Last-Hitting Times and Williams' Decomposition of the Bessel Process of Dimension 3 at its Ultimate Minimum

In this note we shortly recall the importance of last-hitting times in theory and applications of optimal stopping. As a small contribution to this domain we then propose a concise proof of David Williams' decomposition of the Bessel Process of dimension 3 (BES(3)), starting from r > 0 at its ultimate minimum. This discussion is strongly motivated by our interest in properties of last hitting times in general, and here specifically, directly linked with the forthcoming reading guide of Nikeghbali and Platen on this subject.