Ito's Lamma

In very simple terms, Ito's lemma is a formula that allows us to compute the differential of a function of a stochastic process. In normal functions, we can use the chain rule to compute the differential of a function. However, in stochastic calculus, we need to use Ito's lemma.

Suppose we have a stochastic process $X_t$ that follows a stochastic differential equation (SDE).

$$dX_t = \mu(X_t, t) dt + \sigma(X_t, t) dW_t$$

where $W_t$ is a Wiener process (or Brownian motion), $\mu(X_t, t)$ is the drift term, and $\sigma(X_t, t)$ is the diffusion term.

Now, let's say we have a function $f(X_t, t)$ (twice differentiable) that we want to differentiate with respect to time. Ito's lemma states that the differential of $f(X_t, t)$ is given by:

$$\frac{\Delta f(t)}{dt} dt = \frac{\partial f}{\partial t} dt + \frac{1}{2}\frac{\partial^2f}{\partial t^2} (dt)^2 + \cdots$$

and so with $x$, the total derivative of $f$ will be

$$\begin{align*} df &= f_t dt + f_x dx \\ &=\lim_{dx, dt\rightarrow 0, 0} \frac{\partial f}{\partial t} dt + \frac{\partial f}{\partial x} dx + \frac{1}{2} \left(\frac{\partial^2f}{\partial t^2} (dt)^2 + \frac{\partial^2f}{\partial x^2} (dx)^2 \right) + \cdots \end{align*}$$

Than substitute $x = X_t$, in the limit $dt \to 0$, the following terms tend to zero faster than $dt$ are:

1. $(dt)^2$

2. $dtdW_t$

3. $(dx)^3$

Noted that $(dB_t)^2 = \mathcal{O}(dt)$ due to quadratic variation of a Wiener process. # TODO, Hance

$$df = \lim_{dt \to 0} \left(\frac{\partial f}{\partial t} + \mu_t \frac{\partial f}{\partial x} + \frac{\sigma_t^2}{2} \frac{\partial^2 f}{\partial x^2} \right) dt + \sigma_t \frac{\partial f}{\partial x} dW_t$$

That's it!