Stochastic Calculus Primer — Section 18: Calculus & Optimization

Brownian motion has paths that are continuous but not differentiable. To do calculus on processes driven by Brownian motion, you need stochastic calculus.

The basic building block is the Itô integral $\int_0^T \sigma(s)\, dW(s)$ , which extends the Riemann integral to stochastic integrands. Its key property: $E[\int_0^T \sigma(s)\, dW(s)] = 0$ — the integral is a martingale.

Itô's Lemma is the chain rule. For $X_t$ with $dX_t = \mu\, dt + \sigma\, dW_t$ :

df(X_t) = f'(X_t)\, dX_t + \frac{1}{2}\sigma^2 f''(X_t)\, dt

The extra term $\frac{1}{2}\sigma^2 f''$ captures the contribution from squared Brownian increments, which scale like $dt$ , not $(dt)^2$ .

Stochastic calculus is the language of continuous-time finance: SDEs for asset prices, derivative pricing PDEs (Black-Scholes), term-structure models, and risk-neutral measure changes (Girsanov's theorem) all live here.

Brownian motion has paths that are continuous but not differentiable. To do calculus on processes driven by Brownian motion, you need stochastic calculus.

Itô's Lemma is the chain rule. For $X_t$ with $dX_t = \mu\, dt + \sigma\, dW_t$ :

df(X_t) = f'(X_t)\, dX_t + \frac{1}{2}\sigma^2 f''(X_t)\, dt

The extra term $\frac{1}{2}\sigma^2 f''$ captures the contribution from squared Brownian increments, which scale like $dt$ , not $(dt)^2$ .