# Ito Lemma and identifying martingale parts

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

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...

• 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 – TheBridge Nov 19 '18 at 7:24
• Yes, is that possible? – AIM_BLB Nov 30 '18 at 8:41