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A Journey in the World of Stochastic Processes

  • Dmitrii Silvestrov
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 271)

Abstract

This paper presents a survey of research results obtained by the author and his collaborators in the areas of limit theorems for Markov-type processes and randomly stopped stochastic processes, renewal theory and ergodic theorems for perturbed stochastic processes, quasi-stationary distributions for perturbed stochastic systems, methods of stochastic approximation for price processes, asymptotic expansions for nonlinearly perturbed semi-Markov processes and applications of the above results to queuing systems, reliability models, stochastic networks, bio-stochastic systems, perturbed risk processes, and American-type options.

Keywords

Limit theorem Markov-type process Random stopping Perturbed renewal equation Coupling Quasi-stationary distribution American type option Perturbed semi-Markov process 

References

  1. 1.
    Silvestrov, D.S.: A generalisation of Pólya’s theorem. Dokl. Akad. Nauk Ukr. SSR, Ser A (10), 906–909 (1968)Google Scholar
  2. 2.
    Silvestrov, D.S.: Limit theorems for a non-recurrent walk connected with a Markov chain. Ukr. Math. Zh. 21, 790–804 (1969). (English translation in Ukr. Math. J. 21, 657–669)Google Scholar
  3. 3.
    Silvestrov, D.S.: Limit Theorems for Semi-Markov Processes and their Applications to Random Walks. Candidate of Science dissertation, Kiev State University, 205 pp. (1969)Google Scholar
  4. 4.
    Silvestrov, D.S.: Limit theorems for semi-Markov processes and their applications. 1, 2. Teor. Veroyatn. Math. Stat. 3; Part 1: 155–172; Part 2: 173–194 (1970) (English translation in Theory Probab. Math. Stat. 3, Part 1: 159–176; Part 2: 177–198)Google Scholar
  5. 5.
    Silvestrov, D.S.: Remarks on the limit of composite random function. Teor. Veroyatn. Primen. 17, 707–715 (1972). (English translation in Theory Probab. Appl. 17, 669–677)Google Scholar
  6. 6.
    Silvestrov, D.S.: The convergence of weakly dependent processes in the uniform topology. 1, 2. Teor. Veroyatn. Math. Stat. Part 1: 6, 104–117; Part 2: 7, 132–145 (1972) (English translation in Theory Probab. Math. Stat. Part 1: 6, 109–119; Part 2: 7, 125–138)Google Scholar
  7. 7.
    Silvestrov, D.S.: Limit Theorems for Composite Random Functions. Doctor of Science dissertation, Kiev State University, 395 pp. (1972)Google Scholar
  8. 8.
    Silvestrov, D.S.: Limit Theorems for Composite Random Functions, 318 pp. Vysshaya Shkola and Izdatel’stvo Kievskogo Universiteta, Kiev (1974)Google Scholar
  9. 9.
    Silvestrov, D.S., Mirzahmedov, M.A., Tursunov, G.T.: On the applications of limit theorems for composite random functions to certain problems in statistics. Theor. Veroyatn. Math. Stat. 14, 124–137 (1976) (English translation in Theory Probab. Math. Stat. 14, 133–147)Google Scholar
  10. 10.
    Dorogovtsev, A.Yu., Silvestrov, D.S. Skorokhod, A.V., Yadrenko, M.I.: Probability Theory. A Collection of Problems. Vishcha Shkola, Kiev, 384 pp. (1976) (2nd extended edition: Vishcha Shkola, Kiev, 432 pp. (1980); English translation: Probability Theory: Collection of Problems. Translations of mathematical monographs, vol. 163, 347 pp. American Mathematical Society (1997))Google Scholar
  11. 11.
    Silvestrov D.S., Tursunov, G.T.: General limit theorems for sums of controlled random variables. 1, 2. Teor. Veroyatn. Math. Stat. Part 1: 17, 120–134 (1977); Part 2: 20, 116–126 (1979) (English translation in Theory Probab. Math. Stat. Part 1: 17, 131–146; Part 2: 20, 131–141)Google Scholar
  12. 12.
    Silvestrov, D.S.: The renewal theorem in a series scheme. 1, 2. Teor. Veroyatn. Math. Stat. Part 1: 18, 144–161 (1978); Part 2: 20, 97–116 (1979) (English translation in Theory Probab. Math. Stat. Part 1: 18, 155–172; Part 2: 20, 113–130)Google Scholar
  13. 13.
    Kaplan, E.I., Silvestrov, D.S.: Theorems of the invariance principle type for recurrent semi-Markov processes with arbitrary phase space. Teor. Veroyatn. Primen. 24, 529–541 (1979) (English translation in Theory Probab. Appl. 24, 537–547)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Silvestrov, D.S.: Semi-Markov Processes with a Discrete State Space. Library for an Engineer in Reliability, 272 pp. Sovetskoe Radio, Moscow (1980)Google Scholar
  15. 15.
    Silvestrov, D.S.: Remarks on the strong law of large numbers for accumulation processes. Teor. Veroyatn. Math. Stat. 22, 118–130 (1980) (English translation in Theory Probab. Math. Stat. 22, 131–143)Google Scholar
  16. 16.
    Silvestrov, D.S.: Mean hitting times for semi-Markov processes and queueing networks. Elektronische Inf. Kybern. 16, 399–415 (1980)MathSciNetGoogle Scholar
  17. 17.
    Kaplan, E.I., Silvestrov, D.S.: General limit theorems for sums of controlled random variables with an arbitrary state space for the controlling sequence. Litov. Math. Sbornik, 20(4), 61–72 (1980) (English translation in Lith. Math. J. 20, 286–293)Google Scholar
  18. 18.
    Silvestrov, D.S.: Theorems of large deviations type for entry times of a sequence with mixing. Teor. Veroyatn. Math. Stat. 24, 129–135 (1981) (English translation in Theory Probab. Math. Stat. 24, 145–151)Google Scholar
  19. 19.
    Silvestrov, D.S., Khusanbayev, Ya.M.: General limit theorems for random processes with conditionally independent increments. Theor. Veroyatn. Math. Stat. 27, 130–139 (1982) (English translation in Theory Probab. Math. Stat. 27, 147–155)Google Scholar
  20. 20.
    Kaplan, E.I., Motsa, A.I., Silvestrov, D.S.: Limit theorems for additive functionals defined on asymptotically recurrent Markov chains. 1, 2. Teor. Veroyatn. Math. Stat. Part 1: 27, 34–51 (1982); Part 2: 28, 31–40 (1983) (English translation in Theory Probab. Math. Stat. Part 1: 27, 39–54; Part 2: 28, 35–43)Google Scholar
  21. 21.
    Korolyuk, D.V., Silvestrov D.S.: Entry times into asymptotically receding domains for ergodic Markov chains. Teor. Veroyatn. Primen. 28, 410–420 (1983) (English translation in Theory Probab. Appl. 28, 432–442)CrossRefGoogle Scholar
  22. 22.
    Silvestrov, D.S.: Invariance principle for the processes with semi-Markov switch-overs with an arbitrary state space. In: Itô, K., Prokhorov, YuV (eds.) Proceedings of the Fourth USSR–Japan Symposium on Probability Theory and Mathematical Statistics, Tbilisi, 1982. Lecture notes in mathematics, vol. 1021, 617–628 pp. Springer, Berlin (1983)Google Scholar
  23. 23.
    Silvestrov, D.S.: Method of a single probability space in ergodic theorems for regenerative processes. 1 – 3. Math. Operat. Stat. Ser. Optim. Part 1: 14, 285–299 (1983); Part 2: 15, 601–612 (1984); Part 3: 15, 613–622 (1984)Google Scholar
  24. 24.
    Silvestrov, D.S., Velikiĭ, YuA: Necessary and sufficient conditions for convergence of attainment times. In: Zolotarev, V.M., Kalashnikov, V.V. (eds.) Stability Problems for Stochastic Models. Trudy Seminara, 129–137 pp. VNIISI, Moscow (1988). (English translation in J. Soviet. Math. 57, 3317–3324)Google Scholar
  25. 25.
    Silvestrov, D.S.: A Software of Applied Statistics, 240 pp. Finansi and Statistika, Moscow (1988)Google Scholar
  26. 26.
    Sillvestrov, D.S., Brusilovskiĭ, I.L.: An invariance principle for sums of stationary connected random variables satisfying a uniform strong mixing condition with a weight. Teor. Veroyatn. Math. Stat. 40, 99–107 (1989) (English translation in Theory Probab. Math. Stat. 40, 117–127)Google Scholar
  27. 27.
    Silvestrov, D.S., Semenov, N.A., Marishchuk, V.V.: Packages of applied programs of statistical analysis, 174 pp. Kiev, Tekhnika (1990)Google Scholar
  28. 28.
    Sillvestrov, D.S.: The invariance principle for accumulation processes with semi-Markov switchings in a scheme of arrays. Teor. Veroyatn. Primen. 36(3), 505–520 (1991) (English translation in Theory Probab. Appl. 36(3), 519–535)Google Scholar
  29. 29.
    Silvestrov, D.S., Abadov, Z.A.: Uniform representations of exponential moments of sums of random variables defined on a Markov chain and for distributions of passage times. 1, 2. Teor. Veroyatn. Math. Stat. Part 1: 45, 108–127 (1991); Part 2: 48, 175–183 (1993) (English translation in Theory Probab. Math. Stat. Part 1: 45, 105–120; Part 2: 48, 125–130)Google Scholar
  30. 30.
    Silvestrov, D.S.: Coupling for Markov renewal processes and the rate of convergence in ergodic theorems for processes with semi-Markov switchings. Acta Appl. Math. 34, 109–124 (1994)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Velikiĭ, Yu.A., Motsa, A.I., Silvestrov, D.S.: Dual processes and ergodic type theorems for Markov chains in the triangular array scheme. Teor. Veroyatn. Primen. 39, 716–730 (1994) (English translation in Theory Probab. Appl. 39, 642–653)Google Scholar
  32. 32.
    Silvestrov, D.S.: Exponential asymptotic for perturbed renewal equations. Teor. Ǐmovirn. Math. Stat. 52, 143–153 (1995) (English translation in Theory Probab. Math. Stat. 52, 153–162)Google Scholar
  33. 33.
    Silvestrov, D.S., Silvestrova, E.D.: Elsevier’s Dictionary of Statistical Terminology. English-Russian, Russian-English, 496 pp. Elsevier, Amsterdam (1995)Google Scholar
  34. 34.
    Silvestrov, D.S.: Recurrence relations for generalised hitting times for semi-Markov processes. Ann. Appl. Probab. 6, 617–649 (1996)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Englund, E., Silvestrov, D.: Mixed large deviation and ergodic theorems for regenerative processes with discrete time. In: Jagers, P., Kulldorff, G., Portenko, N., Silvestrov, D. (eds.) Proceedings of the Second Scandinavian–Ukrainian Conference in Mathematical Statistics, vol. I, Umeå (1997). Theory Stoch. Process. 3(19)(1–2), 164–176 (1997)Google Scholar
  36. 36.
    Silvestrov, D.S., Teugels, J.L.: Limit theorems for extremes with random sample size. Adv. Appl. Probab. 30, 777–806 (1998)MathSciNetCrossRefGoogle Scholar
  37. 37.
    Silvestrov, D.S., Stenflo, Ö.: Ergodic theorems for iterated function systems controlled by regenerative sequences. J. Theor. Probab. 11, 589–608 (1998)MathSciNetCrossRefGoogle Scholar
  38. 38.
    Gyllenberg, M., Silvestrov, D.S.: Nonlinearly perturbed regenerative processes and pseudo-stationary phenomena for stochastic systems. Stoch. Process. Appl. 86, 1–27 (2000)MathSciNetCrossRefGoogle Scholar
  39. 39.
    Gyllenberg, M., Silvestrov, D.S.: Cramér-Lundberg approximation for nonlinearly perturbed risk processes. Insur. Math. Econ. 26, 75–90 (2000)CrossRefGoogle Scholar
  40. 40.
    Silvestrov, D.S.: A perturbed renewal equation and an approximation of diffusion type for risk processes. Teor. Ǐmovirn. Math. Stat. 62, 134–144 (2000) (English translation in Theory Probab. Math. Stat. 62, 145–156)Google Scholar
  41. 41.
    Silvestrov, D.S.: Generalized exceeding times, renewal and risk processes. Theory Stoch. Process. 6(22)(3–4), 125–182 (2000)Google Scholar
  42. 42.
    Silvestrov, D.S., Teugels, J.L.: Limit theorems for mixed max-sum processes with renewal stopping. Ann. Appl. Probab. 14(4), 1838–1868 (2004)MathSciNetCrossRefGoogle Scholar
  43. 43.
    Mishura, Yu.S., Silvestrov, D.S.: Limit theorems for stochastic Riemann-Stieltjes integrals. Theory Stoch. Process. 10(26)(1–2), 122–140 (2004)Google Scholar
  44. 44.
    Silvestrov D.S.: Limit Theorems for Randomly Stopped Stochastic Processes. Probability and its applications, xiv+398 pp. Springer, London (2004)CrossRefGoogle Scholar
  45. 45.
    Jönsson, H., Kukush, A.G., Silvestrov, D.S.: Threshold structure of optimal stopping strategies for American type option. 1, 2. Theor. Ǐmovirn. Math. Stat. Part 1: 71, 82–92 (2004); Part 2: 72, 42–53 (2005) (English translation in Theory Probab. Math. Stat. Part 1: 71, 93–103; Part 2: 72, 47–58)Google Scholar
  46. 46.
    Silvestrov, D.S.: Upper bounds for exponential moments of hitting times for semi-Markov processes. Commun. Stat. Theory Methods 33(3), 533–544 (2005)MathSciNetCrossRefGoogle Scholar
  47. 47.
    Silvestrov, D.S., Drozdenko, M.O.: Necessary and sufficient conditions for weak convergence of first-rare-event times for semi-Markov processes. 1, 2. Theory Stoch. Process. 12(28)(3–4); Part 1: 151–186; Part 2: 187–202 (2006)Google Scholar
  48. 48.
    Ni, Y., Silvestrov, D., Malyarenko, A.: Exponential asymptotics for nonlinearly perturbed renewal equation with non-polynomial perturbations. J. Numer. Appl. Math. 1(96), 173–197 (2008)zbMATHGoogle Scholar
  49. 49.
    Gyllenberg, M., Silvestrov, D.S.: Quasi-Stationary Phenomena in Nonlinearly Perturbed Stochastic Systems. De Gruyter expositions in mathematics, vol. 44, ix+579 pp. Walter de Gruyter, Berlin (2008)Google Scholar
  50. 50.
    Silvestrov, D., Jönsson, H., Stenberg, F.: Convergence of option rewards for Markov type price processes modulated by stochastic indices. 1, 2. Theor. Ǐmovirn. Math. Stat. Part 1: 79, 149–165 (2008); Part 2: 80, 138–155 (2009) (Also in Theory Probab. Math. Stat. Part 1: 79, 153–170; Part 2: 80, 153–172)Google Scholar
  51. 51.
    Lundgren, R., Silvestrov, D.S.: Optimal stopping and reselling of European options. In: Rykov, V., Balakrishan, N., Nikulin, M. (eds.) Mathematical and Statistical Models and Methods in Reliability, Chap. 29, 371–390. Birkhäuser, New York (2010)CrossRefGoogle Scholar
  52. 52.
    Silvetsrov, D., Lundgren, R.: Convergence of option rewards for multivariate price processes. Theor. Ǐmovirn. Math. Stat. 85, 102–116 (2011) (Also in Theory Probab. Math. Stat. 85, 115–131)Google Scholar
  53. 53.
    Ekheden, E., Silvestrov, D.: Coupling and explicit rates of convergence in Cramér-Lundberg approximation for reinsurance risk processes. Commun. Stat. Theory Methods 40(19–20), 3524–3539 (2011)CrossRefGoogle Scholar
  54. 54.
    Silvestrov, D., Petersson, M.: Exponential expansions for perturbed discrete time renewal equations. In: Karagrigoriou, A., Lisnianski, A., Kleyner, A., Frenkel, I. (eds.) Applied Reliability Engineering and Risk Analysis. Probabilistic Models and Statistical Inference, Chap. 23, 349–362. Wiley, Chichester (2013)Google Scholar
  55. 55.
    Silvestrov, D.: Improved asymptotics for ruin probabilities. In: Silvestrov, D., Martin-Löf, A. (eds.) Modern Problems in Insurance Mathematics, Chap. 5, 37–68. EAA series, Springer, Cham (2014)Google Scholar
  56. 56.
    Silvestrov, D., Manca, R., Silvestrova, E.: Computational algorithms for moments of accumulated Markov and semi-Markov rewards. Commun. Stat. Theory Methods 43(7), 1453–1469 (2014)MathSciNetCrossRefGoogle Scholar
  57. 57.
    Silvestrov D.S.: American-Type Options. Stochastic Approximation Methods, Volume 1. De Gruyter studies in mathematics, vol. 56, x+509 pp. Walter de Gruyter, Berlin (2014)Google Scholar
  58. 58.
    Silvestrov D.S.: American-Type Options. Stochastic Approximation Methods, Volume 2. De Gruyter studies in mathematics, vol. 57, xi+558 pp. Walter de Gruyter, Berlin (2015)Google Scholar
  59. 59.
    Silvestrov, D., Li, Y.: Stochastic approximation methods for American type options. Commun. Stat. Theory Methods 45(6), 1607–1631 (2016)MathSciNetCrossRefGoogle Scholar
  60. 60.
    Silvestrov, D.: Necessary and sufficient conditions for convergence of first-rare-event times for perturbed semi-Markov processes. Theor. Ǐmovirn. Math. Stat. 95, 119–137 (2016) (Also in Theory Probab. Math. Stat. 95, 135–151)Google Scholar
  61. 61.
    Silvestrov, D., Silvestrov, S.: Asymptotic expansions for stationary distributions of perturbed semi-Markov processes. In: Silvestrov, S., Ranc̆ić, M. (eds.) Engineering Mathematics II. Algebraic, Stochastic and Analysis Structures for Networks, Data Classification and Optimization. Springer proceedings in mathematics and statistics, vol. 179, Chap. 10, 151–222, Springer. Cham (2016)Google Scholar
  62. 62.
    Silvestrov, D., Silvestrov, S.: Asymptotic expansions for stationary distributions of nonlinearly perturbed semi-Markov processes. 1, 2. Method. Comput. Appl. Probab. Part 1:  https://doi.org/10.1007/s11009-017-9605-0, 20 pp.; Part 2:  https://doi.org/10.1007/s11009-017-9607-y, 20 pp. (2017)
  63. 63.
    Silvestrov, D., Silvestrov, S.: Asymptotic expansions for power-exponential moments of hitting times for nonlinearly perturbed semi-Markov processes. Theor. Ǐmovirn. Math. Stat. 97, 171–187 (2017)Google Scholar
  64. 64.
    Silvestrov, D., Manca, R.: Reward algorithms for semi-Markov processes. Method. Comput. Appl. Probab. 19(4), 1191–1209 (2017)MathSciNetCrossRefGoogle Scholar
  65. 65.
    Silvestrov, D., Silvestrov, S.: Nonlinearly Perturbed Semi-Markov Processes. Springer briefs in probability and mathematical statistics, xiv+143 pp. Springer, Cham (2017)CrossRefGoogle Scholar
  66. 66.
    Silvestrov, D.: Individual ergodic theorems for perturbed alternating regenerative processes. In: Silvestrov, S., Ranc̆ić, M., Malyarenko, A. (eds.) Stochastic Processes and Applications. Springer proceedings in mathematics & statistics, vol. 271, Chap. 3. Springer, Cham (2018)Google Scholar
  67. 67.
    Silvestrov, D., Petersson, M., Hössjer, O.: Nonlinearly perturbed birth-death-type models. In: Silvestrov, S., Ranc̆ić, M., Malyarenko, A. (eds.) Stochastic Processes and Applications. Springer proceedings in mathematics & statistics, vol. 271, Chap. 11. Springer, Cham (2018)Google Scholar
  68. 68.

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© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Department of MathematicsStockholm UniversityStockholmSweden

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