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Could someone provide rigorous proof of Ito formula for time dependent function $f(t,x)\in C^{1,2}$ ? I can find some proofs in the books but they are for functions of the form $f(x)$ of $f(t,B_t)$, but not for $f(t,X_t)$, where $X_t$ is Ito process.

Anton Sorokovskiy
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    You can find such a proof in K.L. Chung and R.J. Williams, Introduction to Stochastic Integration, 2014. See Theorem 5.1. Note, they state the theorem for $f(V_t,M_t)$, where $V_t$ is a continuous finite variation process and $M_t$ a continuous local martingale. Observe, the time process $t\mapsto t$ is of finite variation and Itô processes are continuous (local) martingales. – Mark Sep 21 '21 at 10:05
  • @Mark Itô processes are not necessarily local martingales. They are the sum of a local martingale and a finite variation process. An Itô process is only a local martingale if its finite variation part is indistinguishable from zero. – Shiva Sep 27 '21 at 02:24
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    @Shiva, you are completely right, I commented too fast! My mistake. Fortunately, this should not be a too big of an issue regarding OP's question, as one can easily generalise to a setting where $f(t,x,y)\in C^{1,1,2}$ and hence consider $f(t,A_t,M_t)$, where $X=A+M$. – Mark Sep 28 '21 at 08:14

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