We consider a stochastic differential equation:
$dx\left( t\right) =f\left( x\left( t\right) \right) dt+\sigma\left( x\left( t\right) \right) dW\left( t\right)$
where $x:\mathbb{R}\rightarrow\mathbb{R}^{n}$, and $\sigma:\mathbb{R}% ^{n}\rightarrow\mathbb{R}^{n\times m}$. If $g:\mathbb{R}^{n}\rightarrow \mathbb{R}$ is twice continuously differentiable. I have some problems with the following:
(i) Is $g_{x}\left( x\left( t\right) \right) ^{T}\sigma\left( x\left( t\right) \right) $ a martingale? ($g_{x}$ represents the gradient of $g$ with respect to $x$).
(ii) If $g_{x}\left( x\left( t\right) \right) ^{T}\sigma\left( x\left( t\right) \right) $ is Ito-integrable, the random process $f_{x}\left( x\left( t\right) \right) \sigma\left( x\left( t\right) \right) $ is mean-square integrable from $0$ to $t$?