Asymptotic of Non-Crossings probability of Additive Wiener Fields

Abstract

Let $W_i=\{W_i(t_i),t_i\in {\mathbb{R}}_{+}\},i=1,2,\dots ,d$ are independent Wiener processes. $W=\{W(\mathbf{t}),t\in {\mathbb{R}}_{+}^d\}$ 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 $$P_f=P\{\sum _{i=1}^d{W}_i(t_i)+f(\mathbf{t})\le u(\mathbf{t}),\mathbf{t}\in {\mathbb{R}}_{+}^d\},$$ where $u:{\mathbb{R}}_{+}^d\to {\mathbb{R}}_{+}$ 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 $γ\to \IF$, where $\underline{f}$ is the projection of $f$ on some closed convex subset of $\kHC$.

12 pages. arXiv admin note: text overlap with arXiv:1402.2620 by other authors

References in corpus (1)

• Boundary crossing probabilities for $(q,d)$-Slepian-processes