A stochastic ordinary differential equation (SODE) is an ordinary differential equation with a random forcing, usually given by a white noise ζ (t). White noise is chosen so that the random forces ζ (t) are uncorrelated at distinct times t. For example, adding noise to the ODE du/dt = -λu, we consider the SODE

for parameters λ, σ > 0 and an initial condition u0 ∈ ℝ. As we saw in §6.3, ζ (t) = dW (t)/dt for a Brownian motion W(t) and we rewrite (8.1) in terms of W(t) by integrating over [0, t]. Consider

which is written in short as

The solution is a stochastic process {u(t): t > 0} such that (8.2) holds for t ≥ 0 and is known as the Ornstein–Uhlenbeck process.

More generally, we introduce a vector-valued function f: ℝd → ℝd, known as the drift, and a matrix-valued function G: ℝd → ℝd × m>, known as the diffusion, and consider SODEs of the form

also written for brevity as

where u0 ∈ ℝd is the initial condition and W (t) = [W1(t),…, Wm (t)]T for iid Brownian motions Wi(t). The last term in (8.3) is a stochastic integral and it needs careful definition.