# Boundary non-crossing probabilities of Gaussian processes: sharp bounds and asymptotics

## TL;DR

This paper derives sharp bounds and precise asymptotics for boundary non-crossing probabilities of Gaussian processes, especially for large drifts, using optimization in the reproducing kernel Hilbert space.

## Contribution

It provides the first sharp bounds and two-term asymptotics for boundary crossing probabilities of Gaussian processes with arbitrary index sets.

## Key findings

- Sharp upper and lower bounds for boundary non-crossing probabilities.
- Precise logarithmic asymptotics for large drifts in boundary crossing probabilities.
- Application of optimization in reproducing kernel Hilbert space to derive asymptotics.

## Abstract

We study boundary non-crossing probabilities $$ P_{f,u} := \mathrm P\big(\forall t\in \mathbb T\ X_t + f(t)\le u(t)\big) $$ for continuous centered Gaussian process $X$ indexed by some arbitrary compact separable metric space $\mathbb T$. We obtain both upper and lower bounds for $P_{f,u}$. The bounds are matching in the sense that they lead to precise logarithmic asymptotics for the large-drift case $P_{y f,u}$, $y \to+\infty$, which are two-term approximations (up to $o(y)$). The asymptotics are formulated in terms of the solution $\tilde f$ to the constrained optimization problem $$ \|h\|_{\mathbb H_X}\to \min, \quad h\in \mathbb H_X, h\ge f $$ in the reproducing kernel Hilbert space $\mathbb H_X$ of $X$. Several applications of the results are further presented.