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November  2017, 22(9): 3615-3628. doi: 10.3934/dcdsb.2017182

Governing equations for Probability densities of stochastic differential equations with discrete time delays

Yayun Zheng and  Xu Sun ,

School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, Hubei, China

* Corresponding author: Xu Sun

Received  September 2016 Revised  April 2017 Published  July 2017

Fund Project: The authors are supported by National Natural Science Foundation of China grants 11531006.

The time evolution of probability densities for solutions to stochastic differential equations (SDEs) without delay is usually described by Fokker-Planck equations, which require the adjoint of the infinitesimal generator for the solutions. However, Fokker-Planck equations do not exist for stochastic delay differential equations (SDDEs) since the solutions to SDDEs are not Markov processes and have no corresponding infinitesimal generators. In this paper, we address the open question of finding governing equations for probability densities of SDDEs with discrete time delays. In the governing equation, densities for SDDEs with discrete time delays are expressed in terms of those for SDEs without delay. The latter have been well studied and can be obtained by solving the corresponding Fokker-Planck equations. The governing equation is given in a simple form that facilitates theoretical analysis and numerical computation. Some example are presented to illustrate the proposed governing equations.

Keywords: Stochastic differential equations, Brownian motions, probability density, discrete time delay, stochastic delay differential equations.
Mathematics Subject Classification: Primary:60H10, 65C50;Secondary:34K50.
Citation: Yayun Zheng, Xu Sun. Governing equations for Probability densities of stochastic differential equations with discrete time delays. Discrete & Continuous Dynamical Systems - B, 2017, 22 (9) : 3615-3628. doi: 10.3934/dcdsb.2017182
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J. Bao, G. Yin and C. Yuan, Stationary distributions for retarded stochastic differential equations without dissipativity, Stochastics, 89 (2017), 530-549.  doi: 10.1080/17442508.2016.1267180.  Google Scholar

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J. Bao, G. Yin, C. Yuan and L. Wang, Stationary distributions for retarded stochastic differential equations without dissipativity, Applicable Analysis, 93 (2014), 2330-2349.  doi: 10.1080/00036811.2014.952291.  Google Scholar

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A. Beuter, J. Belair, C. Labrie and J. Belair, Feedback and delays in neurological diseases: A modeling study using dynamical systems, Bulletin of Mathematical Biology, 55 (1993), 525-541.   Google Scholar

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V. I. Bogachev, N. V. Krylov, M. Rockner and S. V. Shaposhnikov, Fokker-Planck-Kolmogorov Equations, American Mathematical Society, 2015. doi: 10.1090/surv/207.  Google Scholar

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V. I. Bogachev, M. Rockner and S. V. Shapshnikov, Positive densities of transition probabilities of diffusion processes, SIAM Theory Probab. Appl., 53 (2009), 194-215.  doi: 10.1137/S0040585X97983523.  Google Scholar

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J. H. Crawford III, E. I. Verriest and T. C. Lieuwen, Exact statistics for linear time delayed oscillators subjected to Gaussian excitation, Journal of Sound and Vibration, 332 (2013), 5929-5938.   Google Scholar

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E. B. Davies, Heat Kernels and Spectral Theory, Cambridge University Press, 1989. doi: 10.1017/CBO9780511566158.  Google Scholar

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J. Duan, An Introduction to Stochastic Dynamics, Cambridge University Press, 2015.  Google Scholar

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X. Gu and W. Q. Zhu, Time-delayed stochastic optimal control of strongly non-linear systems with actuator saturation by using stochastic maximum principle, International Journal of Non-Linear Mechanics, 58 (2014), 199-205.  doi: 10.1016/j.ijnonlinmec.2013.09.011.  Google Scholar

[12]

S. Guillouzic, I. L'Heureux and A. Longtin, Small delay approximation of stochastic delay differential equations, Phys. Rev. E, 59 (1999), 3970-3982.  doi: 10.1103/PhysRevE.59.3970.  Google Scholar

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S. Guillouzic, Fokker-Planck approach to Stochastic Delay Differential Equations, Ph. D. thesis, University of Ottawa, Canada, 2000. Google Scholar

[14]

F. C. Klebaner, Introduction to Stochastic Calculus with Applications, Imperial College Press, 2nd Edition, 2005. doi: 10.1142/p386.  Google Scholar

[15]

S. Kusuoka and D. Stroock, Applications of the Malliavin calculus part Ⅰ, Tuniguchi Symp. SA Katata, 32 (1984), 271-306.  doi: 10.1016/S0924-6509(08)70397-0.  Google Scholar

[16]

Y. K. Lin and G. Q. Cai, Probabilistic Structural Dynamics: Advanced Theory and Applications, Springer, 2005. Google Scholar

[17]

A. Longtin, Stochastic delay-differential equations, In: Complex Time-Delay Systems, edited by F. Atay, Springer, Berlin, (2010), 177-195.  Google Scholar

[18]

X. Mao, Stochastic Differential Equations and Applications, Second edition. Horwood Publishing Limited, Chichester, 2008. doi: 10.1533/9780857099402.  Google Scholar

[19]

S. A. Mohammed, Stochastic Functional Differential Equations, Pitman, 1984.  Google Scholar

[20]

J. R. Norris and D. W. Stroock, Estimates on the fundamental solution to heat flows with uniformly elliptic coefficients, Proc. London Math. Soc.(3), 62 (1991), 373-402.  doi: 10.1112/plms/s3-62.2.373.  Google Scholar

[21]

D. Nualart, The Malliavin Calculus and Related Topics, Springer, 2nd Edition, 2006.  Google Scholar

[22]

P. Protter, Stochastic Integration and Differential Equations, Springer, 2nd Edition, 2004.  Google Scholar

show all references

References:
[1]

D. G. Aronson, Non-negative solutions of linear parabolic equations, Ann. Sci. Norm. Sup. Pisa, 22 (1968), 607-694.   Google Scholar

[2]

J. Bao, G. Yin and C. Yuan, Stationary distributions for retarded stochastic differential equations without dissipativity, Stochastics, 89 (2017), 530-549.  doi: 10.1080/17442508.2016.1267180.  Google Scholar

[3]

J. Bao, G. Yin, C. Yuan and L. Wang, Stationary distributions for retarded stochastic differential equations without dissipativity, Applicable Analysis, 93 (2014), 2330-2349.  doi: 10.1080/00036811.2014.952291.  Google Scholar

[4]

D. R. Bell and S. A. Mohammed, The Malliavin calculus and stochastic delay equations, Journal of Functional Analysis, 99 (1991), 75-99.  doi: 10.1016/0022-1236(91)90052-7.  Google Scholar

[5]

A. Beuter, J. Belair, C. Labrie and J. Belair, Feedback and delays in neurological diseases: A modeling study using dynamical systems, Bulletin of Mathematical Biology, 55 (1993), 525-541.   Google Scholar

[6]

V. I. Bogachev, N. V. Krylov, M. Rockner and S. V. Shaposhnikov, Fokker-Planck-Kolmogorov Equations, American Mathematical Society, 2015. doi: 10.1090/surv/207.  Google Scholar

[7]

V. I. Bogachev, M. Rockner and S. V. Shapshnikov, Positive densities of transition probabilities of diffusion processes, SIAM Theory Probab. Appl., 53 (2009), 194-215.  doi: 10.1137/S0040585X97983523.  Google Scholar

[8]

J. H. Crawford III, E. I. Verriest and T. C. Lieuwen, Exact statistics for linear time delayed oscillators subjected to Gaussian excitation, Journal of Sound and Vibration, 332 (2013), 5929-5938.   Google Scholar

[9]

E. B. Davies, Heat Kernels and Spectral Theory, Cambridge University Press, 1989. doi: 10.1017/CBO9780511566158.  Google Scholar

[10]

J. Duan, An Introduction to Stochastic Dynamics, Cambridge University Press, 2015.  Google Scholar

[11]

X. Gu and W. Q. Zhu, Time-delayed stochastic optimal control of strongly non-linear systems with actuator saturation by using stochastic maximum principle, International Journal of Non-Linear Mechanics, 58 (2014), 199-205.  doi: 10.1016/j.ijnonlinmec.2013.09.011.  Google Scholar

[12]

S. Guillouzic, I. L'Heureux and A. Longtin, Small delay approximation of stochastic delay differential equations, Phys. Rev. E, 59 (1999), 3970-3982.  doi: 10.1103/PhysRevE.59.3970.  Google Scholar

[13]

S. Guillouzic, Fokker-Planck approach to Stochastic Delay Differential Equations, Ph. D. thesis, University of Ottawa, Canada, 2000. Google Scholar

[14]

F. C. Klebaner, Introduction to Stochastic Calculus with Applications, Imperial College Press, 2nd Edition, 2005. doi: 10.1142/p386.  Google Scholar

[15]

S. Kusuoka and D. Stroock, Applications of the Malliavin calculus part Ⅰ, Tuniguchi Symp. SA Katata, 32 (1984), 271-306.  doi: 10.1016/S0924-6509(08)70397-0.  Google Scholar

[16]

Y. K. Lin and G. Q. Cai, Probabilistic Structural Dynamics: Advanced Theory and Applications, Springer, 2005. Google Scholar

[17]

A. Longtin, Stochastic delay-differential equations, In: Complex Time-Delay Systems, edited by F. Atay, Springer, Berlin, (2010), 177-195.  Google Scholar

[18]

X. Mao, Stochastic Differential Equations and Applications, Second edition. Horwood Publishing Limited, Chichester, 2008. doi: 10.1533/9780857099402.  Google Scholar

[19]

S. A. Mohammed, Stochastic Functional Differential Equations, Pitman, 1984.  Google Scholar

[20]

J. R. Norris and D. W. Stroock, Estimates on the fundamental solution to heat flows with uniformly elliptic coefficients, Proc. London Math. Soc.(3), 62 (1991), 373-402.  doi: 10.1112/plms/s3-62.2.373.  Google Scholar

[21]

D. Nualart, The Malliavin Calculus and Related Topics, Springer, 2nd Edition, 2006.  Google Scholar

[22]

P. Protter, Stochastic Integration and Differential Equations, Springer, 2nd Edition, 2004.  Google Scholar

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