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Necessary and sufficient conditions for ergodicity of CIR model driven by stable processes with Markov switching

 Department of Applied Mathematics, Donghua University, Shanghai 201620, China

* Corresponding author: Jinying Tong.

Received  April 2017 Revised  September 2017 Published  February 2018

Fund Project: The author Zhenzhong Zhang is supported by the Humanities and Social Sciences Fund of Ministry of Education of China (No. 17YJA910004). The author Jinying Tong is supported by the National Natural Science Foundation of China (Nos. 11401093 and 11471071).

In this paper, we consider long time behavior of the Cox-Ingersoll-Ross (CIR) interest rate model driven by stable processes with Markov switching. Under some assumptions, we prove an ergodicity-transience dichotomy, namely, the interest rate process is either ergodic or transient. The sufficient and necessary conditions for ergodicity and transience of such interest model are given under some assumptions. Finally, an application to interval estimation of the interest rate processes is presented to illustrate our results.

Citation: Zhenzhong Zhang, Enhua Zhang, Jinying Tong. Necessary and sufficient conditions for ergodicity of CIR model driven by stable processes with Markov switching. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2018053
References:
 [1] M.Abramowitz and I.A.Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, John Wiley and Sons Incorporated, New York, 1984. [2] D.Applebaum, Lévy Processes and Stochastic Calculus, 2$^{nd}$ edition, Cambridge University Press, Cambridge, 2009. [3] A. Arapostathis, A. Biswas and L. Caffarelli, The Dirichlet problem for stable like operators and related probabilistic representations, Commun. Part. Diff. Eq., 41 (2016), 1472-1511. [4] A.Berman and R.J.Plemmons, Nonnegative Matrices in the Mathematical Science, SIAM Press classics Series, Philadelphia, 1994. [5] Z. Chen and J. Wang, Ergodicity for time-changed symmetric stable processes, Stoch. Proc. Appl., 124 (2014), 2799-2823. doi: 10.1016/j.spa.2014.04.003. [6] A. Clauset, C. R. Shalizi and M. E. J. Newman, Power-law distributions in empirical data, SIAM Rev., 51 (2009), 661-703. doi: 10.1137/070710111. [7] J. C. Cox, J. E. Ingersoll and S. A. Ross, A theory of the term structure of interest rates, Econometrica, 53 (1985), 385-407. doi: 10.2307/1911242. [8] N.Fournier, On pathwise uniqueness for stochastic differential equations driven by stable Lévy processes, Ann.Inst.Henri Poincaré Probab.Stat., 49 (2013), 138-159. [9] K. Handa, Ergodic properties for $α$-CIR models and a class of generalized Fleming-Viot processes, Electron. J. Probab., 19 (2014), 1-25. [10] Y. Jiao, C. Ma and S. Scotti, Alpha-CIR model with branching processes in sovereign interest rate modelling, Financ. Stoch., 21 (2017), 789-813. doi: 10.1007/s00780-017-0333-7. [11] R.Khasminskii, Stochastic Stability of Differential Equations, Springer, Berlin, 2012. [12] X. Li, A. Gray, D. Jiang and X. Mao, Sufficient and necessary conditions of stochastic permanence and extinction for stochastic logistic populations under regime switching, J. Math. Anal. Appl., 376 (2011), 11-28. doi: 10.1016/j.jmaa.2010.10.053. [13] Z. Li and C. Ma, Asymptptic properties of estimators in a stable Cox-Ingersoll-Ross model, Stoch. Proc. Appl., 125 (2015), 3196-3233. doi: 10.1016/j.spa.2015.03.002. [14] B. B. Mandelbrot, The variation of certain speculative prices, J. Bus., 36 (1963), 394-419. [15] X. Mao, Stability of stochastic differential equations with Markovian switching, Stoch. Proc. Appl., 79 (1999), 45-67. doi: 10.1016/S0304-4149(98)00070-2. [16] X. Mao, G. Yin and C. Yuan, Stabilization and destabilization of hybrid systems of stochastic differential equations, Automatica, 43 (2007), 264-273. doi: 10.1016/j.automatica.2006.09.006. [17] M. Pinsky and R. Pinsky, Transience recurrence and central limit theorem behavior for diffusions in random temporal environments, Ann. Probab., 21 (1993), 433-452. doi: 10.1214/aop/1176989410. [18] G.Samorodnitsky and M.S.Taqqu, Stable Non-Gaussian Random Processes: Stochastic modeling, Chapman & Hall, New York, 1994. [19] N. Sandrić, Long-time behavior of stable-like processes, Stoch. Proc. Appl., 123 (2013), 1276-1300. doi: 10.1016/j.spa.2012.12.004. [20] D. R. Smith, Markov-switching and stochastic volatility diffusion models of short-term interest rates, J. Bus. Econ. Stat., 20 (2002), 183-197. doi: 10.1198/073500102317351949. [21] J.Tong and Z.Zhang, Exponential ergodicity of CIR interest rate model with random switching, Stoch.Dynam., 17 (2017), 1750037, 20pp. [22] J. T. Wu, Markov regimes switching with monetary fundamental-based exchange rate model, Asia Pac. Man. Rev., 20 (2015), 79-89. doi: 10.1016/j.apmrv.2014.12.009. [23] Z. Zhang, J. Tong and L. Hu, Long-term behavior of stochastic interest rate models with Markov switching, Insur. Math. Econ., 70 (2016), 320-326. doi: 10.1016/j.insmatheco.2016.06.017. [24] N. Zhou and R. Mamon, An accessible implementation of interest rate models with Markov-switching, Expert Syst. Appl., 39 (2012), 4679-4689. doi: 10.1016/j.eswa.2011.09.053.

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References:
 [1] M.Abramowitz and I.A.Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, John Wiley and Sons Incorporated, New York, 1984. [2] D.Applebaum, Lévy Processes and Stochastic Calculus, 2$^{nd}$ edition, Cambridge University Press, Cambridge, 2009. [3] A. Arapostathis, A. Biswas and L. Caffarelli, The Dirichlet problem for stable like operators and related probabilistic representations, Commun. Part. Diff. Eq., 41 (2016), 1472-1511. [4] A.Berman and R.J.Plemmons, Nonnegative Matrices in the Mathematical Science, SIAM Press classics Series, Philadelphia, 1994. [5] Z. Chen and J. Wang, Ergodicity for time-changed symmetric stable processes, Stoch. Proc. Appl., 124 (2014), 2799-2823. doi: 10.1016/j.spa.2014.04.003. [6] A. Clauset, C. R. Shalizi and M. E. J. Newman, Power-law distributions in empirical data, SIAM Rev., 51 (2009), 661-703. doi: 10.1137/070710111. [7] J. C. Cox, J. E. Ingersoll and S. A. Ross, A theory of the term structure of interest rates, Econometrica, 53 (1985), 385-407. doi: 10.2307/1911242. [8] N.Fournier, On pathwise uniqueness for stochastic differential equations driven by stable Lévy processes, Ann.Inst.Henri Poincaré Probab.Stat., 49 (2013), 138-159. [9] K. Handa, Ergodic properties for $α$-CIR models and a class of generalized Fleming-Viot processes, Electron. J. Probab., 19 (2014), 1-25. [10] Y. Jiao, C. Ma and S. Scotti, Alpha-CIR model with branching processes in sovereign interest rate modelling, Financ. Stoch., 21 (2017), 789-813. doi: 10.1007/s00780-017-0333-7. [11] R.Khasminskii, Stochastic Stability of Differential Equations, Springer, Berlin, 2012. [12] X. Li, A. Gray, D. Jiang and X. Mao, Sufficient and necessary conditions of stochastic permanence and extinction for stochastic logistic populations under regime switching, J. Math. Anal. Appl., 376 (2011), 11-28. doi: 10.1016/j.jmaa.2010.10.053. [13] Z. Li and C. Ma, Asymptptic properties of estimators in a stable Cox-Ingersoll-Ross model, Stoch. Proc. Appl., 125 (2015), 3196-3233. doi: 10.1016/j.spa.2015.03.002. [14] B. B. Mandelbrot, The variation of certain speculative prices, J. Bus., 36 (1963), 394-419. [15] X. Mao, Stability of stochastic differential equations with Markovian switching, Stoch. Proc. Appl., 79 (1999), 45-67. doi: 10.1016/S0304-4149(98)00070-2. [16] X. Mao, G. Yin and C. Yuan, Stabilization and destabilization of hybrid systems of stochastic differential equations, Automatica, 43 (2007), 264-273. doi: 10.1016/j.automatica.2006.09.006. [17] M. Pinsky and R. Pinsky, Transience recurrence and central limit theorem behavior for diffusions in random temporal environments, Ann. Probab., 21 (1993), 433-452. doi: 10.1214/aop/1176989410. [18] G.Samorodnitsky and M.S.Taqqu, Stable Non-Gaussian Random Processes: Stochastic modeling, Chapman & Hall, New York, 1994. [19] N. Sandrić, Long-time behavior of stable-like processes, Stoch. Proc. Appl., 123 (2013), 1276-1300. doi: 10.1016/j.spa.2012.12.004. [20] D. R. Smith, Markov-switching and stochastic volatility diffusion models of short-term interest rates, J. Bus. Econ. Stat., 20 (2002), 183-197. doi: 10.1198/073500102317351949. [21] J.Tong and Z.Zhang, Exponential ergodicity of CIR interest rate model with random switching, Stoch.Dynam., 17 (2017), 1750037, 20pp. [22] J. T. Wu, Markov regimes switching with monetary fundamental-based exchange rate model, Asia Pac. Man. Rev., 20 (2015), 79-89. doi: 10.1016/j.apmrv.2014.12.009. [23] Z. Zhang, J. Tong and L. Hu, Long-term behavior of stochastic interest rate models with Markov switching, Insur. Math. Econ., 70 (2016), 320-326. doi: 10.1016/j.insmatheco.2016.06.017. [24] N. Zhou and R. Mamon, An accessible implementation of interest rate models with Markov-switching, Expert Syst. Appl., 39 (2012), 4679-4689. doi: 10.1016/j.eswa.2011.09.053.
Computer simulation of a single path of $X_t$ with initial value $X_0 = 0.3,r_0 = 1$ and different coefficients $\alpha = 1.25$(up), $\alpha = 1.75$(down)
Computer simulation of a single path of $X_t$ with initial value $X_0 = 0.3,r_0 = 1$ and $\alpha = 1.75$.
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