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Suppose that $X_t$ is a càdlàg semi-martingale with decomposition $$ X_t= X_0+ B_t + M_t. $$ I know that using the Ito lemma for any $C^2$-function $f$, $$ f(X_t)= f(X_0)\\ + \int_{0^+}^tf_x(X_{s-})dB_s + \int_{0^+}^t \frac{f_{xx}}{2}(X_{s-})[M]_t\\ +\int_{0^+}^tf_x(X_{s-})dM_s\\ \\ + \sum_{0<s\leq t}\left(f(X_{s-}) - f(X_s)\right) +f_x(X_{s-})\Delta B_s +\frac1{2}f_x(X_{s-})(\Delta M)^2_s \\ + \sum_{0<s\leq t} f_x(X_{s-})\Delta M_s $$

My partial Solution Suppose that $X_t = X_0 +\int_0^t \mu(t,X_t)dt + \int_0^t \sigma(t,X_t)dW_t$, then $$ \int_0^t\left(f_x(X_t)\mu(t,X_t) + f_{xx}\frac{\sigma(t,X_t)}{2}\right)dt $$ is the drift part and $$ \int_0^tf_{x}\sigma(t,X_t)dW_t, $$ is the local martingale part of $f(X_t)$. However, I am having trouble identifying which part is which in the general setting. I know that from the continuous case that the second line of the Ito Lemma is finite-variation and that the third is a martingale, but I am not sure which portions are local martingale or fv in the last 3 lines...

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    $\begingroup$ You should be more clear and direct about the question you are asking, which as I get it is : is it possible, directly from Ito's lemma, to identify the (local?)-martingale part of the process $f(X_t)$ ? Regards $\endgroup$ – TheBridge Nov 19 '18 at 7:24
  • $\begingroup$ Yes, is that possible? $\endgroup$ – AIM_BLB Nov 30 '18 at 8:41

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