StochasticDiffEq.jl is a component package in the DifferentialEquations ecosystem. It holds the stochastic differential equations solvers and utilities. While completely independent and usable on its own, users interested in using this functionality should check out DifferentialEquations.jl.
StochasticDiffEq.jl is part of the JuliaDiffEq common interface, but can be used independently of DifferentialEquations.jl. The only requirement is that the user passes an StochasticDiffEq.jl algorithm to
solve. For example, we can solve the SDE tutorial from the docs using the
using StochasticDiffEq α=1 β=1 u₀=1/2 f(u,p,t) = α*u g(u,p,t) = β*u dt = 1//2^(4) tspan = (0.0,1.0) prob = SDEProblem(f,g,u₀,(0.0,1.0)) sol =solve(prob,SRIW1())
That example uses the out-of-place syntax
f(u,p,t), while the inplace syntax (more efficient for systems of equations) is shown in the Lorenz example:
function lorenz(du,u,p,t) du = 10.0(u-u) du = u*(28.0-u) - u du = u*u - (8/3)*u end function σ_lorenz(du,u,p,t) du = 3.0 du = 3.0 du = 3.0 end prob_sde_lorenz = SDEProblem(lorenz,σ_lorenz,[1.0,0.0,0.0],(0.0,10.0)) sol = solve(prob_sde_lorenz) plot(sol,vars=(1,2,3))
The problems default to diagonal noise. Non-diagonal noise can be added by setting
f = (du,u,p,t) -> du.=1.01u g = function (du,u,p,t) du[1,1] = 0.3u du[1,2] = 0.6u du[1,3] = 0.9u du[1,4] = 0.12u du[2,1] = 1.2u du[2,2] = 0.2u du[2,3] = 0.3u du[2,4] = 1.8u end prob = SDEProblem(f,g,ones(2),(0.0,1.0),noise_rate_prototype=zeros(2,4))
Colored noise can be set using an
AbstractNoiseProcess. For example, we can set the underlying noise process to a
μ = 1.0 σ = 2.0 W = GeometricBrownianMotionProcess(μ,σ,0.0,1.0,1.0) # ... # Define f,g,u0,tspan for a SDEProblem # ... prob = SDEProblem(f,g,u0,tspan,noise=W)
StochasticDiffEq.jl also handles solving random ordinary differential equations. This is shown in the RODE tutorial.
using StochasticDiffEq function f(u,p,t,W) 2u*sin(W) end u0 = 1.00 tspan = (0.0,5.0) prob = RODEProblem(f,u0,tspan) sol = solve(prob,RandomEM(),dt=1/100)