Ito's Lemma

Let’s consider a function of Brownian motion (which is a particular case of stochastic process), as for instance:

f (t, W (t)) = \exp ((μ - \frac{σ^{2}}{2}) t + σ W (t)) .

Using Ito's Lemma for a function $f (t, W (t))$ , we have:

d f = (\frac{\partial f}{\partial t} + \frac{1}{2} \frac{\partial^{2} f}{\partial W^{2}}) d t + \frac{\partial f}{\partial W} d W .

Then, for the given example,

f (t, W (t)) = \exp (A t + σ W (t)), where A = μ - \frac{σ^{2}}{2},

we have

Now plug into Ito’s formula:

d S = (A f + \frac{1}{2} σ^{2} f) d t + σ f d W = f ((A + \frac{1}{2} σ^{2}) d t + σ d W)

Recall $A = μ - \frac{σ^{2}}{2}$ , so:

A + \frac{1}{2} σ^{2} = μ

Therefore:

d S = f (μ d t + σ d W) = S (t) (μ d t + σ d W)