-
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
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 |
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.
References:
[1] |
D. Applebaum, Lévy Processes and Stochastic Calculus, Cambridge Studies in Advanced Mathematics, 116, Cambridge University Press, Cambridge, 2009.
doi: 10.1017/CBO9780511809781. |
[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. |
[3] |
P. Becker-Kern, M. 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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[9] |
P. Billingsley, Convergence of Probability Measures, John Wiley & Sons, Inc., New York-London-Sydney, 1968. |
[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. |
[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. |
[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. |
[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. |
[14] |
D. R. Cox, Renewal Theory, Methuen & Co. Ltd., London; John Wiley & Sons, Inc., New York, 1962. |
[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. |
[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. |
[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. |
[18] |
R. Garra, E. Orsingher and F. Polito,
State-dependent fractional point processes, J. Appl. Probab., 52 (2015), 18-36.
doi: 10.1239/jap/1429282604. |
[19] |
R. Garra, A. Giusti, F. 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. |
[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. |
[21] |
B. V. Gnedenko and A. N. Kolmogorov, Limit Distributions for Sums of Independent Random Variables, Addison-Wesley Educational Publishers Inc., Cambridge, Mass., 1954. |
[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. |
[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. |
[24] |
A. Gut, Probability: A Graduate Course, Springer Texts in Statistics, 75, Springer, New York, 2013.
doi: 10.1007/978-1-4614-4708-5. |
[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. |
[26] |
M. E. Hernández-Hernández, V. 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. |
[27] |
J. Jacod,
Systèmes régenératifs and processus semi-markoviens, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 31 (1974/75), 1-23.
doi: 10.1007/BF00538712. |
[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. |
[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. |
[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. |
[31] |
A. Kumar, E. Nane and P. Vellaisamy,
Time-changed Poisson processes, Statist. Probab. Lett., 81 (2011), 1899-1910.
doi: 10.1016/j.spl.2011.08.002. |
[32] |
T. G. Kurtz,
Comparison of semi-Markov and Markov processes, Ann. Math. Statist., 42 (1971), 991-1002.
doi: 10.1214/aoms/1177693327. |
[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. |
[34] |
N. N. Leonenko, E. 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. |
[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. |
[36] |
F. Mainardi, R. Gorenflo and E. Scalas,
A fractional generalization of the Poisson process, Vietnam J. Math., 32 (2007), 53-64.
|
[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. |
[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. |
[39] |
M. M. Meerschaert, E. 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. |
[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. |
[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. |
[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. |
[43] |
T. Michelitsch, G. Maugin, S. Derogar and A. Nowakowski, Sur une généralisation de l'opérateur fractionnaire, preprint, arXiv: 1111.1898v1. |
[44] |
J. R. Norris, Markov Chains, Cambridge Series in Statistical and Probabilistic Mathematics, 2, Cambridge University Press, Cambridge, 1998.
doi: 10.1017/CBO9780511810633. |
[45] |
E. Orsingher and F. Polito,
Fractional pure birth processes, Bernoulli, 16 (2010), 858-881.
doi: 10.3150/09-BEJ235. |
[46] |
E. Orsingher and F. Polito,
On a fractional linear birth-death process, Bernoulli, 17 (2011), 114-137.
doi: 10.3150/10-BEJ263. |
[47] |
E. Orsingher, F. 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. |
[48] |
E. Orsingher, C. 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. |
[49] |
E. Orsingher, C. 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. |
[50] |
A. Pachon, F. 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. |
[51] |
A. Pachon, L. 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. |
[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. |
[53] |
R. Pyke,
Markov renewal processes with finitely many states, Ann. Math. Statist., 32 (1961), 1243-1259.
doi: 10.1214/aoms/1177704864. |
[54] |
R. Pyke,
Markov renewal processes with infinitely many states, Ann. Math. Statist., 35 (1964), 1746-1764.
|
[55] |
M. Raberto, F. Rapallo and E. Scalas, Semi-Markov graph dynamics, PLoS ONE, 6 (2011).
doi: 10.1371/journal.pone.0023370. |
[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. |
[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. |
[58] |
K. Sato, Lèvy Processes and Infinitely Divisible Distributions, Cambridge Studies in Advanced Mathematics, 68, Cambridge University Press, Cambridge, 1999. |
[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. |
[60] |
A. V. Skorohod,
Limit theorems for stochastic processes, Teor. Veroyatnost. i Primenen., 1 (1956), 289-319.
doi: 10.1137/1101022. |
[61] |
A. V. Skorohod,
Limit theorems for stochastic processes with independent increments, Teor. Veroyatnost. i Primenen., 2 (1957), 145-177.
|
[62] |
W. L. Smith,
Regenerative stochastic processes, Proc. Roy. Soc. London Ser. A, 232 (1955), 6-31.
doi: 10.1098/rspa.1955.0198. |
[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. |
[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. |
[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. |
[66] |
W. Whitt,
Some useful functions for functional limit theorems, Math. Oper. Res., 5 (1980), 67-85.
doi: 10.1287/moor.5.1.67. |
[67] |
W. Whitt, Stochastic-Process Limits, Springer Series in Operations Research, Springer-Verlag, New York, 2002.
doi: 10.1007/b97479. |
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. |
[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. |
[3] |
P. Becker-Kern, M. 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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[9] |
P. Billingsley, Convergence of Probability Measures, John Wiley & Sons, Inc., New York-London-Sydney, 1968. |
[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. |
[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. |
[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. |
[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. |
[14] |
D. R. Cox, Renewal Theory, Methuen & Co. Ltd., London; John Wiley & Sons, Inc., New York, 1962. |
[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. |
[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. |
[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. |
[18] |
R. Garra, E. Orsingher and F. Polito,
State-dependent fractional point processes, J. Appl. Probab., 52 (2015), 18-36.
doi: 10.1239/jap/1429282604. |
[19] |
R. Garra, A. Giusti, F. 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. |
[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. |
[21] |
B. V. Gnedenko and A. N. Kolmogorov, Limit Distributions for Sums of Independent Random Variables, Addison-Wesley Educational Publishers Inc., Cambridge, Mass., 1954. |
[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. |
[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. |
[24] |
A. Gut, Probability: A Graduate Course, Springer Texts in Statistics, 75, Springer, New York, 2013.
doi: 10.1007/978-1-4614-4708-5. |
[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. |
[26] |
M. E. Hernández-Hernández, V. 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. |
[27] |
J. Jacod,
Systèmes régenératifs and processus semi-markoviens, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 31 (1974/75), 1-23.
doi: 10.1007/BF00538712. |
[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. |
[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. |
[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. |
[31] |
A. Kumar, E. Nane and P. Vellaisamy,
Time-changed Poisson processes, Statist. Probab. Lett., 81 (2011), 1899-1910.
doi: 10.1016/j.spl.2011.08.002. |
[32] |
T. G. Kurtz,
Comparison of semi-Markov and Markov processes, Ann. Math. Statist., 42 (1971), 991-1002.
doi: 10.1214/aoms/1177693327. |
[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. |
[34] |
N. N. Leonenko, E. 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. |
[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. |
[36] |
F. Mainardi, R. Gorenflo and E. Scalas,
A fractional generalization of the Poisson process, Vietnam J. Math., 32 (2007), 53-64.
|
[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. |
[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. |
[39] |
M. M. Meerschaert, E. 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. |
[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. |
[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. |
[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. |
[43] |
T. Michelitsch, G. Maugin, S. Derogar and A. Nowakowski, Sur une généralisation de l'opérateur fractionnaire, preprint, arXiv: 1111.1898v1. |
[44] |
J. R. Norris, Markov Chains, Cambridge Series in Statistical and Probabilistic Mathematics, 2, Cambridge University Press, Cambridge, 1998.
doi: 10.1017/CBO9780511810633. |
[45] |
E. Orsingher and F. Polito,
Fractional pure birth processes, Bernoulli, 16 (2010), 858-881.
doi: 10.3150/09-BEJ235. |
[46] |
E. Orsingher and F. Polito,
On a fractional linear birth-death process, Bernoulli, 17 (2011), 114-137.
doi: 10.3150/10-BEJ263. |
[47] |
E. Orsingher, F. 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. |
[48] |
E. Orsingher, C. 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. |
[49] |
E. Orsingher, C. 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. |
[50] |
A. Pachon, F. 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. |
[51] |
A. Pachon, L. 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. |
[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. |
[53] |
R. Pyke,
Markov renewal processes with finitely many states, Ann. Math. Statist., 32 (1961), 1243-1259.
doi: 10.1214/aoms/1177704864. |
[54] |
R. Pyke,
Markov renewal processes with infinitely many states, Ann. Math. Statist., 35 (1964), 1746-1764.
|
[55] |
M. Raberto, F. Rapallo and E. Scalas, Semi-Markov graph dynamics, PLoS ONE, 6 (2011).
doi: 10.1371/journal.pone.0023370. |
[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. |
[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. |
[58] |
K. Sato, Lèvy Processes and Infinitely Divisible Distributions, Cambridge Studies in Advanced Mathematics, 68, Cambridge University Press, Cambridge, 1999. |
[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. |
[60] |
A. V. Skorohod,
Limit theorems for stochastic processes, Teor. Veroyatnost. i Primenen., 1 (1956), 289-319.
doi: 10.1137/1101022. |
[61] |
A. V. Skorohod,
Limit theorems for stochastic processes with independent increments, Teor. Veroyatnost. i Primenen., 2 (1957), 145-177.
|
[62] |
W. L. Smith,
Regenerative stochastic processes, Proc. Roy. Soc. London Ser. A, 232 (1955), 6-31.
doi: 10.1098/rspa.1955.0198. |
[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. |
[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. |
[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. |
[66] |
W. Whitt,
Some useful functions for functional limit theorems, Math. Oper. Res., 5 (1980), 67-85.
doi: 10.1287/moor.5.1.67. |
[67] |
W. Whitt, Stochastic-Process Limits, Springer Series in Operations Research, Springer-Verlag, New York, 2002.
doi: 10.1007/b97479. |
[1] |
Yueyuan Zhang, Yanyan Yin, Fei Liu. Robust observer-based control for discrete-time semi-Markov jump systems with actuator saturation. Journal of Industrial and Management Optimization, 2021, 17 (6) : 3013-3026. 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 and Continuous Dynamical Systems - B, 2021, 26 (2) : 755-774. 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 and Management Optimization, 2018, 14 (3) : 913-930. doi: 10.3934/jimo.2017082 |
[4] |
Jaydeep Swarnakar. Discrete-time realization of fractional-order proportional integral controller for a class of fractional-order system. Numerical Algebra, Control and Optimization, 2022, 12 (2) : 309-320. doi: 10.3934/naco.2021007 |
[5] |
Agnieszka B. Malinowska, Tatiana Odzijewicz. Optimal control of the discrete-time fractional-order Cucker-Smale model. Discrete and Continuous Dynamical Systems - B, 2018, 23 (1) : 347-357. doi: 10.3934/dcdsb.2018023 |
[6] |
Tadeusz Kaczorek, Andrzej Ruszewski. Analysis of the fractional descriptor discrete-time linear systems by the use of the shuffle algorithm. Journal of Computational Dynamics, 2021, 8 (2) : 153-163. doi: 10.3934/jcd.2021007 |
[7] |
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 and Continuous Dynamical Systems - B, 2020, 25 (12) : 4535-4551. doi: 10.3934/dcdsb.2020111 |
[8] |
Michael C. Fu, Bingqing Li, Rongwen Wu, Tianqi Zhang. Option pricing under a discrete-time Markov switching stochastic volatility with co-jump model. Frontiers of Mathematical Finance, 2022, 1 (1) : 137-160. doi: 10.3934/fmf.2021005 |
[9] |
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 |
[10] |
Baoyin Xun, Kam C. Yuen, Kaiyong Wang. The finite-time ruin probability of a risk model with a general counting process and stochastic return. Journal of Industrial and Management Optimization, 2022, 18 (3) : 1541-1556. doi: 10.3934/jimo.2021032 |
[11] |
Qiuli Liu, Xiaolong Zou. A risk minimization problem for finite horizon semi-Markov decision processes with loss rates. Journal of Dynamics and Games, 2018, 5 (2) : 143-163. doi: 10.3934/jdg.2018009 |
[12] |
Eduardo Liz. A new flexible discrete-time model for stable populations. Discrete and Continuous Dynamical Systems - B, 2018, 23 (6) : 2487-2498. doi: 10.3934/dcdsb.2018066 |
[13] |
Ming Chen, Hao Wang. Dynamics of a discrete-time stoichiometric optimal foraging model. Discrete and Continuous Dynamical Systems - B, 2021, 26 (1) : 107-120. doi: 10.3934/dcdsb.2020264 |
[14] |
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 |
[15] |
Ciprian Preda. Discrete-time theorems for the dichotomy of one-parameter semigroups. Communications on Pure and Applied Analysis, 2008, 7 (2) : 457-463. doi: 10.3934/cpaa.2008.7.457 |
[16] |
Bara Kim, Jeongsim Kim. Explicit solution for the stationary distribution of a discrete-time finite buffer queue. Journal of Industrial and Management Optimization, 2016, 12 (3) : 1121-1133. doi: 10.3934/jimo.2016.12.1121 |
[17] |
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 |
[18] |
Peter Giesl, Zachary Langhorne, Carlos Argáez, Sigurdur Hafstein. Computing complete Lyapunov functions for discrete-time dynamical systems. Discrete and Continuous Dynamical Systems - B, 2021, 26 (1) : 299-336. doi: 10.3934/dcdsb.2020331 |
[19] |
Veena Goswami, Gopinath Panda. Optimal information policy in discrete-time queues with strategic customers. Journal of Industrial and Management Optimization, 2019, 15 (2) : 689-703. doi: 10.3934/jimo.2018065 |
[20] |
H. L. Smith, X. Q. Zhao. Competitive exclusion in a discrete-time, size-structured chemostat model. Discrete and Continuous Dynamical Systems - B, 2001, 1 (2) : 183-191. doi: 10.3934/dcdsb.2001.1.183 |
2021 Impact Factor: 1.497
Tools
Metrics
Other articles
by authors
[Back to Top]