粗糙分数布朗运动最大值处的局部结构与尖锐持久渐近性
Local structure at the maximum and sharp persistence asymptotics of rough fractional Brownian motion
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中文总结 AI 辅助
研究具有赫斯特指数\(0 < H < 1/2\)的分数布朗运动最大值处局部结构，证明重标过程收敛到特定极限切线律，该律有自然解释，还将其应用于持久概率，给出相关渐近公式。
中文摘要 AI 辅助
我们考虑具有赫斯特指数\(0 < H < 1/2\)的分数布朗运动\(B\)及其在\([0,1]\)上的最大值点\(\tau\)。我们证明了重标过程\(a^H(B_{\tau+\cdot/a}-B_\tau)\)在\(C_{\mathrm{loc}}(\mathbb R)\)中收敛到一个极限切线律，该切线律是\(H\)-自相似的，支撑在非正路径且在零点固定，并且具有重根 - 重标不变性。我们还表明该切线律可自然解释为“在整条直线上非正的分数布朗运动”。作为应用，我们考虑分数布朗运动的持久概率。\(B\)的一个倾斜变体产生具有有限左边界和无限右边界的不同切线律，并且我们证明了\[ \mathbb P(B_t \leq 1\text{ 对于所有 }0 \leq t \leq T) \sim \frac{H\mathbb E[M]}{\Gamma(1/H)D_H}T^{-(1 - H)}, \]其中\(D_H \in (0,\infty)\)根据倾斜切线律有明确表示。
英文摘要
We consider a fractional Brownian motion $B$ with Hurst index $0<H<1/2$, and its maximiser $τ$ on $[0,1]$. We show that the rescaled process $a^H(B_{τ+\,\cdot\,/a}-B_τ)$ converges in $C_{\mathrm{loc}}(\mathbb R)$ to a limiting tangent law that is $H$-self-similar, supported on nonpositive paths pinned at zero, and rerooting-rescaling invariant: rerooting the limit process at its maximum on any fixed compact interval separated from zero and rescaling again asymptotically reproduces the same law. We also identify the tangent law as the limit of two-sided finite-grid hard-wall laws as the mesh vanishes and both horizons diverge, which can informally be interpreted as conditioning fractional Brownian motion on a nonpositive path. As an application, we consider persistence probabilities for fractional Brownian motion: a tilted variant of $B$ yields a different tangent law with a finite left horizon and an infinite right horizon and we show that \[ \mathbb P(B_t\leq1\text{ for all }0\leq t\leq T) = \big(C+o(1)\big)\,T^{-(1-H)},\quad \text{as }T\to\infty, \] where the leading order coefficient $C\in(0,\infty)$ has an explicit representation in terms of the expected maximum and the tilted tangent law.
