For a Brownian motion with a constant drift and its maximum process and are diffusion processes by the extensions of Lévy's and Pitman's theorems. We show that is not a Markov process if ∊. We also give other elementary proofs of Lévy's and Pitman's theorems.
Hiroyuki MATSUMOTO. Yukio OGURA. "Markov or non-Markov property of processes." J. Math. Soc. Japan 56 (2) 519 - 540, April, 2004. https://doi.org/10.2969/jmsj/1191418643