In this note we 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. An ultimate minimum of a stochastic process may be seen as a state of a process at a last hitting time. This discussion is strongly motivated by our interest in properties of last hitting times in general, and here specifically, directly linked with the reading guide of Nikeghbali and Platen (2013).
"A New Proof of Williams' Decomposition of the Bessel Process of Dimension Three with a Look at Last-Hitting Times." Bull. Belg. Math. Soc. Simon Stevin 22 (2) 319 - 330, may 2015. https://doi.org/10.36045/bbms/1432840867