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Stochastic delay differential equation



A stochastic delay differential equation or SDDE for short, is a stochastic differential equation where the increment of the process depends on values of the process (and maybe other functions) of the past. These equations can be used to model processes with a memory. An example is the influence of the ocean in a coupled atmoshperic-ocean model of the climate, see for example the “delayed action oscillator” on the ENSO page.

Since the oceans have a high heat capacity, but transport processes can take years, decades or even centuries, the oceans have a “long memory” compared to the atmosphere: The heat feedback from the oceans back to the atmosphere may depend on conditions that date back decades or even centuries.

As in the preceding example, delay differential equations can be used to approximate partial differential equations in the sense that influences due to signals with finite traveling speeds in the full fledged partial differential model are simplified to a delayed influence in a delay differential equation.

Note that delay differential equation are usually infinite dimensional in the sense that a state of a solution needs infinite many numbers to be fully specified.

There does not seem to be much known about SDDE in full generality, formally or numerically; it is an active area of mathematical research.


An Itô stochastic delay differential equation with delay τ\tau is prescribed by

dX t=a(x t,t)dt+b(x t,t)dW t d X_t = a(x_t, t) \; d t + b(x_t, t) d W_t

where W tW_t is the Wiener process in mm dimensions and a,ba, b are functions

a:C([τ,0], d)× + d a: C([-\tau, 0], \mathbb{R}^d) \times \mathbb{R}_+ \to \mathbb{R}^d


b:C([τ,0], d)× + dm b: C([-\tau, 0], \mathbb{R}^d) \times \mathbb{R}_+ \to \mathbb{R}^{d m}

and x tx_t is short hand for the segment

x t={x(t+θ):θ[τ,0]} x_t = \{ x(t + \theta): \theta \in [-\tau, 0] \}

of sample paths.

If the right hand side depends on a continuum of values of a sample path, the equation is sometimes called distributed delay differential equation, and if it depends on a countable set of values only a discrete delay differential equation.

Since SDDE describe an evolution that does not have the Markov property, there is usually not an associated Fokker-Planck equation. It may be possible to approximate the solution of an SDDE with Markov processes, however.


Bistable Potential

The Langevin equation for a bistable potential with a time-delayed linear feedback term is

dX t=(X tX t 3+αX tτ)dt+σdW t d X_t = (X_t - X_t^3 + \alpha \; X_{t - \tau } ) \; d t + \sigma d W_t

See also stochastic resonance and the “delayed action oscillator” on the ENSO page.



See also Numerical methods for delay differential equations.

An overview of several approaches and results can be found here:

  • André Longtin: Stochastic delay-differential equations, in Fatihcan M. Atay (editor): Complex Time-Delay Systems. Theory and Applications, Springer 2010.

A proof of the strong convergence of the Milstein scheme for SDDE is provided by the following paper:

  • P.E. Kloeden and T. Shardlow: The Milstein scheme for stochastic delay differential equations without using anticipative calculus (download).

Model building with delay differential equations:

  • Hal Smith: An introduction to delay differential equations with applications to the life sciences. ZMATH

Control of stochastic delay systems:

  • Harold J. Kushner: Numerical methods for controlled stochastic delay systems. ZMATH