Deriving Law of Motion by Ito's Lemma

Question

I've been trying to derive the law of motion for the stochastic process above using Ito's Lemma, given Geometric Brownian Motion with it's law of motion shown below:

I've managed to take the partial derivative such that I can substitute them into the Ito's Lemma form as shown below:

From this I'm able to simplify down to:

The above is possible because:

However, I am struggling to simplify any further towards an answer for deriving the law of motion. I wonder whether I've miscalculated the partial derivatives or perhaps an error in my simplification thus far, but I can't seem to eliminate the St's which seems necessary. Could I perhaps take the total differential of dUt/Ut of the most simplified version thus far? Thanks for any help in advance.

Answer 1

Since the process $e^{-\mu t}$ is continuously differntiable, then it has finite variation. Thus, Ito's lemma essentially implies the 'normal' product rule:

\begin{align} dU_t &= d(S_te^{-\mu t}) \\ &= e^{-\mu t}dS_t + S_td(e^{-\mu t}) \\ &= e^{-\mu t}\mu S_t dt + e^{-\mu t}\sigma S_t dw_t - \mu e^{-\mu t} S_t dt \\ &= e^{-\mu t}\sigma S_t dw_t \end{align}