what will a brownian motion be if we know some future conditions

Question

Suppose $B_t$ is a Brownian motion, and define $\hat B_t = B_{t-L}$, where L is a fixed and known positive number.

Now suppose we can only observe $\hat B_t$ and we know that at time $t_0$, $B_{t_0} = a$, and during the time ${t_0}-L$ to ${t_0}$, $B_t$ is always larger than $b$ $(b \leq a)$.

What will the dynamic process of $\hat B_t$ like in the time ${t_0}$ to ${t_0}+L$?

Thanks.

Answer 1

We assume $B_{t_{0}}=a$ and that $B_{t}\geq b$ for $t\in [t_{0}-L,t_{0}]$. So we know that $\hat{B}_{t_{0}+L}=B_{t_{0}}=a$ and $\hat{B}_{t}\geq b$ for $t\in [t_{0},t_{0}+L]$.