# Modeling with Stochasticity

All models with `ODESystem`

are deterministic. `SDESystem`

adds another element to the model: randomness. This is a stochastic differential equation which has a deterministic (drift) component and a stochastic (diffusion) component. Let's take the Lorenz equation from the first tutorial and extend it to have multiplicative noise.

```
using ModelingToolkit, StochasticDiffEq
# Define some variables
@parameters σ ρ β
@variables t x(t) y(t) z(t)
D = Differential(t)
eqs = [D(x) ~ σ*(y-x),
D(y) ~ x*(ρ-z)-y,
D(z) ~ x*y - β*z]
noiseeqs = [0.1*x,
0.1*y,
0.1*z]
@named de = SDESystem(eqs,noiseeqs,t,[x,y,z],[σ,ρ,β])
u0map = [
x => 1.0,
y => 0.0,
z => 0.0
]
parammap = [
σ => 10.0,
β => 26.0,
ρ => 2.33
]
prob = SDEProblem(de,u0map,(0.0,100.0),parammap)
sol = solve(prob,SOSRI())
```