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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:

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

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From this I'm able to simplify down to:

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The above is possible because:

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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.

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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}

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