This question was asked here, but it did not get enough attention, so I'm crossposting it to MO.

Let $u: \mathbb{R}_+ \times \mathbb{R}^d$ be a bounded $C^2$ function whose first and second partial derivatives are uniformly bounded (or, more generally, have at most polynomial growth as $|x| \to \infty)$ on $[0, T] \times \mathbb{R}^d$, for any $0 \le T < \infty$. For any $t \ge 0$ and any $x \in \mathbb{R}^d$ it is not hard to see that,$$E^x u(t, W_t) = u(0, x) + E^x \int_0^y \left( {\partial\over{\partial s}} + {1\over2}\Delta_s\right) u(s, W_s)\,ds.$$$($$W_t$ is a Brownian motion process which takes the value $x$ at $t=0$ so the dependence of $x$ is implicit here, most of the references will consider $W_t$ a standard Brownian motion which means it takes the value $0$ at $t=0$ and in the above formula we will have $x+W_t$ instead. $\Delta_s$ here means the sum of all the mixed derivatives.$)$

My question is, how do we conclude that under $P^x$ the process$$u(t, W_t) - u(0, x) - \left({\partial\over{\partial t}} + {1\over2}\Delta_x\right)u(t, W_t)$$is a martingale?

This question was asked here, but it did not get enough attention, so I'm crossposting it to MO.

Let $u: \mathbb{R}_+ \times \mathbb{R}^d$ be a bounded $C^2$ function whose first and second partial derivatives are uniformly bounded (or, more generally, have at most polynomial growth as $|x| \to \infty)$ on $[0, T] \times \mathbb{R}^d$, for any $0 \le T < \infty$. For any $t \ge 0$ and any $x \in \mathbb{R}^d$ it is not hard to see that,$$E^x u(t, W_t) = u(0, x) + E^x \int_0^y \left( {\partial\over{\partial s}} + {1\over2}\Delta_s\right) u(s, W_s)\,ds.$$$($$W_t$ is a Brownian motion process which takes the value $x$ at $t=0$ so the dependence of $x$ is implicit here, most of the references will consider $W_t$ a standard Brownian motion which means it takes the value $0$ at $t=0$ and in the above formula we will have $x+W_t$ instead. $\Delta_s$ here means the sum of all the mixed derivatives.$)$

My question is, how do we conclude that under $P^x$ the process$$u(t, W_t) - u(0, x) - \left({\partial\over{\partial t}} + {1\over2}\Delta_x\right)u(t, W_t)$$is a martingale?

This question was asked here, but it did not get enough love, so I'm asking it again.

Let $u: \mathbb{R}_+ \times \mathbb{R}^d$ be a bounded $C^2$ function whose first and second partial derivatives are uniformly bounded (or, more generally, have at most polynomial growth as $|x| \to \infty)$ on $[0, T] \times \mathbb{R}^d$, for any $0 \le T < \infty$. For any $t \ge 0$ and any $x \in \mathbb{R}^d$ it is not hard to see that,$$E^x u(t, W_t) = u(0, x) + E^x \int_0^y \left( {\partial\over{\partial s}} + {1\over2}\Delta_s\right) u(s, W_s)\,ds.$$$($$W_t$ is a Brownian motion process which takes the value $x$ at $t=0$ so the dependence of $x$ is implicit here, most of the references will consider $W_t$ a standard Brownian motion which means it takes the value $0$ at $t=0$ and in the above formula we will have $x+W_t$ instead. $\Delta_s$ here means the sum of all the mixed derivatives.$)$

My question is, how do we conclude that under $P^x$ the process$$u(t, W_t) - u(0, x) - \left({\partial\over{\partial t}} + {1\over2}\Delta_x\right)u(t, W_t)$$is a martingale?

This question was asked here, but it did not get enough attention, so I'm crossposting it to MO.

Let $u: \mathbb{R}_+ \times \mathbb{R}^d$ be a bounded $C^2$ function whose first and second partial derivatives are uniformly bounded (or, more generally, have at most polynomial growth as $|x| \to \infty)$ on $[0, T] \times \mathbb{R}^d$, for any $0 \le T < \infty$. For any $t \ge 0$ and any $x \in \mathbb{R}^d$ it is not hard to see that,$$E^x u(t, W_t) = u(0, x) + E^x \int_0^y \left( {\partial\over{\partial s}} + {1\over2}\Delta_s\right) u(s, W_s)\,ds.$$$($$W_t$ is a Brownian motion process which takes the value $x$ at $t=0$ so the dependence of $x$ is implicit here, most of the references will consider $W_t$ a standard Brownian motion which means it takes the value $0$ at $t=0$ and in the above formula we will have $x+W_t$ instead. $\Delta_s$ here means the sum of all the mixed derivatives.$)$

My question is, how do we conclude that under $P^x$ the process$$u(t, W_t) - u(0, x) - \left({\partial\over{\partial t}} + {1\over2}\Delta_x\right)u(t, W_t)$$is a martingale?