While trying to solve a problem, I realized I might need to compute the distribution of $W_T^{*} = \sup_{t \leq T} W_t$ conditional to $\mathcal{F}_A$, where $T > A > 0$ is a deterministic time and $\mathcal{F}$ is the natural filtration of the Brownian motion $W$.
I would really appreciate any help on this one ! Thank you so much and have a wonderful new year's eve !
Since
$$W_T^* = \max \left\{W_A^*, W_A+ \sup_{0 \leq t \leq T-A} (W_{t+A}-W_A) \right\}$$
we have
$$\mathbb{E}(f(W_T^*) \mid \mathcal{F}_A) = \mathbb{E} \left( f \left[ \max \left\{W_A^*, W_A+ \sup_{t \leq T-A} (W_{t+A}-W_A) \right\} \right] \mid \mathcal{F}_A \right).$$
The random variables $W_A$ and $W_A^*$ are $\mathcal{F}_A$-measurable and the process $$B_t := W_{t+A}-W_A$$ is independent from $\mathcal{F}_A$; therefore, we obtain
$$\mathbb{E}(f(W_T^*) \mid \mathcal{F}_A) = \mathbb{E} \left( f \left[ \max \left\{x, y+ \sup_{t \leq T-A} B_t \right\} \right] \right) \bigg|_{x=W_A^*, y=W_A}. \tag{1}$$
The restarted process $(B_t)_{t \geq 0}$ is a Brownian motion and therefore we know from the reflection principle that
$$\sup_{t \leq T-A} B_t \sim |B_{T-A}|.$$
Since $B_{T-A}$ is Gaussian with mean $0$ and variance $T-A$, we obtain
$$\begin{align*} \mathbb{E}(f(W_T^*) \mid \mathcal{F}_A) &= \mathbb{E}(f[\max\{x,y+|B_{T-A}|\}]) \bigg|_{x=W_A^*,y=W_A} \\ &= \sqrt{\frac{2}{\pi(T-A)}} \int_{u \geq 0} f(\max\{W_A^*,W_A+u\}) \exp \left(- \frac{u^2}{2(T-A)} \right) \, du. \end{align*}$$