A converse to a theorem of P. Lévy
By a well-known theorem of \textit{P. Lévy} [Processus stochastiques et mouvement brownien (1948; Zbl 0034.226)], if \((X_ t)\) is a standard Brownian motion on \({\mathbb{R}}\) with \(X_ 0=0\) and if \(H_ t=\min_{u\leq t}X_ u\), then \((Y_ t)=(X_ t-H_ t)\) is a Brownian motion with 0 as a reflecting lower boundary. More generally, if X is allowed to have nonzero drift or a reflecting lower boundary at \(A<0\), then the process \(Y=X-H\) is still a diffusion process. We prove the converse result: If X is a diffusion on an interval \(I\subset {\mathbb{R}}\) which contains 0 as an interior point, and if \((Y_ t)=(X_ t-H_ t)\) is a time homogeneous strong Markov process (when \(X_ 0=0)\), then X must be a Brownian motion on I (with drift \(\mu\), variance parameter \(\sigma^ 2>0\), killing rate \(c\geq 0\) and reflection at inf I in case inf I\(>-\infty)\).
scientific article; zbMATH DE number 4064205