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

Lindskog, Filip; Majumder, Abhishek Pal

Bernoulli 26, No. 4, 2572-2604 (2020).

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 \(X\perp \!\!\!\!\perp (A,B)\).

MSC:

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

Keywords:

Cox Ingersoll Ross; long time behavior; Ornstein Uhlenbeck; regime switching; SIS epidemic model

Software:

MCQueue

References:

[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