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Wiki gives a version of the Ito's lemma for the Ito proccess when we differentiate a function $f(t,X_t)$ of time and some diffusion process.

In the general case of multivariate semimartingale (possibly discontinuous) it is written for the function without the time parameter $f(X_t)$. Can we extend it to $f(t,X_t)$ when $X_t$ is a semimartingale? I checked several books and everywhere we have $f(X_t)$ for the semimartingale case. What if we have $f(X_t,Y_t)$ where $X_t$ is a diffusion/semimartingale and $Y_t$ is some other process?

paul
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1 Answers1

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(I'm not allowed to comment, so sorry if this is not the answer you are looking for).

Yes, Ito Lemma works for a vector of semimartingales as Wikipedia said. A semimartingale is a sum of a local martingale (which can be entirely null) and a process with finite variation paths, for example, $A_t(\omega)=t$, for all $t\geq0$ and all $\omega \in \Omega$. Hence it already encompasses a lot. For example, to me $f(t,X_t)$ is a subcase of $f(A_t,X_t)$, with $A_t$ defined above, with both $A_t$ and $X_t$ being semimartingales.

ir7
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