Stochastic Differential Equations

Drift plus noise, dX = (X) dt + (X) dW — and solving GBM with Itô's lemma.

The idea

A stochastic differential equation describes a random process by its change over each instant: a predictable push plus a random kick.

Definition (Stochastic differential equation).

A stochastic differential equation for a process $X$ has the form $dX = \mu(X)\,dt + \sigma(X)\,dW,$ where $dW$ is the increment of a Brownian motion. The coefficient $\mu(X)$ is the drift and $\sigma(X)$ is the diffusion.

The drift is the average rate of change, and the diffusion sets how strongly the Brownian noise shakes the process. Either coefficient may depend on the current value of $X$, so a single equation can describe a motion whose behaviour changes as it moves.

The form follows that of an ordinary differential equation, which prescribes a rate of change and asks us to reconstruct the function. Most quantities — a growing population, a moving body — are easier to describe locally than globally, and adding noise does not change that.

To solve an SDE is to produce the process itself: an explicit formula for $X(t)$ in terms of $t$ and $W(t)$. The tool is Itô's Lemma, and the standard strategy is a change of variables: find a function of $X$ whose own equation has no $X$ on the right-hand side, then integrate that equation directly.

Ways to work on it

• Walkthrough. What a stochastic differential equation prescribes, solved in full for geometric Brownian motion.
• Practice. Read the growth of a process's logarithm, mean, and median off its equation.
• Hardest. Follow a geometric Brownian motion from its equation to the probability it ends above its start.

Not sure where to start? Take the ten-question placement test.