Suppose that $X_t$ is an $\mathbb{R}$-valued semi-martingale with decomposition $X_0+A_t+M_t$, where $A_t$ is finite variation and $M_t$ is a local martingale. What would be the semi-martingale decomposition for $$ e^{k\cdot X_t}X_t, $$ where $k>0$?
2026-03-25 19:00:50.1774465250
Weird Ito Transformation
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Hint: apply ito's formula on the $f(x,t) = xe^{kx}$, where $x = X_{t}$ and then simply split the $dt$ parts and the stochastic parts