Self-similar processes with independent increments associated with Lévy and Bessel processes

Wolfe (Stochastic Process. Appl. 12(3) (1982) 301) and Sato (Probab. Theory Related Fields 89(3) (1991) 285) gave two different representations of a random variable X1 with a self-decomposable distribution in terms of processes with independent increments. This paper shows how either of these representations follows easily from the other, and makes these representations more explicit when X1 is either a first or last passage time for a Bessel process.