Overview
Itos lemma is one of the fundamental theorems of the stochastic calculus. It gives a method for finding analytic solutions of stochastic integrals.
Ito Lemma
The basic form of Itos lemma provides an Ito process for a function which is dependent on time and a Brownian Motion
{% df(t, W(t)) = \frac{\partial f}{\partial t}dt + \frac{\partial f}{\partial W}dW + \frac{1}{2} \frac{\partial f^2}{\partial ^2 W} dt%}
Ocasionally youll see a more general form of the equation, stated as
{% df(t, W(t)) = \frac{\partial f}{\partial t}dt + \frac{\partial f}{\partial W}dW + \frac{1}{2} \frac{\partial f^2}{\partial ^2 W} d[W] %}
Here, the last term is stated in terms of the quadratic variation of {% W %}.
However, for Brownian motion, the quadratic variation is equation to {% t %}.
{% [W] = t %}
Note, some authors will use the notation {% \langle W \rangle %} to note the variation of {% W %}.
Extended Ito Lemma
When the function is a function of time and an Ito process, the lemma takes on an extended form.
{% dX(t) = \mu(X, t) dt + \sigma(X,t) dW %}
{% df(t, X(t)) = ( \frac{\partial f}{\partial t} + \mu \frac{\partial f}{\partial x} + \frac{1}{2} \sigma^2 \frac{\partial ^2 f}{\partial x^2} )dt
+ \sigma \frac{\partial f}{\partial x} dW(t)
%}