Let $W_{0\leq s \leq T}$ be a standard Wiener process. Define the Ito integral
$$
X_t= x_0+\int_0^t g(i)ds+ \int_0^tdW_s
$$
with $x_0$ given and where $i\in \{1,2,...n\}$ is some parameter that affects the drift term.
Suppose that $i$ is unknown and we can only observe $X$. Let $p_0^k\equiv Pr(i=k)$ be the prior that the parameter equals $k$ and $p_t^k(X_t)\equiv Pr(i=k|X_t)$ be the Bayesian updated probability that the parameter is $k$ after observing $X_t$. Finally, let $E_t[\cdot]$ be the expectation operator given the filtration generated by $X$ up to time $t$.
Is the following true? $$ E_t[X_T]= X_t + E_t\bigg[\int_t^TdX_s\bigg] $$ $$ =X_t+ \sum_{k=1}^n p_t^k(X_t)E_t\bigg[\int_t^TdX_s\bigg|i=k\bigg]. $$ I am having a bit of trouble showing this formally. In discrete time, using iterated expectations, we would get a similar result but I am not sure in the continuous time setting.