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SRB measures for some diffeomorphisms with dominated splittings as zero noise limits
Exponential stability of SDEs driven by fBm with Markovian switching
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 |
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.
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. |


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