# Using Itos lemma to find an SDE

## Homework Statement

Using Ito's lemma to find the SDE satisfied by ##U## given that ##U = ln(Y)## and ##Y## satisfies:
$$dY = \frac{1}{2Y}dt + dW$$

$$Y(0) = Y_0$$

Ito's lemma.

## 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.