SDDE Solvers

StochasticDelayDiffEq.jl is deprecated

The separate StochasticDelayDiffEq.jl package is deprecated as of the OrdinaryDiffEq v7 / SciMLBase v3 ecosystem release. SDDE problems are now solved directly via DelayDiffEq.jl, which has supported them for some time. Replace using StochasticDelayDiffEq with using DelayDiffEq (and add using StochasticDiffEq if you rely on SDE algorithm names such as EM(), RKMil(), or SRIW1()). The MethodOfSteps(alg) wrapper and the SDDEProblem constructor are unchanged. See the OrdinaryDiffEq v7 migration guide for details.

solve(prob::AbstractSDDEProblem, alg; kwargs)

Solves the SDDE defined by prob using the algorithm alg. If no algorithm is given, a default algorithm will be chosen.

Packages

The solvers on this page are distributed across the packages below. Add the package(s) you need to your environment.

Package	Methods	Good for
`DelayDiffEq`	`MethodOfSteps`, `SDDEProblem`	SDDE driver - reuses an SDE algorithm as the inner solver.
`StochasticDiffEq`	Umbrella for SDE solvers; pulls in `StochasticDiffEq*` sublibs	Provides the inner SDE algorithm names (EM, RKMil, SRIW1, ...).

Recommended Methods

The recommended method for SDDE problems are the SDE algorithms. On SDEs you simply reuse the same algorithm as the SDE solver, and DelayDiffEq.jl will convert it to an SDDE solver. The recommendations for SDDE solvers match those of SDEs, except that only up to strong order 1 is recommended. Also note that order 1 is currently only attainable if there is no delay term in the diffusion function $g$: delays in the drift function $f$ are compatible with first order convergence. Theoretical issues with higher order methods (1.5+) on SDDEs is currently unknown.

Note that adaptive time stepping utilizes the same rejection sampling with memory technique as SDEs, but no proof of convergence is known for SDDEs.

Example

using DelayDiffEq, StochasticDiffEq

function hayes_modelf(du, u, h, p, t)
    τ, a, b, c, α, β, γ = p
    du .= a .* u .+ b .* h(p, t - τ) .+ c
end
function hayes_modelg(du, u, h, p, t)
    τ, a, b, c, α, β, γ = p
    du .= α .* u .+ γ
end
h(p, t) = (ones(1) .+ t);
tspan = (0.0, 10.0)

pmul = [1.0, -4.0, -2.0, 10.0, -1.3, -1.2, 1.1]
padd = [1.0, -4.0, -2.0, 10.0, -0.0, -0.0, 0.1]

prob = SDDEProblem(hayes_modelf, hayes_modelg, [1.0], h, tspan, pmul;
    constant_lags = (pmul[1],));
sol = solve(prob, RKMil())