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

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, 2018, 23 (6) : 2433-2455. 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.  Google Scholar

[2]

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

[3]

A. ArapostathisA. 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. ClausetC. 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. CoxJ. 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. JiaoC. 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. LiA. GrayD. 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. MaoG. 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. ZhangJ. 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. ArapostathisA. 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. ClausetC. 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. CoxJ. 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. JiaoC. 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. LiA. GrayD. 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. MaoG. 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. ZhangJ. 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)
Figure 2.  Computer simulation of a single path of $X_t$ with initial value $X_0 = 0.3,r_0 = 1$ and $\alpha = 1.75$.
[1]

Lei Liu, Li Wu. Multiplicity of closed characteristics on $ P $-symmetric compact convex hypersurfaces in $ \mathbb{R}^{2n} $. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020378

[2]

Federico Rodriguez Hertz, Zhiren Wang. On $ \epsilon $-escaping trajectories in homogeneous spaces. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 329-357. doi: 10.3934/dcds.2020365

[3]

Mathew Gluck. Classification of solutions to a system of $ n^{\rm th} $ order equations on $ \mathbb R^n $. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5413-5436. doi: 10.3934/cpaa.2020246

[4]

Yichen Zhang, Meiqiang Feng. A coupled $ p $-Laplacian elliptic system: Existence, uniqueness and asymptotic behavior. Electronic Research Archive, 2020, 28 (4) : 1419-1438. doi: 10.3934/era.2020075

[5]

Aihua Fan, Jörg Schmeling, Weixiao Shen. $ L^\infty $-estimation of generalized Thue-Morse trigonometric polynomials and ergodic maximization. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 297-327. doi: 10.3934/dcds.2020363

[6]

Denis Bonheure, Silvia Cingolani, Simone Secchi. Concentration phenomena for the Schrödinger-Poisson system in $ \mathbb{R}^2 $. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020447

[7]

Lihong Zhang, Wenwen Hou, Bashir Ahmad, Guotao Wang. Radial symmetry for logarithmic Choquard equation involving a generalized tempered fractional $ p $-Laplacian. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020445

[8]

Mokhtar Bouloudene, Manar A. Alqudah, Fahd Jarad, Yassine Adjabi, Thabet Abdeljawad. Nonlinear singular $ p $ -Laplacian boundary value problems in the frame of conformable derivative. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020442

[9]

Thabet Abdeljawad, Mohammad Esmael Samei. Applying quantum calculus for the existence of solution of $ q $-integro-differential equations with three criteria. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020440

[10]

Wenjun Liu, Yukun Xiao, Xiaoqing Yue. Classification of finite irreducible conformal modules over Lie conformal algebra $ \mathcal{W}(a, b, r) $. Electronic Research Archive, , () : -. doi: 10.3934/era.2020123

[11]

Zuliang Lu, Fei Huang, Xiankui Wu, Lin Li, Shang Liu. Convergence and quasi-optimality of $ L^2- $norms based an adaptive finite element method for nonlinear optimal control problems. Electronic Research Archive, 2020, 28 (4) : 1459-1486. doi: 10.3934/era.2020077

[12]

Siyang Cai, Yongmei Cai, Xuerong Mao. A stochastic differential equation SIS epidemic model with regime switching. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020317

[13]

Laurence Cherfils, Stefania Gatti, Alain Miranville, Rémy Guillevin. Analysis of a model for tumor growth and lactate exchanges in a glioma. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020457

[14]

Laurent Di Menza, Virginie Joanne-Fabre. An age group model for the study of a population of trees. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020464

[15]

Weiwei Liu, Jinliang Wang, Yuming Chen. Threshold dynamics of a delayed nonlocal reaction-diffusion cholera model. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020316

[16]

Yining Cao, Chuck Jia, Roger Temam, Joseph Tribbia. Mathematical analysis of a cloud resolving model including the ice microphysics. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 131-167. doi: 10.3934/dcds.2020219

[17]

Zhouchao Wei, Wei Zhang, Irene Moroz, Nikolay V. Kuznetsov. Codimension one and two bifurcations in Cattaneo-Christov heat flux model. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020344

[18]

Shuang Chen, Jinqiao Duan, Ji Li. Effective reduction of a three-dimensional circadian oscillator model. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020349

[19]

Barbora Benešová, Miroslav Frost, Lukáš Kadeřávek, Tomáš Roubíček, Petr Sedlák. An experimentally-fitted thermodynamical constitutive model for polycrystalline shape memory alloys. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020459

[20]

Cuicui Li, Lin Zhou, Zhidong Teng, Buyu Wen. The threshold dynamics of a discrete-time echinococcosis transmission model. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020339

2019 Impact Factor: 1.27

Metrics

  • PDF downloads (164)
  • HTML views (456)
  • Cited by (1)

Other articles
by authors

[Back to Top]