# Using Itos lemma to find an SDE

1. Feb 11, 2013

### jend23

1. The problem statement, all variables and given/known data
Using Ito's lemma to find the SDE satisfied by $U$ given that $U = ln(Y)$ and $Y$ satisfies:
$$dY = \frac{1}{2Y}dt + dW$$

$$Y(0) = Y_0$$

2. Relevant equations

Ito's lemma.

3. The attempt at a solution

If $U \equiv U(Y,t)$ and $dY = a(Y,t)dt + b(Y,t)dW$

Then $dU = \left(\frac{\partial U}{\partial t} + a(Y,t)\frac{\partial U}{\partial Y} + \frac{1}{2}b(Y,t)^2\frac{\partial^2U}{\partial Y^2}\right)dt + b(Y,t)\frac{\partial U}{\partial Y}dW$

Here $U = ln(Y)$ and $dY = \frac{1}{2Y}dt + dW$

$\frac{\partial U}{\partial t}=0$, $\frac{\partial U}{\partial Y} = \frac{1}{Y}$, $\frac{\partial^2 U}{\partial Y^2} = \frac{-1}{Y^2}$

Therefore:

$$dU = \left(0 + \frac{1}{2Y^2} - \frac{1}{2Y^2} \right)dt + \frac{1}{Y}dW$$

$$dU = \frac{1}{Y}dW$$

I *think* i've arrived at the right answer but given that I haven't done any kind of mathematics for nearly two decades, I wonder if somebody would be so kind as to critique my approach and let me know if the answer is right?

Any help, much appreciated.