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: |
D. G. Aronson
, Non-negative solutions of linear parabolic equations, Ann. Sci. Norm. Sup. Pisa, 22 (1968)
, 607-694.
![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.
![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() |
|
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.
![]() |
|
E. B. Davies,
Heat Kernels and Spectral Theory, Cambridge University Press, 1989.
doi: 10.1017/CBO9780511566158.![]() ![]() ![]() |
|
J. Duan,
An Introduction to Stochastic Dynamics, Cambridge University Press, 2015.
![]() ![]() |
|
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.![]() ![]() |
|
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.![]() ![]() |
|
S. Guillouzic,
Fokker-Planck approach to Stochastic Delay Differential Equations, Ph. D. thesis, University of Ottawa, Canada, 2000.
![]() |
|
F. C. Klebaner,
Introduction to Stochastic Calculus with Applications, Imperial College Press, 2nd Edition, 2005.
doi: 10.1142/p386.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
Y. K. Lin and G. Q. Cai,
Probabilistic Structural Dynamics: Advanced Theory and Applications, Springer, 2005.
![]() |
|
A. Longtin, Stochastic delay-differential equations, In: Complex Time-Delay Systems, edited by F. Atay, Springer, Berlin, (2010), 177-195.
![]() ![]() |
|
X. Mao,
Stochastic Differential Equations and Applications, Second edition. Horwood Publishing Limited, Chichester, 2008.
doi: 10.1533/9780857099402.![]() ![]() ![]() |
|
S. A. Mohammed,
Stochastic Functional Differential Equations, Pitman, 1984.
![]() ![]() |
|
J. R. Norris
and D. W. Stroock
, Estimates on the fundamental solution to heat flows with uniformly elliptic coefficients, Proc. London Math. Soc., 62 (1991)
, 373-402.
doi: 10.1112/plms/s3-62.2.373.![]() ![]() ![]() |
|
D. Nualart,
The Malliavin Calculus and Related Topics, Springer, 2nd Edition, 2006.
![]() ![]() |
|
P. Protter,
Stochastic Integration and Differential Equations, Springer, 2nd Edition, 2004.
![]() ![]() |