The first exit time for a Bessel process from the minimum and maximum random domains

Consider two exit probabilities of the Bessel process B(s) where hi(x),i=1,2 are reversible nondecreasing lower semi-continuous convex functions on [0,[infinity]) with hi(0),i=1,2 finite. W1(s) and W2(s) are independent standard Brownian motions and independent of {B(s)[set membership, variant]Rd,t>=0}. Based on the specific relationship between and , very useful estimates for the asymptotics of logP([dot operator]) are given by using Gaussian technique, respectively.