Fluctuations for Brownian bridge expansions and convergence rates of Lévy area approximations
5th May 2021, 4:00 pm – 5:00 pm
We start by deriving a polynomial expansion for Brownian motion expressed in terms of shifted Legendre polynomials by considering Brownian motion conditioned to have vanishing iterated time integrals of all orders. We further discuss the fluctuations for this expansion and show that they converge in finite dimensional distributions to a collection of independent zero-mean Gaussian random variables whose variances follow a scaled semicircle. We then link the asymptotic convergence rates of approximations for Brownian Lévy area which are based on the Fourier series expansion and the polynomial expansion of the Brownian bridge to these limit fluctuations. We close with the observation that the Lévy area 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.