Fix $d,k \in \mathbb{N}$. Let $\,b\colon \mathbb{R}^d \to \mathbb{R}^d\,$ and $\,\sigma\colon \mathbb{R}^d \to \mathbb{R}^{d \times k}\,$ be locally Lipschitz functions such that the Itô SDE
$$dX_t=b(X_t)dt+\sigma(X_t)dW_t $$
has global existence and uniqueness of solutions, where $(W_t)_{t \geq 0}$ is a $k$-dimensional Wiener process. Let $f\colon\mathbb{R}^d \to \mathbb{R}$ be a smooth function. Then Itô's lemma gives that for a solution $(X_t)_{t \geq 0}$ of the above SDE,
$$ f(X_t) \ = \ f(X_0) \, + \int_0^t \nabla f(X_s) \boldsymbol{\cdot} b(X_s) \, + \, \tfrac{1}{2}\mathrm{Tr}(\, \sigma(X_s)^T \, D^2\!f(X_s) \, \sigma(X_s) \,) \, ds $$ $$ + \, \underbrace{\int_0^t \nabla f(X_s)^T \sigma(X_s) \, dW_s}_{Y_t} $$
for each $t \geq 0$. My question is:
Assume $X_0$ is independent of $(W_t)_{t \geq 0}$. Are there reasonably straightforwardly verifiable (and reasonably broadly applicable) conditions on $b$, $\sigma$, $f$ and the law of $X_0$, that are sufficient to guarantee that (for a given $t>0$) $$ \mathbb{E}[Y_t] \; \overset{\mathrm{def}}{=} \; \mathbb{E}\left[ \int_0^t \nabla f(X_s)^T \sigma(X_s) \, dW_s \right] \, = \, 0 \ ? $$
As in Itō Integral has expectation zero, if I understand correctly, $\mathbb{E}[Y_t]$ is guaranteed to be $0$ if
$$ \mathbb{E}\left[ \int_0^t \left|\nabla f(X_s)^T \sigma(X_s)\right|^{\,2} \, ds \right] \, < \, \infty \, , $$
but I see no straightforward way to verify this condition.
