Some explicit distributions related to the first exit time from a bounded interval for certain functionals of Brownian motion (Q867100)

Let \((B_t)_{t\geq 0}\) be standard Brownian motion starting at \(y\) and set \(X_t=x+ \int^t_0 V(B_s)\,ds\) for \(x\in (a,b)\), with \(V(y)=y^\gamma\) if \(y\geq 0\), \(V(y)=-K(-y)^\gamma\) if \(y\leq 0\), where \(\gamma\) and \(K\) are some given positive constants. Set \(\tau_{ab }=\inf \{t>0:X_t \notin (a,b)\}\). The author provides some formulas for the probability distribution of the random variable \(B_{\tau_{ab}}\) as well as for the probability \(\mathbb P \{X_{\tau_{ab}}=a\) (or \(b\))\}. The formulas corresponding to the particular cases \(x=a\) or \(b\) are explicitely expressed by means of hypergeometric functions.