The first exit time for a Bessel process from the minimum and maximum random domains (Q734693)

Let \(h_i(\cdot)\), \(i= 1,2\) reversible non-decreasing lower semicontinuous convex function on \([0,\infty)\) with \(h_i(0)\), \(i= 1,2\) finite and let \(W_i(\cdot)\), \(W_2(\cdot)\) are independent standard Brownian motions and independent of \(\{B(t)\in\mathbb{R}^d,\,t\geq 0\}\), where \(|B(s)|\) is a Bessel process. Considering some relations between \(h^{-1}_1(\cdot)\) and \(h^{-1}_2(\cdot\), the author establishes asymptotic estimates for \(\log P(\cdot)\), where \[ P\Biggl(|B(s)|\leq \min_{i=1,2} \{h^{-1}_i(h_i(0)+ 1+ W_i(s))\},\,0\leq s\leq t\Biggr) \] and \[ P\Biggl(|B(s)|\leq \max_{i=1,2} \{h^{-1}_i(h_i(0)+ 1+ W_i(s))\},\, 0\leq s\leq t\Biggr). \]