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.