Itô Calculus & Stochastic Differential Equations
Calculus of Brownian motion — and how it priced an option.
For Brownian motion $B_t$, define the stochastic integral $\int_0^t \sigma(s, X_s)\, dB_s$ as the $L^2$ limit of left-endpoint Riemann sums (Itô convention).
The basic differentials satisfy
$$dB_t \cdot dB_t = dt, \qquad dB_t \cdot dt = 0, \qquad dt \cdot dt = 0.$$Itô's lemma for $f(t, X_t)$ with $dX_t = \mu\, dt + \sigma\, dB_t$:
$$df = \left(\partial_t f + \mu\, \partial_x f + \tfrac{1}{2}\sigma^2 \partial_{xx} f\right) dt + \sigma\, \partial_x f\, dB_t.$$The extra "$\tfrac{1}{2}\sigma^2 \partial_{xx} f \, dt$" comes from $(dB)^2 = dt$.
Canonical SDEs:
- Geometric Brownian motion $dS_t = \mu S_t\, dt + \sigma S_t\, dB_t$ — solution $S_t = S_0 \exp((\mu - \tfrac{1}{2}\sigma^2)t + \sigma B_t)$. Models log-normal stock prices.
- Ornstein–Uhlenbeck $dX_t = -\theta X_t\, dt + \sigma\, dB_t$ — mean-reverting.
The Black–Scholes equation for option prices follows from Itô + no-arbitrage:
$$\partial_t V + \tfrac{1}{2}\sigma^2 S^2 \partial_{SS} V + r S \partial_S V - r V = 0.$$It earned the 1997 Nobel in Economics (Merton & Scholes; Black had died).