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Under $\mathbb{Q}$, let $\mathrm{d}S_t=rS_t\mathrm{d}t+\sigma S_t\mathrm{d}B_t$. Define the function $f(t,x)=e^{-\beta t}x^3$ with partial derivatives $f_t(t,x)=-\beta f(t,x)$, $f_x(t,x)=3e^{-\beta t}x^2$ and $f_{xx}(t,x)=6e^{-\beta t}x$. You are interested in the process $X_t=f(t,S_t)=e^{-\beta t}S_t^3$. By Ito's Lemma, \begin{align*} \mathrm{d}f(t,S_t) &...


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