First Exit Time from a Bounded Interval for a Certain Class of Additive Functionals of Brownian Motion

Let (B t) t≥0 be standard Brownian motion starting at y, X t = x + ∫ t 0 V(B s) ds for x ∈ (a, b), with V(y) = y γ if y≥0, V(y)=−K(−y) γ if y≤0, where γ>0 and K is a given positive constant. Set τ ab=inf{t>0: X t∉(a, b)} and σ 0=inf{t>0: B t=0}. In this paper we give several informations about the random variable τ ab. We namely evaluate the moments of the random variables $$B_{\tau _{ab} } and B_{\tau _{ab} \wedge \sigma _0 } $$ , and also show how to calculate the expectations $${\mathbb{E}}\left( {\tau _{ab}^m B_{\tau _{ab} }^n } \right) and {\mathbb{E}}\left( {\left( {\tau _{ab} \wedge \sigma _0 } \right)^m B_{\tau _{ab} \wedge \sigma _0 }^n } \right)$$ . Then, we explicitly determine the probability laws of the random variables $$B_{{\tau }_{ab} } and B_{{\tau }_{ab} \wedge \sigma _0 }$$ as well as the probability $${\mathbb{P}}\left\{ {X_{\tau _{ab} } = a\left( {or b} \right)} \right\}$$ by means of special functions.