I am reading a paper on the Fokker-Planck equation and this comes up. Let us consider the SDE in $\mathbb{R}^n$ \begin{equation} dX_t = b(X_t,t)dt + \sqrt{2}dB_t, \qquad t\in [0,T],\\ X_0 = x_0 \end{equation} where $B_t$ is a standard Brownian motion in $n$-dimension. My question is, if $\varphi:\mathbb{R}^n\times[0,T]\to \mathbb{R}$ is bounded and $\mathrm{C}^2$ in space, $\mathrm{C}^1$ in time, then \begin{equation} t\mapsto \int_0^t D\varphi(X_s,s)\cdot dB_s \end{equation} is a martingale? Here $D\varphi(x,s)$ is the gradient in $x$ of $\varphi$.
I am talking about page 35-36 in this note: https://www.ceremade.dauphine.fr/~cardaliaguet/MFGcours2018.pdf