Asymptotic of Non-Crossings probability of Additive Wiener Fields

TL;DR

This paper investigates the asymptotic behavior of the probability that an additive Wiener field with a trend does not cross a boundary, providing bounds and asymptotic relations for large trends in high-dimensional stochastic processes.

Contribution

It derives bounds and asymptotic relations for non-crossing probabilities of additive Wiener fields with trends in high dimensions, extending understanding of boundary crossing in stochastic processes.

Findings

Established upper and lower bounds for non-crossing probabilities.

Proved asymptotic equivalence of probabilities for large trend functions.

Connected the asymptotics to projections in the reproducing kernel Hilbert space.

Abstract

Let $W_i=\{W_i(t_i), t_i\in \R_+\}, i=1,2,\ldots,d$ are independent Wiener processes. $W=\{W(\mathbf{t}),t\in \R_+^d\}$ be the additive Wiener field define as the sum of $W_i$. For any trend $f$ in $\kHC$ (the reproducing kernel Hilbert Space of $W$), we derive upper and lower bounds for the boundary non-crossing probability $$P_f=P\{\sum_{i=1}^{d}W_i(t_i) +f(\mathbf{t})\leq u(\mathbf{t}), \mathbf{t}\in\R_+^d\},$$ where $u: \R_+^d\rightarrow \R_+$ is a measurable function. Furthermore, for large trend functions $\gamma f>0$, we show that the asymptotically relation $\ln P_{\gamma f}\sim \ln P_{\gamma \underline{f}}$ as $\gamma \to \IF$, where $\underline{f}$ is the projection of $f$ on some closed convex subset of $\kHC$.