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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 $k>0$?

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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

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  • $\begingroup$ $X_t$ is a semi-martingale, there need not be a "dt" part... $\endgroup$
    – user355356
    Nov 17, 2018 at 18:57
  • $\begingroup$ What do you mean? dt parts will belong to the new $A_t$ representation. Your "$X_t$ is a semi-martingale" statement is irrelevant $\endgroup$
    – Makina
    Nov 17, 2018 at 19:10
  • $\begingroup$ Oh you mean the finite variation part? $\endgroup$
    – user355356
    Nov 17, 2018 at 19:17
  • $\begingroup$ Yes, I mean exactly that $\endgroup$
    – Makina
    Nov 17, 2018 at 19:19
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    $\begingroup$ I believe that this website isn't supposed to serve as a homework solving place. For educational purposes it is better to give hints rather than the solution, don't you think? Definition of "answer to a question" is not necessarily a solution to the problem. I opted for this type of answer under the assumption that you did not just blatantly ask others to solve your homework question (even though you did) but maybe wanted some help/hits to lead you in the right direction $\endgroup$
    – Makina
    Nov 17, 2018 at 19:40

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