用于 Lévy 面积逼近及 Riemann zeta 函数特殊值的 Brown 桥展开
Brownian bridge expansions for Lévy area approximations and particular values of the Riemann zeta function
AI总结：
本文利用 Fourier 与 Brown 桥多项式展开构造 Lévy 面积逼近，证明其在仅用独立正态向量时优于 Kloeden-Platen-Wright 方法，并通过涨落分析独立导出 Riemann zeta 函数的正偶数值。
摘要：
我们研究基于 Fourier 级数展开以及相关 Brown 桥的多项式展开的 Brown 运动 Lévy 面积逼近。通过比较这些 Lévy 面积逼近的渐近收敛速率，我们看到由 Brown 桥多项式展开得到的逼近比 Kloeden-Platen-Wright 逼近更精确，同时仍然只使用独立正态随机向量。随后，我们将这些逼近的渐近收敛速率与 Brown 桥相应级数展开的极限涨落联系起来。此外，这一点本身也具有意义：我们用于识别 Brown 桥的 Karhunen-Loève 展开和 Fourier 级数展开之涨落过程的分析，被推广用于独立推导 Riemann zeta 函数在正偶数处的取值。
英文摘要：
We study approximations for the Lévy area of Brownian motion which are based on the Fourier series expansion and a polynomial expansion of the associated Brownian bridge. Comparing the asymptotic convergence rates of the Lévy area approximations, we see that the approximation resulting from the polynomial expansion of the Brownian bridge is more accurate than the Kloeden-Platen-Wright approximation, whilst still only using independent normal random vectors. We then link the asymptotic convergence rates of these approximations to the limiting fluctuations for the corresponding series expansions of the Brownian bridge. Moreover, and of interest in its own right, the analysis we use to identify the fluctuation processes for the Karhunen-Loève and Fourier series expansions of the Brownian bridge is extended to give a stand-alone derivation of the values of the Riemann zeta function at even positive integers.
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