Let (W) be a Brownian motion, (x, u, sigma) real numbers, (X_{t}=x exp left(u t+sigma W_{t} ight))

Let \(W\) be a Brownian motion, \(x, u, \sigma\) real numbers, \(X_{t}=x \exp \left(u t+\sigma W_{t}\right)\) and \(M_{t}^{X}=\sup _{s \leq t} X_{s}\). Prove that the process \(\left(Y_{t}=M_{t}^{X} / X_{t}, t \geq 0\right)\) is a Markov process. This fact (proved by Lévy) is used in particular in Shepp and Shiryaev [787] for the valuation of Russian options and in Guo and Shepp [412] for perpetual lookback American options.