First exit time from a bounded interval for a certain class of additive functionals of Brownian motion
Let \((B_t)_{t\geq 0}\) be a standard Brownian motion starting from \(y\) and \(X_t\) be the process defined by \(X_t=x+\int^t_0V(B_s)ds\) for \(x\in (a, b)\), with \(V(z)=z^{\gamma}\) if \(z\geq 0\), \(V(z)=-K(-z)^{\gamma}\) if \(z<0\), where \(\gamma\) and \(K\) are given positive constants. Set \(\tau_{ab}=\inf\{t>0: X_t\notin(a, b)\}\), and \(\sigma_0= \inf\{t>0: B_t=0\}\). The author studies (1) the moments of the random variables \(B_{\tau_{ab}}\) and \(B_{\tau_{ab}\wedge\sigma_0}\); (2) the joint moments of \(\tau_{ab}\) and \(B_{\tau_{ab}}\), and those of \(\tau_{ab}\wedge\sigma_0\) and \(B_{\tau_{ab}\wedge\sigma_0}\); (3) the laws of the random variables \(B_{\tau_{ab}}\) and \(B_{\tau_{ab}\wedge\sigma_0}\).
scientific article; zbMATH DE number 1552708