American Institute of Mathematical Sciences

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August  2018, 23(6): 2433-2455. doi: 10.3934/dcdsb.2018053

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

Zhenzhong Zhang , Enhua Zhang and  Jinying Tong ,

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.

Keywords: CIR model, symmetric $α$-stable process, Markov switching, stationary distribution, transience.
Mathematics Subject Classification: Primary: 60G52, 60J27.
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, 2018, 23 (6) : 2433-2455. doi: 10.3934/dcdsb.2018053
References:
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D.Applebaum, Lévy Processes and Stochastic Calculus, 2$^{nd}$ edition, Cambridge University Press, Cambridge, 2009.  Google Scholar

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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.   Google Scholar

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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.  Google Scholar

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N.Fournier, On pathwise uniqueness for stochastic differential equations driven by stable Lévy processes, Ann.Inst.Henri Poincaré Probab.Stat., 49 (2013), 138-159.  Google Scholar

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K. Handa, Ergodic properties for $α$-CIR models and a class of generalized Fleming-Viot processes, Electron. J. Probab., 19 (2014), 1-25.   Google Scholar

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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.  Google Scholar

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R.Khasminskii, Stochastic Stability of Differential Equations, Springer, Berlin, 2012.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[14]

B. B. Mandelbrot, The variation of certain speculative prices, J. Bus., 36 (1963), 394-419.   Google Scholar

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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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[18]

G.Samorodnitsky and M.S.Taqqu, Stable Non-Gaussian Random Processes: Stochastic modeling, Chapman & Hall, New York, 1994.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

J.Tong and Z.Zhang, Exponential ergodicity of CIR interest rate model with random switching, Stoch.Dynam., 17 (2017), 1750037, 20pp.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

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.  Google Scholar

[2]

D.Applebaum, Lévy Processes and Stochastic Calculus, 2$^{nd}$ edition, Cambridge University Press, Cambridge, 2009.  Google Scholar

[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.   Google Scholar

[4]

A.Berman and R.J.Plemmons, Nonnegative Matrices in the Mathematical Science, SIAM Press classics Series, Philadelphia, 1994.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

K. Handa, Ergodic properties for $α$-CIR models and a class of generalized Fleming-Viot processes, Electron. J. Probab., 19 (2014), 1-25.   Google Scholar

[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.  Google Scholar

[11]

R.Khasminskii, Stochastic Stability of Differential Equations, Springer, Berlin, 2012.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

B. B. Mandelbrot, The variation of certain speculative prices, J. Bus., 36 (1963), 394-419.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

G.Samorodnitsky and M.S.Taqqu, Stable Non-Gaussian Random Processes: Stochastic modeling, Chapman & Hall, New York, 1994.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

J.Tong and Z.Zhang, Exponential ergodicity of CIR interest rate model with random switching, Stoch.Dynam., 17 (2017), 1750037, 20pp.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

Figure 1.  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)
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Figure 2.  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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