Let
- $(\Omega,\mathcal A,\operatorname P)$ be a complete probability space
- $(\mathcal F_t)_{t\ge0}$ be a filtration on $(\Omega,\mathcal A)$
- $W$ be an $\mathcal F$-Brownian motion on $(\Omega,\mathcal A,\operatorname P)$
- $X$ be a continuous $\mathcal F$-adapted process on $(\Omega,\mathcal A,\operatorname P)$ with $$X_t=X_0+\int_0^t\mu_s\:{\rm d}s+\int_0^t\sigma_s\:{\rm d}W_s\;\;\;\text{for all }t\ge0\text{ almost surely}\tag1$$ for some $\mathcal F$-progressive processes $\mu,\sigma$ with $$\int_0^t|\mu_s|+|\sigma_s|^2\:{\rm d}s<\infty\;\;\;\text{almost surely for all }t\ge0\tag2$$
Now, let $f\in C^2(\mathbb R)$. By the Itō formula, $$f(X_t)=f(X_0)+\int_0^tf'(X_s)\:{\rm d}X_s+\frac12\int_0^tf''(X_s)\:{\rm d}[X]_s\;\;\;\text{for all }t\ge0\text{ almost surely}\tag3.$$
Which assumption do we need to impose, if we want that $$\left(\int_0^t\sigma_sf'(X_s)\:{\rm d}W_s\right)_{t\ge0}$$ is an $\mathcal F$-martingale?
One possible assumption would be that $f$ has compact support and that $\mu$ and $\sigma$ are bounded on $\left\{(\omega,t)\in\Omega\times[0,\infty):X_t(\omega)\in\operatorname{supp}f\right\}$.
I'm particularly interested in the case $\mu_t=\tilde\mu(t,X_t)$ and $\sigma_t=\tilde\sigma(t,X_t)$ for all $t\ge0$ for some Borel measurable $\tilde\mu,\tilde\sigma:[0,\infty)\times\mathbb R\to\mathbb R$.
