Ito's Lemma

Ito's lemma is the chain rule of stochastic calculus. If $X_t$ follows $dX_t = \mu \, dt + \sigma \, dW_t$ and $f(t, x)$ is a smooth function, then: $$df = \frac{\partial f}{\partial t} dt + \frac{\partial f}{\partial x} dX_t + \frac{1}{2} \frac{\partial^2 f}{\partial x^2} (dX_t)^2$$ Using $(dW_t)^2 = dt$, $dW_t \cdot dt = 0$, $(dt)^2 = 0$: $$df = \left(\frac{\partial f}{\partial t} + \mu \frac{\partial f}{\partial x} + \frac{\sigma^2}{2} \frac{\partial^2 f}{\partial x^2}\right) dt + \sigma \frac{\partial f}{\partial x} dW_t$$ Intuition: In ordinary calculus, the Taylor expansion's second-order term $\frac{1}{2}f''(dx)^2$ vanishes because $(dx)^2$ is infinitesimally small compared to $dx$. But Brownian motion is "rougher" than smooth paths — $(dW)^2$ is of order $dt$, not $(dt)^2$. So the second-order term survives! This is the fundamental surprise of stochastic calculus. The classic application — deriving GBM: Let $dS = \mu S \, dt + \sigma S \, dW$ and apply Ito's lemma to $f(S) = \ln S$: - $f' = 1/S$, $f'' = -1/S^2$ - $d(\ln S) = \frac{1}{S}(\mu S \, dt + \sigma S \, dW) + \frac{1}{2}(-1/S^2)(\sigma S)^2 dt$ - $d(\ln S) = (\mu - \sigma^$$\frac{2}{2}$) dt + \sigma \, dW The $-\sigma^$$\frac{2}{2}$ correction appears naturally — it's not an ad hoc adjustment, it's Ito's lemma at work. Concrete example: Let $f(W_t) = W_t^2$. By Ito's lemma ($f'= 2W_t$, $f''=2$): $$d(W_t^2) = 2W_t \, dW_t + \frac{1}{2}(2)(dW_t)^2 = 2W_t \, dW_t + dt$$ So $W_t^2 = 2\int_0^t W_s \, dW_s + t$, which gives $E[W_t^2] = t$. The extra $dt$ term is pure Ito. When to use: Any time you need to find the dynamics of a function of a stochastic process. Deriving the Black-Scholes PDE. Computing expectations of transformed processes. This is the workhorse of quantitative finance.