Suppose the stochastic equation： \begin{equation*} d X(u)=\beta(u,X(u))d u+\gamma(u,X(u))d W(u). \end{equation*} Suppose $X(T)$ is the solution of above stochastic differential equation with initial condition $X(t)=x$ and $h(x)$ is a Borel-measurable function. Denote by $$g(t,x)=E^{t,x}h(X(T))$$ We assume $E^{t,x}|h(X(T))|<\infty$

Let $X(u)$ is the solution of above stochastic differential equation with initial condition given at time $0.$

Use the markov property of $X(t),$ we have existing $g(t,x)$ s.t $$E[h(X(T))|\mathcal{F}(t)]=g(t,X(t))$$

My question is are those two $g(t,x)$ same? Since we want to use Feynman-Kac equation, but I am not sure whether it is true for first $$g(t,x)=E^{t,x}h(X(T)).$$

Suppose the stochastic equation： \begin{equation*} d X(u)=\beta(u,X(u))d u+\gamma(u,X(u))d W(u). \end{equation*} Suppose $X(T)$ is the solution of above stochastic differential equation with initial condition $X(t)=x$ and $h(x)$ is a Borel-measurable function. Denote by $$g(t,x)=E^{t,x}h(X(T))$$ We assume $E^{t,x}|h(X(T))|<\infty$

Let $X(u)$ is the solution of above stochastic differential equation with initial condition given at time $0.$

Use the markov property of $X(t),$ we have existing $g(t,x)$ s.t $$E[h(X(T))|\mathcal{F}(t)]=g(t,X(t))$$

My question is are those two $g(t,x)$ same? Since we want to use Feynman-Kac equation, but I am not sure whether it is true for first $$g(t,x)=E^{t,x}h(X(T)).$$ since the proof of Feynman-Kac equation needs the martingale of $g(t,X(t)),$ but I don't think here $g(t,X(t))$ is martingale?