Sharp barrier estimates for Bessel bridges

TL;DR

This paper provides precise asymptotic estimates for the probability that a Bessel bridge stays below a linear barrier, including error bounds and behavior under parameter limits, with extensions to perturbed barriers.

Contribution

It introduces sharp barrier estimates for Bessel bridges, detailing asymptotic behaviors, error terms, and effects of barrier perturbations, advancing understanding of their probabilistic properties.

Findings

Asymptotic probability estimates as T→∞

Error bounds for barrier crossing probabilities

Behavior under large parameter limits

Abstract

In this article, we derive precise estimates for the probability that a Bessel bridge of dimension $d \ge 0$ and end points $x$ and $a+bT-j$ stays below the linear barrier $a + bt$ for all $t \in [0,T]$. We identify the leading order term as well as the asymptotic error for this probability as $T\to \infty$, depending on $a,b,j,x$. We also derive the behaviour of such leading term as we allow $a,j\to \infty$, and obtain precise bounds for all error terms. Finally, we establish a complementary result where the linear barrier is perturbed by a small concave function.