Strong convergence rates for markovian representations of fractional processes

Philipp Harms

doi:10.3934/dcdsb.2020367

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2021, Volume 26, Issue 10: 5567-5579. Doi: 10.3934/dcdsb.2020367

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Strong convergence rates for markovian representations of fractional processes

Philipp Harms

Department of Mathematical Stochastics, University of Freiburg, Germany

Received: February 2019

Revised: August 2020

Early access: December 2020

Published: October 2021

The author gratefully acknowledges support in the form of a Junior Fellowship of the Freiburg Institute of Advances Studies

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Abstract

Many fractional processes can be represented as an integral over a family of Ornstein–Uhlenbeck processes. This representation naturally lends itself to numerical discretizations, which are shown in this paper to have strong convergence rates of arbitrarily high polynomial order. This explains the potential, but also some limitations of such representations as the basis of Monte Carlo schemes for fractional volatility models such as the rough Bergomi model.

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Mathematics Subject Classification: Primary: 60G22; Secondary: 60G15, 65C05, 91G60.

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Figure 1. Volterra Brownian motion of Hurst index $ H\in(0,1/2) $ can be represented as an integral $ W^H_t = \int_0^\infty Y_t(x) x^{-1/2-H}dx $ over a Gaussian random field $ Y_t(x) $. The smoothness of the random field in the spatial dimension $ x $ allows one to approximate this integral efficiently using high order quadrature rules

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Figure 2. Dependence of the approximations on the number $ n $ of quadrature intervals and the Hurst index $ H $. Left: varying the number $ n\in\{2,5,10,20,40\}$ = of quadrature intervals with fixed parameters $ H = 0.1 $, $ m = 5 $. Right: varying the Hurst index $ H\in\{0.1,0.2,0.3,0.4\}$ = with fixed parameters $ n = 40 $, $ m = 5 $

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Figure 3. The upper bound $ 2Hm/3 $ on the convergence rate established in Remark 6.2 for $ m $-point interpolatory quadrature closely matches the numerically observed one (here: at $ t = 1 $, computed analytically from the covariance functions of the Gaussian processes $ W^H $ and $ W^{H,n} $). Left: relative error $ e = \|W^H_1-W^{H,n}_1\|_{L^2(\Omega)}/\|W^H_1\|_{L^2(\Omega)} $ for $ m\in\{2,3,\dots,20\}$ = with $ H = 0.1 $. Right: slopes of the lines in the left plot (dots) and predicted convergence rate (line)

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Table 1. Complexity of several numerical methods for sampling a fractional process $ (W^H_{i/k})_{i\in\{1,\dots,k\}} $ with Hurst index $ H\in(0,1/2) $ at $ k $ equidistant time points

Method	Structure	Error	Complexity
Cholesky	Static	0	$ k^3 $
Hosking, Dieker [28,17]	Recursive	0	$ k^2 $
Dietrich, Newsam [18]	Static	0	$ k\log k $
Bennedsen, Lunde, Pakkanen [12]	Recursive	$ k^{-H} $	$ k \log k $
Carmona, Coutin, Montseny [15]	Recursive	$ \epsilon $	$ k\epsilon^{-3/(4H)} $
This paper	Recursive	$ \epsilon $	$ k\epsilon^{-1/r} $ for $ r\in(0,\infty) $

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