# When is the local martingale in the Itō formula a (strict) martingale?

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 $$\tilde\mu,\tilde\sigma:[0,\infty)\times\mathbb R\to\mathbb R$$.