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

Abstract

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.