标题： 加性维纳场非交叉概率的渐近性 显示英文标题

标题： Asymptotic of Non-Crossings probability of Additive Wiener Fields

摘要： 设$W_i=\{W_i(t_i), t_i\in \R_+\}, i=1,2,\ldots,d$为独立的维纳过程。$W=\{W(\mathbf{t}),t\in \R_+^d\}$为加性维纳场，定义为$W_i$的和。 对于任何趋势$f$在$\kHC$（$W$的再生核希尔伯特空间）中，我们推导了边界不交叉概率$$P_f=P\{\sum_{i=1}^{d}W_i(t_i) +f(\mathbf{t})\leq u(\mathbf{t}), \mathbf{t}\in\R_+^d\},$$的上下界，其中$u: \R_+^d\rightarrow \R_+$是一个可测函数。 此外，对于大的趋势函数$\gamma f>0$，我们证明了渐近关系$\ln P_{\gamma f}\sim \ln P_{\gamma \underline{f}}$当$\gamma \to \IF$，其中$\underline{f}$是$f$在$\kHC$的某个闭凸子集上的投影。 显示英文摘要

摘要： 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 $\kHC$.