Motivating Ito’s Formula

Ito’s formula, which characterises the stochastic differential, has been mentioned by various textbooks and courses, but now for the first time (after James Norris’s first lecture for the Stochastic Calculus course) I think I finally have a reasonable idea of what’s going on. The reasons I was initially confused help to explain what the motivation is:

What processes can we consider? Well, initially continuous time, time-homogeneous Markov processes in $\mathbb{R}^d$ with continuous paths. It could be space-homogeneous as well if desired. By the theory of decomposition of Levy processes (ie what we are considering), the continuous paths property gives that such a process must be a Brownian motion with drift. This has the property that $X_{t+dt}-X_t\sim N(b(X_t)dt,a(X_t)dt)$ where $a(X_t)$ is the diffusivity, that is, the intensity of the Brownian component, and $b(X_t)$ is the drift.
What is the stochastic differential? Well, for a process as above, we define: $dX_t:= X_{t+dt}-X_t-N(b(X_t)dt,a(X_t)dt)$ . This is non-deterministic: that’s reasonable since X is a stochastic process. And, a normal differential is meaningful only when you integrate, so similarly the stochastic differential is only meaningful when you take an expectation.
Write $N_t$ for the Brownian noise. Then $\mathbb{E}[d(f(X_t))|\mathcal{F}_t]=\mathbb{E}[f(X_{t+dt})-f(X_t)|\mathcal{F}_t]$ , so by Taylor: $=\mathbb{E}[f'(X_t)(b(X_t)dt+N_t)+\frac12 f''(X_t)N_t^2 +o(dt)]$ , remembering that $N_t=O(\sqrt{dt})$ .
This is generally written as $\mathbb{E}[d(f(X_t))|\mathcal{F}_t]=Lf(X_t)dt$ where $Lf(x)=b(x)f'(x)+\frac12 a(x)f''(x)$ . Now note that $\mathbb{E}[dX_t|\mathcal{F}_t]=b(X_t)dt$ and $\mathbb{E}[dX_tdX_t|\mathcal{F}_t]=a(X_t)dt$ , so it is reasonable that we might ‘cancel the expectations’ to get: $d(f(X_t))=f'(X_t)dX_t+\frac12 f''(X_t)dX_tdX_t$ .
Use a suitable tensor product or $dX_tdX_t^T$ when d>1.
This is (a version of) Ito’s Lemma.