Markov or non-Markov property of cM-X processes

Abstract

For a Brownian motion with a constant drift X and its maximum process M, M-X and 2M-X are diffusion processes by the extensions of Lévy's and Pitman's theorems. We show that cM-X is not a Markov process if c∈ \bm{R}\backslash{0, 1, 2}. We also give other elementary proofs of Lévy's and Pitman's theorems.