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November  2019, 39(11): 6467-6483. doi: 10.3934/dcds.2019280

Exponential stability of SDEs driven by fBm with Markovian switching

Litan Yan 1,2, , Wenyi Pei 1, and  Zhenzhong Zhang 2,3,,

1. 

College of Information Science and Technology, Donghua University, Shanghai 201620, China
2. 

Department of Statistics, Donghua University, Shanghai 201620, China
3. 

Institute for Nonlinear Science, Donghua University, Shanghai 201620, China

* Corresponding author: Tel: +86 021 67792412. E-mail: zzzhang@dhu.edu.cn(Z. Zhang)

Received  December 2018 Revised  May 2019 Published  August 2019

In this paper, we focus on the exponential stability of stochastic differential equations driven by fractional Brownian motion (fBm) with Hurst parameter $ H\in(1/2, 1) $. Based on the generalized Itô formula and representation of the fBm, some sufficient conditions for exponential stability of a class of SDEs with additive fractional noise are given. Besides, we present a criterion on the exponential stability for the fractional Ornstein-Uhlenbeck process with Markov switching. A numerical example is provided to illustrate our results.

Keywords: Stochastic differential equation, exponential stability, fractional Brownian motion, Markovian switching, Poisson equation.
Mathematics Subject Classification: Primary: 60H10, 60G22; Secondary: 93E15, 93C30.
Citation: Litan Yan, Wenyi Pei, Zhenzhong Zhang. Exponential stability of SDEs driven by fBm with Markovian switching. Discrete and Continuous Dynamical Systems, 2019, 39 (11) : 6467-6483. doi: 10.3934/dcds.2019280
References:
[1]

E. Alòs, O. Mazet and D. Nualart, Stochastic calculus with respect to Gaussian processes, Ann. Probab, 29 (2001), 766-801.  doi: 10.1214/aop/1008956692.

[2]

W. J. Anderson, Continuous-Time Markov Chains: An Applications-Oriented Approach, Springer Series in Statistics: Probability and its Applications, Springer-Verlag, New York, 1991.

[3]

J.-B. Bardet, H. Guérin and F. Malrieu, Long time behavior of diffusions with Markov switching, ALEA Lat. Am. J. Probab. Math. Stat, 7 (2010), 151-170. 

[4]

A. Berman and R. J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1994. doi: 10.1137/1.9781611971262.

[5]

F. Biagini, Y. Z. Hu, B. ∅ksendal and T. S. Zhang, Stochastic Calculus for Fractional Brownian Motion and Applications, Probability and its Applications (New York), Springer-Verlag London, 2008. doi: 10.1007/978-1-84628-797-8.

[6]

D. C. Brody, J. Syroka and M. Zervos, Dynamical pricing of weather derivatives, Quant. Finance, 2 (2002), 189-198.  doi: 10.1088/1469-7688/2/3/302.

[7]

B. Cloez and M. Hairer, Exponential ergodicity for Markov processes with random switching, Bernoulli, 21 (2015), 505-536.  doi: 10.3150/13-BEJ577.

[8]

T. E. Duncan, B. Maslowski and B. Pasik-Duncan, Stochastic equations in Hilbert space with a multiplicative fractional Gaussian noise, Stochastic Process. Appl., 115 (2005), 1357-1383.  doi: 10.1016/j.spa.2005.03.011.

[9]

T. E. Duncan, Y. Z. Hu and B. Pasik-Duncan, Stochastic calculus for fractional Brownian motion. I. Theory, SIAM J. Control Optim., 38 (2000), 582-612.  doi: 10.1137/S036301299834171X.

[10]

T. E. Duncan, B. Pasik-Duncan and B. Maslowski, Fractional Brownian motion and stochastic equations in Hilbert spaces, Stoch. Dyn., 2 (2002), 225-250.  doi: 10.1142/S0219493702000340.

[11]

M. K. Ghosh, A. Arapostahis and S. I. Marcus, Ergodic control of switching diffusions, SIAM J. Control Optim., 35 (1997), 1952-1988.  doi: 10.1137/S0363012996299302.

[12]

J. D. Hamilton, A new approach to the economic analysis of nonstationary time series and the business cycle, Econometrica, 57 (1989), 357-384.  doi: 10.2307/1912559.

[13]

H. Holden, B. ∅ksendal, J. Ubøe and T. S. Zhang, Stochastic Partial Differential Equations, 2nd edition, Universitext, Springer, New York, 2010. doi: 10.1007/978-0-387-89488-1.

[14]

Y. Z. Hu, Itô Stochastic differential equations driven by fractional Brownian motion of Hurst parameter $H>1/2$, Stochastics, 90 (2018), 720-761.  doi: 10.1080/17442508.2017.1415342.

[15]

G. Huang, H. M. Jansen, M. Mandjes, P. Spreij and K. De. Turck, Markov-modulated Ornstein-Uhlenbeck processes, Adv. in Appl. Probab., 48 (2016), 235-254.  doi: 10.1017/apr.2015.15.

[16]

I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus, 2nd edition, Graduate Texts in Mathematics, 113. Springer, New York, 1991. doi: 10.1007/978-1-4612-0949-2.

[17]

M. L. Li and F. Q. Deng, Almost sure stability with general decay rate of neutral stochasti delayed hybrid systems with Lévy noise, Nonlinear Anal. Hybrid Syst., 24 (2017), 171-185.  doi: 10.1016/j.nahs.2017.01.001.

[18]

X. R. Mao, Stability of stochastic differential equations with Markovian switching, Stochastic Process. Appl., 79 (1999), 45-67.  doi: 10.1016/S0304-4149(98)00070-2.

[19] X. R. Mao and C. G. Yuan, Stochastic Differential Equations with Markovian Switching, Imperial College Press, London, 2006.  doi: 10.1142/p473.
[20]

J. Mémin, Y. Mishura and E. Valkeila, Inequalities for the moments of Wiener integrals with respect to a fractional Brownian motion, Statist. Probab. Lett., 51 (2001), 197-206.  doi: 10.1016/S0167-7152(00)00157-7.

[21]

Y. S. Mishura, Stochastic Calculus for Fractional Brownian Motion and Related Processes, Lecture Notes in Mathematics, 1929. Springer-Verlag, Berlin, 2008. doi: 10.1007/978-3-540-75873-0.

[22]

D. Nualart and A. Rășcanu, Differential equations driven by fractional Brownian motion, Collect. Math., 53 (2002), 55-81. 

[23]

A. Rathinasamy and M. Balachandran, Mean-square stability of Milstein method for linear hybrid stochastic delay integro-differential equations, Nonlinear Anal. Hybrid Syst., 2 (2008), 1256-1263.  doi: 10.1016/j.nahs.2008.09.015.

[24]

S. Cong, On almost sure stability conditions of linear switching stochastic differential systems, Nonlinear Anal. Hybrid Syst., 22 (2016), 108-115.  doi: 10.1016/j.nahs.2016.03.010.

[25]

I. Simonsen, Measuring anti-correlations in the Nordic electricity spot market by wavelets, Physica A: Statistical Mechanics and its Applications, 322 (2003), 597-606.  doi: 10.1016/S0378-4371(02)01938-6.

[26]

A. V. Skorohod, Asymptotic Methods in the Theory of Stochastic Differential Equations, Translations of Mathematical Monographs, 78. American Mathematical Society, Providence, RI, 1989.

[27]

L. Tan, Exponential stability of fractional stochastic differential equations with distributed delay, Adv. Difference Equ., 2014 (2014), 8 pp. doi: 10.1186/1687-1847-2014-321.

[28]

G. G. Yin and C. Zhu, Hybrid Switching Diffusions: Properties and Applications, Stochastic Modelling and Applied Probability, 63. Springer, New York, 2010. doi: 10.1007/978-1-4419-1105-6.

[29]

C. G. Yuan and X. R. Mao, Asymptotic stability in distribution of stochastic differential equations with Markovian switching, Stochastic Process. Appl., 103 (2003), 277-291.  doi: 10.1016/S0304-4149(02)00230-2.

[30]

C. G. Yuan and X. R. Mao, Stability of stochastic delay hybrid systems with jumps, Eur. J. Control, 16 (2010), 595-608.  doi: 10.3166/ejc.16.595-608.

[31]

Z. Z. Zhang, J. Y. Tong and L. J. Hu, Long-term behavior of stochastic interest rate models with Markov switching, Insurance Math. Econom., 70 (2016), 320-326.  doi: 10.1016/j.insmatheco.2016.06.017.

[32]

W. N. Zhou, J. Yang, X. Q. Yang, A. D. Dai, H. S. Liu and J. Fang, pth Moment exponential stability of stochastic delayed hybrid systems with Lévy noise, Appl. Math. Model., 39 (2015), 5650-5658.  doi: 10.1016/j.apm.2015.01.025.

show all references

References:
[1]

E. Alòs, O. Mazet and D. Nualart, Stochastic calculus with respect to Gaussian processes, Ann. Probab, 29 (2001), 766-801.  doi: 10.1214/aop/1008956692.

[2]

W. J. Anderson, Continuous-Time Markov Chains: An Applications-Oriented Approach, Springer Series in Statistics: Probability and its Applications, Springer-Verlag, New York, 1991.

[3]

J.-B. Bardet, H. Guérin and F. Malrieu, Long time behavior of diffusions with Markov switching, ALEA Lat. Am. J. Probab. Math. Stat, 7 (2010), 151-170. 

[4]

A. Berman and R. J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1994. doi: 10.1137/1.9781611971262.

[5]

F. Biagini, Y. Z. Hu, B. ∅ksendal and T. S. Zhang, Stochastic Calculus for Fractional Brownian Motion and Applications, Probability and its Applications (New York), Springer-Verlag London, 2008. doi: 10.1007/978-1-84628-797-8.

[6]

D. C. Brody, J. Syroka and M. Zervos, Dynamical pricing of weather derivatives, Quant. Finance, 2 (2002), 189-198.  doi: 10.1088/1469-7688/2/3/302.

[7]

B. Cloez and M. Hairer, Exponential ergodicity for Markov processes with random switching, Bernoulli, 21 (2015), 505-536.  doi: 10.3150/13-BEJ577.

[8]

T. E. Duncan, B. Maslowski and B. Pasik-Duncan, Stochastic equations in Hilbert space with a multiplicative fractional Gaussian noise, Stochastic Process. Appl., 115 (2005), 1357-1383.  doi: 10.1016/j.spa.2005.03.011.

[9]

T. E. Duncan, Y. Z. Hu and B. Pasik-Duncan, Stochastic calculus for fractional Brownian motion. I. Theory, SIAM J. Control Optim., 38 (2000), 582-612.  doi: 10.1137/S036301299834171X.

[10]

T. E. Duncan, B. Pasik-Duncan and B. Maslowski, Fractional Brownian motion and stochastic equations in Hilbert spaces, Stoch. Dyn., 2 (2002), 225-250.  doi: 10.1142/S0219493702000340.

[11]

M. K. Ghosh, A. Arapostahis and S. I. Marcus, Ergodic control of switching diffusions, SIAM J. Control Optim., 35 (1997), 1952-1988.  doi: 10.1137/S0363012996299302.

[12]

J. D. Hamilton, A new approach to the economic analysis of nonstationary time series and the business cycle, Econometrica, 57 (1989), 357-384.  doi: 10.2307/1912559.

[13]

H. Holden, B. ∅ksendal, J. Ubøe and T. S. Zhang, Stochastic Partial Differential Equations, 2nd edition, Universitext, Springer, New York, 2010. doi: 10.1007/978-0-387-89488-1.

[14]

Y. Z. Hu, Itô Stochastic differential equations driven by fractional Brownian motion of Hurst parameter $H>1/2$, Stochastics, 90 (2018), 720-761.  doi: 10.1080/17442508.2017.1415342.

[15]

G. Huang, H. M. Jansen, M. Mandjes, P. Spreij and K. De. Turck, Markov-modulated Ornstein-Uhlenbeck processes, Adv. in Appl. Probab., 48 (2016), 235-254.  doi: 10.1017/apr.2015.15.

[16]

I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus, 2nd edition, Graduate Texts in Mathematics, 113. Springer, New York, 1991. doi: 10.1007/978-1-4612-0949-2.

[17]

M. L. Li and F. Q. Deng, Almost sure stability with general decay rate of neutral stochasti delayed hybrid systems with Lévy noise, Nonlinear Anal. Hybrid Syst., 24 (2017), 171-185.  doi: 10.1016/j.nahs.2017.01.001.

[18]

X. R. Mao, Stability of stochastic differential equations with Markovian switching, Stochastic Process. Appl., 79 (1999), 45-67.  doi: 10.1016/S0304-4149(98)00070-2.

[19] X. R. Mao and C. G. Yuan, Stochastic Differential Equations with Markovian Switching, Imperial College Press, London, 2006.  doi: 10.1142/p473.
[20]

J. Mémin, Y. Mishura and E. Valkeila, Inequalities for the moments of Wiener integrals with respect to a fractional Brownian motion, Statist. Probab. Lett., 51 (2001), 197-206.  doi: 10.1016/S0167-7152(00)00157-7.

[21]

Y. S. Mishura, Stochastic Calculus for Fractional Brownian Motion and Related Processes, Lecture Notes in Mathematics, 1929. Springer-Verlag, Berlin, 2008. doi: 10.1007/978-3-540-75873-0.

[22]

D. Nualart and A. Rășcanu, Differential equations driven by fractional Brownian motion, Collect. Math., 53 (2002), 55-81. 

[23]

A. Rathinasamy and M. Balachandran, Mean-square stability of Milstein method for linear hybrid stochastic delay integro-differential equations, Nonlinear Anal. Hybrid Syst., 2 (2008), 1256-1263.  doi: 10.1016/j.nahs.2008.09.015.

[24]

S. Cong, On almost sure stability conditions of linear switching stochastic differential systems, Nonlinear Anal. Hybrid Syst., 22 (2016), 108-115.  doi: 10.1016/j.nahs.2016.03.010.

[25]

I. Simonsen, Measuring anti-correlations in the Nordic electricity spot market by wavelets, Physica A: Statistical Mechanics and its Applications, 322 (2003), 597-606.  doi: 10.1016/S0378-4371(02)01938-6.

[26]

A. V. Skorohod, Asymptotic Methods in the Theory of Stochastic Differential Equations, Translations of Mathematical Monographs, 78. American Mathematical Society, Providence, RI, 1989.

[27]

L. Tan, Exponential stability of fractional stochastic differential equations with distributed delay, Adv. Difference Equ., 2014 (2014), 8 pp. doi: 10.1186/1687-1847-2014-321.

[28]

G. G. Yin and C. Zhu, Hybrid Switching Diffusions: Properties and Applications, Stochastic Modelling and Applied Probability, 63. Springer, New York, 2010. doi: 10.1007/978-1-4419-1105-6.

[29]

C. G. Yuan and X. R. Mao, Asymptotic stability in distribution of stochastic differential equations with Markovian switching, Stochastic Process. Appl., 103 (2003), 277-291.  doi: 10.1016/S0304-4149(02)00230-2.

[30]

C. G. Yuan and X. R. Mao, Stability of stochastic delay hybrid systems with jumps, Eur. J. Control, 16 (2010), 595-608.  doi: 10.3166/ejc.16.595-608.

[31]

Z. Z. Zhang, J. Y. Tong and L. J. Hu, Long-term behavior of stochastic interest rate models with Markov switching, Insurance Math. Econom., 70 (2016), 320-326.  doi: 10.1016/j.insmatheco.2016.06.017.

[32]

W. N. Zhou, J. Yang, X. Q. Yang, A. D. Dai, H. S. Liu and J. Fang, pth Moment exponential stability of stochastic delayed hybrid systems with Lévy noise, Appl. Math. Model., 39 (2015), 5650-5658.  doi: 10.1016/j.apm.2015.01.025.

Figure 1.  A single path of solution
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