I have a Gaussian process on the form
\begin{align} X(t) &= X_0 + \mu(t) + \int_{0}^{t}\sigma(u)dW(u), \end{align} where $W(t)$ is a brownian motion and the initial condition $X(0) = X_0$ is given and $\sigma : \mathbb{R} \rightarrow \mathbb{R}$ is a deterministic function.
I want to find the dynamics of $X(t)$ for $t \geq T$ when additionally \begin{align} X(T) = X_t \end{align} is known.
My intuition says that as $X(t)$ is an Ito diffusion, it has the Markov property, thus the dynamics conditional on $X(T) = X_T$ shold be \begin{align} X_T + \mu(t) \int_{T}^t \sigma(u)dW(u). \end{align}
Is this in fact correct? If so, how could this be formalized? Does one appeal to the independent increments of the process?