How to use the markovian property and use the uniqueness of SDEs Let $T>0$ be fixed.
Consider the stochastic differential equation
$$
(8.21) \quad 
\left\{
\begin{array}{l}
\displaystyle{{\rm d}X(t)=2(X(t) \vee 0)^{1/2}{\rm d}B(t)+\left(3-\frac{2X(t)}{T-t}\right){\rm d}t} \\
X(0)=0.
\end{array}
\right.
$$
Furthermore, $X(t)$ defines a time-dependent Markov process.
To be precise, for $0 \leq s<T$ and $x \in [0,\infty)$, let $W_{s,x}$, be the totality of all continuous paths $w:[s,T) \ni t \mapsto w(t) \in [0,\infty)$ such that $w(s)=x$ and $w(t)>0$ for all $t \in (s,T)$, ${\mathscr B}(W_{s,x})$ be the $\sigma$-field on $W_{s,x}$ generated by Borel cylinder sets and ${\mathscr B}_{t}(W_{s,x})$, $s<t<T$, be the sub $\sigma$-field generated by Borel cylinder sets depending only on the interval $[s,t]$.
Let $\hat{P}_{0,0}$ be the probability law on $(W_{0,0},{\mathscr B}(W_{0,0}))$ of the solution $X(t)$ of (8.21) and more generally $\hat{P}_{s,x}$, be the probability law on $(W_{s,x},{\mathscr B}(W_{s,x}))$ of the unique solution $\{X(t)\}_{t \in [s,T)}$ of
$$
(8.23) \quad 
\left\{
\begin{array}{l}
\displaystyle{{\rm d}X(t)=2(X(t) \vee 0)^{1/2}{\rm d}B(t)+\left(3-\frac{2X(t)}{T-t}\right){\rm d}t} \\
X(s)=x.
\end{array}
\right.
$$
The Markovian property of $\hat{P}_{0,0}$ is now formulated as follows; for $0<s<t$ and $f \in B([0,\infty))$ (i.e., $B([0,\infty))$ is the totality of all bounded Borel measurable functions on $[0,\infty)$.),
$$
(8.24) \quad 
\hat{{\mathbb E}}_{0,0}[f(w(t))|{\mathscr B}_{s}(W_{0,0})]=\hat{{\mathbb E}}_{s,w(s)}[f(w'(t))] \quad \text{a.a.} \, w(\hat{P}_{0,0}).
$$
Moregenerally, for $0 \leq u<s<t$, $x \in [0,\infty)$ and $f \in B([0,\infty))$,
$$
(8.25) \quad 
\hat{{\mathbb E}}_{u,x}[f(w(t))|{\mathscr B}_{s}(W_{u,x})]=\hat{{\mathbb E}}_{s,w(s)}[f(w'(t))] \quad \text{a.a.} \, w(\hat{P}_{u,x}).
$$
This is described on page 240 of the following book: N. Ikeda and S. Watanabe. Stochastic differential equations and diffusion processes, 2nd edn. North-Holland, Amsterdam, (1981).
Question: I want to know the proof of (8.25).
 A: Let $A \in {\mathscr B}_{s}(W_{u,x})$.
By the markovian property of $\hat{P}_{u,x}$, we obtain
$$
(1) \quad 
\hat{{\mathbb E}}_{u,x}[f(w(t)){\bf 1}_{A}(w)]
=\hat{{\mathbb E}}_{u,x}[\hat{{\mathbb E}}_{u,w(s)}[f(w'(t-s))]{\bf 1}_{A}(w)].
$$
On the other hand, let $w \in W_{u,x}$ and $f \in {\mathscr D}(A)$.
By the ito's formula, a stochastic process
$$
\left(f(w'(t-s))-f(w(s))-\int_{u}^{t-s}(Af)(w'(v)){\rm d}v\right)_{t \in [u+s,u+T)}
$$
is a $(\hat{P}_{u,w(s)},({\mathscr B}_{t}(W_{u,w(s)}))_{t \in [u+s,u+T)})$-martingale.
This implies that a stochastic process
$$
\left(f(w'(t-s))-f(w(s))-\int_{u+s}^{t}(Af)(w'(v-s)){\rm d}v\right)_{t \in [u+s,u+T)}
$$
is a $(\hat{P}_{u,w(s)},({\mathscr B}_{t}(W_{u,w(s)}))_{t \in [u+s,u+T)})$-martingale.
Thus a stochastic process $\{X(t-s)\}_{t \in [u+s,u+T)}$ is a solution to the equation (8.23) with $X(u)=w(s)$.
Hence by the uniqueness of solutions to the equation (8.23), we obtain
$$
(2) \quad 
\hat{{\mathbb E}}_{u,w(s)}[f(w'(t-s))]=\hat{{\mathbb E}}_{s,w(s)}[f(w'(t))].
$$
Consequently, by (1) and (2),
$$
\hat{{\mathbb E}}_{u,x}[f(w(t)){\bf 1}_{A}(w)]
=\hat{{\mathbb E}}_{u,x}[\hat{{\mathbb E}}_{s,w(s)}[f(w'(t))]{\bf 1}_{A}(w)].
$$
This implies that
$$
\hat{{\mathbb E}}_{u,x}[f(w(t))|{\mathscr B}_{s}(W_{u,x})]
=\hat{{\mathbb E}}_{s,w(s)}[f(w'(t))] \quad \text{a.a.} \, w(\hat{P}_{u,x}).
$$
