SDDE Problems

Mathematical Specification of a Stochastic Delay Differential Equation (SDDE) Problem

To define a SDDE Problem, you simply need to give the drift function $f$, the diffusion function g, the initial condition $u_0$ at time point $t_0$, and the history function $h$ which together define a SDDE:

\[du = f(u,h,p,t)dt + g(u,h,p,t)dW_t \qquad (t \geq t_0)\]
\[u(t_0) = u_0,\]
\[u(t) = h(t) \qquad (t < t_0).\]

$f$ should be specified as f(u, h, p, t) (or in-place as f(du, u, h, p, t)) (and $g$ should match). $u_0$ should be an AbstractArray (or number) whose geometry matches the desired geometry of u, and $h$ should be specified as described below. The history function h is accessed for all delayed values. Note that we are not limited to numbers or vectors for $u_0$; one is allowed to provide $u_0$ as arbitrary matrices / higher dimension tensors as well.

Note that this functionality should be considered experimental.

Functional Forms of the History Function

The history function h can be called in the following ways:

  • h(p, t): out-of-place calculation
  • h(out, p, t): in-place calculation
  • h(p, t, deriv::Type{Val{i}}): out-of-place calculation of the ith derivative
  • h(out, p, t, deriv::Type{Val{i}}): in-place calculation of the ith derivative
  • h(args...; idxs): calculation of h(args...) for indices idxs

Note that a dispatch for the supplied history function of matching form is required for whichever function forms are used in the user derivative function f.

Declaring Lags

Lags are declared separately from their use. One can use any lag by simply using the interpolant of h at that point. However, one should use caution in order to achieve the best accuracy. When lags are declared, the solvers can more efficiently be more accurate and thus this is recommended.

Neutral, Retarded, and Algebraic Stochastic Delay Differential Equations

Note that the history function specification can be used to specify general retarded arguments, i.e. h(p,α(u,t)). Neutral delay differential equations can be specified by using the deriv value in the history interpolation. For example, h(p,t-τ, Val{1}) returns the first derivative of the history values at time t-τ.

Note that algebraic equations can be specified by using a singular mass matrix.

Problem Type

Constructors

SDDEProblem(f,g[, u0], h, tspan[, p]; <keyword arguments>)
SDDEProblem{isinplace}(f,g[, u0], h, tspan[, p]; <keyword arguments>)

Parameter isinplace optionally sets whether the function is inplace or not. This is determined automatically, but not inferred.

Parameters are optional, and if not given then a NullParameters() singleton will be used which will throw nice errors if you try to index non-existent parameters. Any extra keyword arguments are passed on to the solvers. For example, if you set a callback in the problem, then that callback will be added in every solve call.

For specifying Jacobians and mass matrices, see the DiffEqFunctions page.

Arguments

  • f: The drift function in the SDDE.
  • g: The diffusion function in the SDDE.
  • u0: The initial condition. Defaults to the value h(p, first(tspan)) of the history function evaluated at the initial time point.
  • h: The history function for the DDE before t0.
  • tspan: The timespan for the problem.
  • p: The parameters with which function f is called. Defaults to NullParameters.
  • constant_lags: A collection of constant lags used by the history function h. Defaults to ().
  • dependent_lags A tuple of functions (u, p, t) -> lag for the state-dependent lags used by the history function h. Defaults to ().
  • neutral: If the DDE is neutral, i.e., if delays appear in derivative terms.
  • order_discontinuity_t0: The order of the discontinuity at the initial time point. Defaults to 0 if an initial condition u0 is provided. Otherwise it is forced to be greater or equal than 1.
  • kwargs: The keyword arguments passed onto the solves.