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

James Foster, Karen Habermann

Research output: Contribution to journalArticlepeer-review

Abstract

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.
Original languageEnglish
Number of pages28
JournalCombinatorics, Probability and Computing
DOIs
Publication statusPublished - 3 Nov 2022

Keywords

  • Brownian motion
  • Karhunen-Loève expansion
  • polynomial approximation
  • Lévy area
  • fluctuations
  • Riemann zeta function

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