Stochastic Differential Equation

A stochastic differential equation (SDE) is an equation that involves increments of a stochastic process (typically the Wiener process) alongside deterministic terms.

The general form is

$$dX=f(X,t)dt+g(X,t)dW$$

where:

• $f(X,t)$ is the drift term (deterministic)
• $g(X,t)$ is the diffusion term (stochastic coefficient)
• $dW$ is an increment of the Wiener process

Example: Ornstein-Uhlenbeck Process

The Ornstein-Uhlenbeck process is the SDE

$$dV=-kVdt+dW$$

with drift $f(V,t)=-kV$ and diffusion $g(V,t)=1$.

SDEs are solved numerically using the Euler-Maruyama scheme, which is the stochastic analogue of the Explicit Euler method.

Wiener process | Ornstein-Uhlenbeck process | Euler-Maruyama scheme | Langevin equation | Explicit Euler method