Invariance formulas for stopping times of squared Bessel process

Abstract

ABSTRACTThe invariance formulas for stopping times of squared Bessel processes with positive index are presented. For a stopping time τ satisfying some relatively mild assumptions and every positive Borel function f, we have the invariance formulawhere and γμ is a gamma random variable independent of σ(τ, R(τ)). The second invariance formula iswhere K is a constant and Δμ is an explicitly given function. The applications of invariance formulas are presented. In particular, new results on the distribution of a killed squared Bessel process, a conditional inverse local time, and the first hitting time of a graph by a squared Bessel process are delivered.