ito lemma heuristics example

Ito Lemma Heuristics

{% Y(t) = \frac{1}{2} W(t)^2 %}

{% dY = \frac{dY}{dW}dW + \frac{1}{2}\frac{d^2Y}{dW^2}dW^2 %}

where

{% \frac{dY}{dW} = W(t) %}

{% \frac{d^2 Y}{dW^2} = 1 %}

then we have

{% dY = \frac{1}{2} dt + W(t)dW(t) %}

{% \int dY = \int \frac{1}{2} dt + \int W(t)dW(t) %}

{% \int_0 ^T W(t)dW(t) = Y(T) - Y(0) -\frac{1}{2}T = Y(T) -\frac{1}{2}T = \frac{1}{2} W(T)^2 - \frac{1}{2}T %}