Brownian Crossings via Regeneration Times

Abstract

Let {Bt, t ≥ 0} be a standard one-dimensional Brownian motion. For each t > 0 let σt be the last entrance time before t into the interval (a,b), dt the time of the first exit from (a,b) after t and \(Y_t := B_t - B_{\sigma_t}\). In this paper we study i) the limit behaviour of the normalised occupation times of the process (Yt), ii) the limiting joint distribution of (t − σt, dt − t) and \((d_t-t, B_{d_t}- B_t)\), conditioned on the event {Bt ∈ (a,b)}, as t → ∞ and iii) derive renewal equations satisfied by the probabilities \( \phi (t) := P_a \{ 0 < t-\sigma_t < u,~ 0< B_t -B_{\sigma_t} < y\} \) and \(\gamma(t) := P_a\{0 < d_t - t < u, ~0 < B_{d_t} -B_{t}< y \}\).