of fractional processes. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020367

[14]

Serena Dipierro, Benedetta Pellacci, Enrico Valdinoci, Gianmaria Verzini. Time-fractional equations with reaction terms: Fundamental solutions and asymptotics. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 257-275. doi: 10.3934/dcds.2020137

[15]

Nguyen Anh Tuan, Donal O'Regan, Dumitru Baleanu, Nguyen H. Tuan. On time fractional pseudo-parabolic equations with nonlocal integral conditions. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020109

[16]

Nguyen Huy Tuan, Vo Van Au, Runzhang Xu. Semilinear Caputo time-fractional pseudo-parabolic equations. Communications on Pure & Applied Analysis, 2021, 20 (2) : 583-621. doi: 10.3934/cpaa.2020282

[17]

Leilei Wei, Yinnian He. A fully discrete local discontinuous Galerkin method with the generalized numerical flux to solve the tempered fractional reaction-diffusion equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020319

[18]

Christopher S. Goodrich, Benjamin Lyons, Mihaela T. Velcsov. Analytical and numerical monotonicity results for discrete fractional sequential differences with negative lower bound. Communications on Pure & Applied Analysis, 2021, 20 (1) : 339-358. doi: 10.3934/cpaa.2020269

[19]

Jean-Claude Saut, Yuexun Wang. Long time behavior of the fractional Korteweg-de Vries equation with cubic nonlinearity. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1133-1155. doi: 10.3934/dcds.2020312

[20]

Hoang The Tuan. On the asymptotic behavior of solutions to time-fractional elliptic equations driven by a multiplicative white noise. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1749-1762. doi: 10.3934/dcdsb.2020318

2019 Impact Factor: 1.27

Metrics

  • PDF downloads (91)
  • HTML views (300)
  • Cited by (1)

[Back to Top]