TY - JOUR

T1 - Brownian bridge expansions for Lévy area approximations and particular values of the Riemann zeta function

AU - Foster, James

AU - Habermann, Karen

PY - 2022/11/3

Y1 - 2022/11/3

N2 - 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.

AB - 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.

KW - Brownian motion

KW - Karhunen-Loève expansion

KW - polynomial approximation

KW - Lévy area

KW - fluctuations

KW - Riemann zeta function

UR - https://arxiv.org/abs/2102.10095

U2 - 10.1017/S096354832200030X

DO - 10.1017/S096354832200030X

M3 - Article

JO - Combinatorics, Probability and Computing

JF - Combinatorics, Probability and Computing

SN - 0963-5483

ER -