Exact long time behavior of some regime switching stochastic processes. (English) Zbl 1455.60106

Summary: Regime switching processes have proved to be indispensable in the modeling of various phenomena, allowing model parameters that traditionally were considered to be constant to fluctuate in a Markovian manner in line with empirical findings. We study diffusion processes of Ornstein-Uhlenbeck type where the drift and diffusion coefficients \(a\) and \(b\) are functions of a Markov process with a stationary distribution \(\pi\) on a countable state space. Exact long time behavior is determined for the three regimes corresponding to the expected drift: \(E_{\pi}a(\cdot )>0, =0,<0\), respectively. Alongside we provide exact time limit results for integrals of form \(\int_0^tb^2(X_s)e^{-2\int_s^ta(X_r)\,dr}\,ds\) for the three different regimes. Finally, we demonstrate natural applications of the findings in terms of Cox-Ingersoll-Ross diffusion and deterministic SIS epidemic models in Markovian environments. The time asymptotic behaviors are naturally expressed in terms of solutions to the well-studied fixed-point equation in law \(X\overset{d}{=}AX+B\) with \(X\perp \!\!\!\!\perp (A,B)\).


60J60 Diffusion processes
60J27 Continuous-time Markov processes on discrete state spaces


Full Text: DOI arXiv Euclid


[1] Abourashchi, N., Clacher, I., Freeman, M.C., Hillier, D., Kemp, M. and Zhang, Q. (2016). Pension plan solvency and extreme market movements: A regime switching approach. Eur. J. Finance 22 1292-1319.
[2] Ang, A. and Timmermann, A. (2012). Regime changes and financial markets. Annu. Rev. Financ. Econ. 4 313-337.
[3] Bardet, J.-B., Guérin, H. and Malrieu, F. (2010). Long time behavior of diffusions with Markov switching. ALEA Lat. Am. J. Probab. Math. Stat. 7 151-170. · Zbl 1276.60084
[4] Basak, G.K., Bisi, A. and Ghosh, M.K. (1996). Stability of a random diffusion with linear drift. J. Math. Anal. Appl. 202 604-622. · Zbl 0856.93102
[5] Behme, A. and Lindner, A. (2015). On exponential functionals of Lévy processes. J. Theoret. Probab. 28 681-720. · Zbl 1323.60063
[6] Benaïm, M. and Lobry, C. (2016). Lotka-Volterra with randomly fluctuating environments or “How switching between beneficial environments can make survival harder”. Ann. Appl. Probab. 26 3754-3785. · Zbl 1358.92075
[7] BenSaida, A. (2015). The frequency of regime switching in financial market volatility. J. Empir. Finance 32 63-79.
[8] Bertoin, J. and Yor, M. (2005). Exponential functionals of Lévy processes. Probab. Surv. 2 191-212. · Zbl 1189.60096
[9] Buraczewski, D., Damek, E. and Mikosch, T. (2016). Stochastic Models with Power-Law Tails. Springer Series in Operations Research and Financial Engineering. Cham: Springer. · Zbl 1357.60004
[10] Buraczewski, D. and Iksanov, A. (2015). Functional limit theorems for divergent perpetuities in the contractive case. Electron. Commun. Probab. 20 no. 10, 14. · Zbl 1307.60026
[11] Cappelletti, D., Pal Majumder, A. and Wiuf, C. (2019). Long time asymptotics of stochastic reaction systems. Preprint. Available at arXiv:1912.00401.
[12] Cloez, B. and Hairer, M. (2015). Exponential ergodicity for Markov processes with random switching. Bernoulli 21 505-536. · Zbl 1330.60094
[13] Cox, J.C., Ingersoll, J.E. Jr. and Ross, S.A. (1985). A theory of the term structure of interest rates. Econometrica 53 385-407. · Zbl 1274.91447
[14] de Saporta, B. and Yao, J.-F. (2005). Tail of a linear diffusion with Markov switching. Ann. Appl. Probab. 15 992-1018. · Zbl 1064.60174
[15] Djehiche, B. and Löfdahl, B. (2018). A hidden Markov approach to disability insurance. N. Am. Actuar. J. 22 119-136. · Zbl 1393.62044
[16] Erdös, P. and Kac, M. (1946). On certain limit theorems of the theory of probability. Bull. Amer. Math. Soc. 52 292-302. · Zbl 0063.01274
[17] Feng, R., Kuznetsov, A. and Yang, F. (2019). Exponential functionals of Lévy processes and variable annuity guaranteed benefits. Stochastic Process. Appl. 129 604-625. · Zbl 1405.60060
[18] Fink, H., Klimova, Y., Czado, C. and Stöber, J. (2017). Regime switching vine copula models for global equity and volatility indices. Econometrics 5 3-17.
[19] Gao, H., Mamon, R., Liu, X. and Tenyakov, A. (2015). Mortality modelling with regime-switching for the valuation of a guaranteed annuity option. Insurance Math. Econom. 63 108-120. · Zbl 1348.91145
[20] Genon-Catalot, V., Jeantheau, T. and Larédo, C. (2000). Stochastic volatility models as hidden Markov models and statistical applications. Bernoulli 6 1051-1079. · Zbl 0966.62048
[21] Gjessing, H.K. and Paulsen, J. (1997). Present value distributions with applications to ruin theory and stochastic equations. Stochastic Process. Appl. 71 123-144. · Zbl 0943.60098
[22] Gray, A., Greenhalgh, D., Mao, X. and Pan, J. (2012). The SIS epidemic model with Markovian switching. J. Math. Anal. Appl. 394 496-516. · Zbl 1271.92030
[23] Gut, A. (2009). Stopped Random Walks: Limit Theorems and Applications, 2nd ed. Springer Series in Operations Research and Financial Engineering. New York: Springer. · Zbl 1166.60001
[24] Hairer, M. (2010). Convergence of Markov processes. Lecture notes.
[25] Hardy, M.R. (2001). A regime-switching model of long-term stock returns. N. Am. Actuar. J. 5 41-53. · Zbl 1083.62530
[26] Hitczenko, P. and Wesolowski, J. (2011). Renorming divergent perpetuities. Bernoulli 17 880-894. · Zbl 1232.60017
[27] Hou, T. and Shao, J. (2019). Heavy tail and light tail of Cox-Ingersoll-Ross processes with regime-switching. Sci. China Math.. https://doi.org/10.1007/s11425-017-9392-5. · Zbl 1459.60161
[28] Jeanblanc, M., Yor, M. and Chesney, M. (2009). Mathematical Methods for Financial Markets. Springer Finance. London: Springer London, Ltd. · Zbl 1205.91003
[29] Lin, X.S., Tan, K.S. and Yang, H. (2009). Pricing annuity guarantees under a regime-switching model. N. Am. Actuar. J. 13 316-338.
[30] Lindskog, F. and Pal Majumder, A. (2020). Supplement to “Exact long time behavior of some regime switching stochastic processes.” https://doi.org/10.3150/20-BEJ1196SUPP
[31] Mao, X. and Yuan, C. (2006). Stochastic Differential Equations with Markovian Switching. London: Imperial College Press. · Zbl 1126.60002
[32] Maulik, K. and Zwart, B. (2006). Tail asymptotics for exponential functionals of Lévy processes. Stochastic Process. Appl. 116 156-177. · Zbl 1090.60046
[33] Norris, J.R. (1998). Markov Chains. Cambridge Series in Statistical and Probabilistic Mathematics 2. Cambridge: Cambridge Univ. Press.
[34] Schilling, R.L. (2005). Measures, Integrals and Martingales. New York: Cambridge Univ. Press. · Zbl 1084.28001
[35] Shao, J. (2014). Ergodicity of one-dimensional regime-switching diffusion processes. Sci. China Math. 57 2407-2414. · Zbl 1360.60153
[36] Shao, J. (2015). Ergodicity of regime-switching diffusions in Wasserstein distances. Stochastic Process. Appl. 125 739-758. · Zbl 1322.60165
[37] Shao, J. (2015). Criteria for transience and recurrence of regime-switching diffusion processes. Electron. J. Probab. 20 no. 63, 15. · Zbl 1327.60017
[38] Shen, Y. and Siu, T.K. (2013). Longevity bond pricing under stochastic interest rate and mortality with regime-switching. Insurance Math. Econom. 52 114-123. · Zbl 1291.91212
[39] Tijms, H.C. (2003). A First Course in Stochastic Models. Chichester: Wiley. · Zbl 1088.60002
[40] Vervaat, W. (1979). On a stochastic difference equation and a representation of nonnegative infinitely divisible random variables. Adv. in Appl. Probab. 11 750-783. · Zbl 0417.60073
[41] Yin, G.G. and Zhu, C. (2010). Hybrid Switching Diffusions: Properties and Applications. Stochastic Modelling and Applied Probability 63. New York: Springer.
[42] Zhang, Z., Tong, J. and Hu, L. (2016). Long-term behavior of stochastic interest rate models with Markov switching. Insurance Math. Econom. 70 320-326. · Zbl 1371.91186
[43] Zhang, Z. · Zbl 1390.60130
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.