Itô's Lemma

Itô's lemma is the chain rule with a twist. The twist is $(d{W}_t{)}^2=dt$, and the entire edifice of stochastic calculus exists to make that one term rigorous. Derive it from a Taylor expansion, then put it to work.

Let $f(t,x)\in C^{1,2}([0,T]\times \mathbb{R})$ and let $W_t$ be a standard Brownian motion.

Part A

Derive Itô's lemma: $df(t,W_t)=\frac{\partial f}{\partial t}dt+\frac{\partial f}{\partial x}d{W}_t+\frac{1}{2}\frac{{\partial }^{2}f}{\partial x^2}(d{W}_t{)}^2.$ Starting from the Taylor expansion of $f(t+\Delta t,x+\Delta x)$, justify why the $(d{W}_t{)}^2=dt$ term survives while $(dt{)}^2$, $dtd{W}_t$, and $(d{W}_t{)}^3$ vanish.

Part B

Apply Itô's lemma to $f(t,x)=x^2$ to find the SDE for $W_t^2$, then verify by taking expectations that $\mathbb{E}[W_t^2]=t$.

Part C

Apply Itô's lemma to $f(t,x)=e^{\sigma x-\frac{1}{2}{\sigma }^{2}t}$ and show that $M_t=e^{\sigma W_t-\frac{1}{2}{\sigma }^{2}t}$ is a martingale (the Doléans-Dade exponential).