Markov or non-Markov property of cM ― X processes

摘要

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 Levy's and Pitman's theorems. We show that cM ― X is not a Markov process if c ∈ R{0, 1, 2}. We also give other elementary proofs of Levy's and Pitman's theorems.