# StochasticDiffEq.jl

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.

## API

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 `SRIW1()`

algorithm:

```
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())
```

The options for `solve`

are defined in the common solver options page and are thoroughly explained in the ODE tutorial.

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[1] = 10.0(u[2]-u[1])
du[2] = u[1]*(28.0-u[3]) - u[2]
du[3] = u[1]*u[2] - (8/3)*u[3]
end
function σ_lorenz(du,u,p,t)
du[1] = 3.0
du[2] = 3.0
du[3] = 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
the `noise_prototype`

:

```
f = (du,u,p,t) -> du.=1.01u
g = function (du,u,p,t)
du[1,1] = 0.3u[1]
du[1,2] = 0.6u[1]
du[1,3] = 0.9u[1]
du[1,4] = 0.12u[2]
du[2,1] = 1.2u[1]
du[2,2] = 0.2u[2]
du[2,3] = 0.3u[2]
du[2,4] = 1.8u[2]
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 `GeometricBrownian`

via:

```
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)
```

## Available Solvers

For the list of available solvers, please refer to the DifferentialEquations.jl SDE Solvers page and the RODE Solvers page.