Let $B_t$ be standard Brownian motion, and let $M_t = \sup_{0 \leq s \leq t} B_s$ be its running maximum. $M_t$ is not a Markov process, but we can augment it with additional information to make it Markov. For instance, $(M_t, B_t)$ and $(M_t, M_t - B_t)$ are Markov processes.

This prompts the question: what else can we augment $M_t$ with to get a Markov process? In particular, consider $$ \begin{aligned} X_t &:= (M_t, t - \sup\{s \leq t \mid B_s = M_t\}) \\ & = (M_t, \text{"time since running maximum was last attained"}). \end{aligned} $$ I'm wondering:

• Is $X_t$ a Markov process?
• Is there a standard name and/or reference for $X_t$?
• Does the conclusion change if instead of $B_t$, we use a different diffusion process?