区间约束布朗路径：精确插值与外推
Interval-Constrained Brownian Paths: Exact Interpolation and Extrapolation
- The Ohio State University（俄亥俄州立大学）
- Foursquare Labs, Inc.（四方实验室公司）
机构由 AI 辅助整理，请以论文原文为准。
中文总结 AI 辅助
本文研究区间约束布朗运动与布朗桥的精确插值和外推，提出统一采样套件，含边界端点并自动切换密度表示，提升接受率，并扩展至多时刻骨架路径模拟。
中文摘要 AI 辅助
我们研究条件限制在固定区间 $[0,a]$ 内的布朗运动和布朗桥过程，重点关注单个时刻的条件分布。对于 $[0,a]$ 中的布朗运动，在给定生存至时间 $t$ 的条件下，我们回顾（并以自包含形式呈现）其在 $t$ 时刻位置的条件密度（精确外推）。对于在 $[0,T]$ 上条件限制在 $[0,a]$ 内的布朗桥，我们将内部时刻的概率密度表示为被杀死转移密度的归一化乘积（精确插值）。这些密度具有对偶互补级数表示，通过雅可比 theta 恒等式相互关联。我们的主要贡献是统一的外推和插值精确采样套件，该套件（i）包含边界端点，（ii）自动在密度表示之间切换以在各区间保持高效的接受率。为此，我们推导了覆盖小时间和大时间区间的简单提议分布族，并采用自动规则选择更紧的包络。我们的方法通过马尔可夫性质扩展到多个时刻离散骨架路径的精确模拟。
英文摘要
We study Brownian motion and Brownian bridge processes conditioned to remain in a fixed interval $[0,a]$, focusing on the conditional distribution at a single time. For a Brownian motion in $[0,a]$, conditioned on survival up to time $t$ we recall (and present in a self-contained form) the conditional density of its position at $t$ (exact extrapolation). For a Brownian bridge conditioned to remain in $[0,a]$ on $[0,T]$ we express the probability density at an interior time as a normalized product of killed transition densities (exact interpolation). These densities admit dual complementary series representations, which are linked via the Jacobi theta identity. Our main contribution is a unified exact sampling suite for both extrapolation and interpolation that (i) includes boundary endpoints and (ii) automatically switches between density representations to keep acceptance rates efficient across regimes. To that end, we derive simple proposal families which cover both small- and large-time regimes, with an automatic rule selecting the tighter envelope. Our procedures extend to exact simulation of discrete skeleton paths at multiple times via the Markov property.
