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)||B(s)|P(|B(s)|≤mini=1,2{hi−1(hi(0)+1+Wi(s))},0≤s≤t),P(|B(s)|≤maxj=1,2{hj−1(hj(0)+1+Wj(s))},0≤s≤t), where hi(x),i=1,2hi(x),i=1,2 are reversible nondecreasing lower semi-continuous convex functions on [0,∞)[0,∞) with hi(0),i=1,2hi(0),i=1,2 finite. W1(s)W1(s) and W2(s)W2(s) are independent standard Brownian motions and independent of {B(s)∈Rd,t≥0}{B(s)∈Rd,t≥0}. Based on the specific relationship between h1−1(x) and h2−1(x), very useful estimates for the asymptotics of logP(⋅)logP(⋅) are given by using Gaussian technique, respectively.