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

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

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.

Citation: Yayun Zheng, Xu Sun. Governing equations for Probability densities of stochastic differential equations with discrete time delays. Discrete and Continuous Dynamical Systems - B, 2017, 22 (9) : 3615-3628. doi: 10.3934/dcdsb.2017182
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D. G. Aronson, Non-negative solutions of linear parabolic equations, Ann. Sci. Norm. Sup. Pisa, 22 (1968), 607-694. 

[2]

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

[3]

J. BaoG. YinC. 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.

[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.

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

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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.

[7]

V. I. BogachevM. 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.

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J. H. Crawford IIIE. 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. 

[9]

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

[10]

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

[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.

[12]

S. GuillouzicI. 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.

[13]

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

[14]

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

[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.

[16]

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

[17]

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

[18]

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

[19]

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

[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.

[21]

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

[22]

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

show all references

References:
[1]

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

[2]

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

[3]

J. BaoG. YinC. 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.

[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.

[5]

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

[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.

[7]

V. I. BogachevM. 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.

[8]

J. H. Crawford IIIE. 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. 

[9]

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

[10]

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

[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.

[12]

S. GuillouzicI. 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.

[13]

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

[14]

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

[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.

[16]

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

[17]

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

[18]

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

[19]

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

[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.

[21]

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

[22]

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

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