Let $X_t=(X_1, \dots , X_T), t \in [0,T] $ be a continuous semimartingale and $f$ a function of class $C^{1,2}$ (continuous and differentiable). Then, $f(t,X)$ is a semimartingale and we have, $\forall t \in T$, the following Ito's lemma:
$$ d [f(t,X_t)] = \frac{\partial f}{\partial t} dt + \frac{\partial f}{\partial X} d X_t + \frac{1}{2} \frac{\partial^2f}{\partial X^2} d \langle X \rangle_t \tag{1}$$
Now, I am unfamiliar with this notation of the lemma and I'm trying to figure out how to bring it back to the notation I know:
$$ d [f(t,X_t)] = \Big( \frac{\partial f}{\partial t} + \mu \frac{\partial f}{\partial X} + \frac{1}{2} \sigma^2 \frac{\partial^2f}{\partial X^2} \Big) dt + \sigma\frac{\partial f}{\partial X} d W_t \tag{2}$$
I guess that the first step is to assume as continuous semimartingale an Ito's process such that $$ d X_t = \mu_t dt + \sigma_t d W_t \tag{3}$$ with
- $d W_t$ Brownian motion
- $\mu=\mu(t, X_t)$
- $\sigma= \sigma(t,X_t)$
I then start to be confused. I know that $ \langle X \rangle_t$ represents the quadratic variation such that $$ \langle X \rangle_t = \langle X,X \rangle_t = \lim_{n \rightarrow \infty} \sum_{i=1}^T \Big(X_t - X_s \Big)^2, \qquad s\leq t \leq T $$
From here I don't know how to go from Equation (1) to Equation (2). Can anyone suggest me a way to do such thing, or alternatively tell me if my assumption are wrong in the first place?
