Suppose $dX_t=b\,dt+\sigma\,dW_t$ and $X_0=x$
$g$ is a measurable function.
What does $\mathbb{E}[g(X_T^{t,Y})]$ mean? Is it a number or R.V.?
It confuses me because I think there are two explanations:
(1) $\mathbb{E}[g(X_T^{t,Y})]=\mathbb{E}[g(X_T)|X_t=Y]=\mathbb{E}^{t,x}[g(X_T)]|_{x=Y}$
(2) Solve $dX_t=b\,dt+\sigma\,dW_t$ and $X_t=Y$, then take the expectation of $X_T$
The result in (1) is a R.V. but in (2) is a number. which one is correct? Thanks in advance.