Weird Ito Transformation 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 $k>0$?

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
