Let $X_t$ be a Brownian motion with positive drift starting at 0. The process with reflection at fixed barrier $b<0$ (sometimes called a "regulated Brownian motion") is:
\begin{equation} \label{fixed-barrier} Y_t = X_t + \max\left\{ 0, \,\, \max_{0 \le n \le t}(b - X_n)\right\}. \end{equation}
Consider instead $Z_t$ constructed by starting at $Z_0=X_0$ and then reflecting $X_t$ at a "trailing barrier" which starts at the same $b <0$ as above, but then increases whenever $Z_t$ attains a new maximum :
\begin{equation} \label{trailing-barrier} Z_t = X_t + \max\left\{ 0, \,\,\max_{0 \le n\ \le t}\left(\max_{0 \le i \le n}(b+\,Z_i - X_n)\right)\right\}. \end{equation}
I want to find the transition density (or say something else useful) for this $Z_t$: a Brownian motion reflected at a trailing barrier, where the barrier is a fixed distance below the running maximum of the reflected process.
Here’s a simulation of $X_t$ (red) and $Z_t$ (blue) with $b=-0.6$. We can see that the reflecting barrier (dashed blue line) often stays fixed for extended periods; it moves only at times when $Z_t$ reaches a new maximum.
Drifting Brownian motion reflected at trailing barrier
The brown stepped line in the version below shows the amount added to $X_t$ to construct $Z_t$.
Reference: Here is the solution for the fixed barrier version https://link.springer.com/article/10.1023/B:CSEM.0000049491.13935.af
