Ito's Lemma

We would like to do stochastic calculus. This is, we would like to learn how to integrate and differentiate functions of Brownian motions. 

We would like to solve stochastic differential equations (SDEs) to find a solution for ST. This solution would tell us the probability distribution of ST. 

 

As with ordinary calculus, the only way to integrate a SDE is to guess a function which it is the derivative of. 

Hence, there is little more to stochastic calculus than knowing how to differentiate functions of stochastic variables. 

Ito's Lemma tells us how to do this. 

 

We define an Ito Process by:           Ito process

 

and take a twice continuously differentiable funtion f(t, Xt), then Ito's Lemma states:

Ito s lemma1

 

It tells us the change in f during an infinitesimal time interval dt. 

The difference between this equation and the chain rule in ordinary calculus is the presence of the final term (dXt)2. 

The proof is straightforward. We first Taylor expand f, and then remember that the expression df only makes sense inside an integral. Then any terms that integrate to zero are dropped. In particular: 

If dXt were not stochastic, we could also drop the term (dXt)2. But the fundamental property of BM states that: (dWt)2 = dt. 

 

Ito lemma2

 

Hence (dXt)2 = (adt + bdW)2 = b2 dt. 

 

Since (dXt)2 is of order dt, it does not integrate to zero and must remain in the equation. 

 

Ito lemma 3