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On moments of integrals with respect to Markov additive processes and of Markov modulated generalized Ornstein-Uhlenbeck processes (2207.11093v2)

Published 22 Jul 2022 in math.PR

Abstract: We establish sufficient conditions for the existence, and derive explicit formulas for the $\kappa$'th moments, $\kappa \geq 1$, of Markov modulated generalized Ornstein-Uhlenbeck processes as well as their stationary distributions. In particular, the running mean, the autocovariance function, and integer moments of the stationary distribution are derived in terms of the characteristics of the driving Markov additive process. Our derivations rely on new general results on moments of Markov additive processes and (multidimensional) integrals with respect to Markov additive processes.

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References (40)
  1. S. Asmussen. Applied Probability and Queues. Springer, 2nd edition, 2003.
  2. S. Asmussen and M. Bladt. Gram-Charlier methods, regime-switching and stochastic volatility in exponential Lévy models. Quantitative Finance, 22:4:675–689, 2022.
  3. O.E. Barndorff-Nielsen and N. Shephard. Non-Gaussian Ornstein-Uhlenbeck-based models and some of their uses in financial economics (with discussion). J. R. Statist. Soc. Ser. B, 63:167–241, 2001.
  4. A. Behme. Distributional properties of solutions of d⁢Vt=Vt−⁢d⁢Ut+d⁢Lt𝑑subscript𝑉𝑡subscript𝑉limit-from𝑡𝑑subscript𝑈𝑡𝑑subscript𝐿𝑡d{V}_{t}={V}_{t-}d{U}_{t}+d{L}_{t}italic_d italic_V start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT = italic_V start_POSTSUBSCRIPT italic_t - end_POSTSUBSCRIPT italic_d italic_U start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT + italic_d italic_L start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT with Lévy noise. Adv. in Appl. Probab., 43(3):688–711, 2011.
  5. Stationary solutions of the stochastic differential equation d⁢Vt=Vt−⁢d⁢Ut+d⁢Lt𝑑subscript𝑉𝑡subscript𝑉limit-from𝑡𝑑subscript𝑈𝑡𝑑subscript𝐿𝑡d{V}_{t}={V}_{t-}d{U}_{t}+d{L}_{t}italic_d italic_V start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT = italic_V start_POSTSUBSCRIPT italic_t - end_POSTSUBSCRIPT italic_d italic_U start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT + italic_d italic_L start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT with Lévy noise. Stoch. Proc. Appl., 121:91–108, 2010.
  6. A. Behme and A. Sideris. Exponential functionals of Markov additive processes. Electron. J. Probab., 25(37):25pp, 2020.
  7. A. Behme and A. Sideris. Markov-modulated generalized Ornstein-Uhlenbeck processes and an application in risk theory. Bernoulli, 28:1309–1339, 2022.
  8. Z. Ben Salah and M. Morales. Lévy systems and the time value of ruin for Markov additive processes. Eur. Actuar. J., 2:289–317, 2012.
  9. D.S. Bernstein. Matrix Mathematics: Theory, Facts, and Formulas. Princeton University Press, 2nd edition, 2009.
  10. M. Bladt and B.F. Nielsen. Matrix-Exponential Distributions in Applied Probability. Springer, 2017.
  11. E. Çinlar. Markov additive processes I. Z. Wahrscheinlichkeitstheorie verw. Gebiete, 24:85–93, 1972.
  12. E. Çinlar. Markov additive processes II. Z. Wahrscheinlichkeitstheorie verw. Gebiete, 24:95–121, 1972.
  13. L. de Haan and R. Karandikar. Embedding a stochastic difference equation into a continuous-time process. Stoch. Proc. Appl., 32:225–235, 1988.
  14. Real self-similar processes started from the origin. Ann. Probab., 45(3):1952–2003, 2017.
  15. Hidden Markov Models: Estimation and Control, volume 29. Springer, New York, 1995.
  16. B. Grigelionis. Additive Markov processes. Lith. Math. J., 18:340–342, 1978.
  17. Matrix Analysis. Cambridge University Press, 1985.
  18. Markov-modulated Ornstein-Uhlenbeck processes. Adv. in Appl. Probab., 48:235–254, 2016.
  19. J. Jacod and A. Shiryaev. Limit theorems for stochastic processes. Springer, 2013.
  20. P. Kevei. Ergodic properties of generalized Ornstein–Uhlenbeck processes. Stoch. Proc. Appl., 128:156–181, 2018.
  21. A continuous-time GARCH process driven by a Lévy process: stationarity and second-order behaviour. J. Appl. Probab., 41(3):601–622, 2004.
  22. The hitting time of zero for a stable process. Electron. J. Probab., 19:1–26, 2014.
  23. Entrance laws at the origin of self-similar Markov processes in high dimensions. Trans. Amer. Math. Soc., 373:6227–6299, 2019.
  24. A. Lindner and R. Maller. Lévy integrals and the stationarity of generalised Ornstein-Uhlenbeck processes. Stoch. Proc. Appl., 115:1701–1722, 2005.
  25. F. Lindskog and A. Pal Majumder. Exact long time behaviour of some regime switching stochastic processes. Bernoulli, 26:2572–2604, 2020.
  26. Ornstein-Uhlenbeck processes and extensions. In T. Andersen, R.A. Davis, J.-P. Kreiß, and T. Mikosch, editors, Handbook of Financial Time Series, pages 421–437. Springer, 2009.
  27. C. Marinelli and M. Röckner. On maximal inequalities for purely discontinuous martingales in infinite dimensions. In C. Donati-Martin, A. Lejay, and A. Rouault, editors, Séminaire de Probabilités XLVI, Lecture Notes in Mathematics, pages 293–315. Springer, 2014.
  28. P.A. Meyer. Inegalités de normes pour les integrales stochastiques. In C. Dellacherie, P. A. Meyer, and M. Weil, editors, Séminaire de Probabilités XII, Lecture Notes in Mathematics, pages 757–762. Springer, 1978.
  29. J.R. Norris. Markov Chains. Cambridge University Press, 1997.
  30. A note on chaotic and predictable representations for Itô-Markov additive processes. Stoch. Anal. Appl., 36(4):622–638, 2018.
  31. J. Paulsen. Risk theory in a stochastic economic environment. Stoch. Proc. Appl., 46:327–361, 1993.
  32. P. Protter. Stochastic Integration and Differential Equations. Springer, 2nd edition, 2005.
  33. D. Revuz and M. Yor. Continuous Martingales and Brownian Motion. Springer, 1999.
  34. P. Salminen and L. Vostrikova. On exponential functionals of processes with independent increments. Theory of Probability & Its Applications, 63(2):267–291, 2018.
  35. K. Sato. Lévy processes and infinitely divisible distributions. Cambridge University Press, 2nd edition, 2013.
  36. Infinite Divisibility of Probability Distributions on the Real Line. Marcel Dekker Inc., 2003.
  37. The stationary distributions of two classes of reflected Ornstein-Uhlenbeck processes. J. Appl. Probab., 46(3):709–720, 2009.
  38. A. Yashin. The expected number of transitions from one state to another: A medico-demographic model. IIASA Working paper, 1982.
  39. Markovian regime-switching market completion using additional Markov jump assets. IMA Journal of Management Mathematics, 23(3):283–305, 07 2011.
  40. Z. Zhang and W. Wang. The stationary distribution of Ornstein Uhlenbeck process with a two-state Markov switching. Communications in Statistics - Simulation and Computation, 36:4783–4794, 2017.
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