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October 2021, 26(10): 5567-5579. doi: 10.3934/dcdsb.2020367
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 Published October 2021 Early access December 2020
Fund Project: The author gratefully acknowledges support in the form of a Junior Fellowship of the Freiburg Institute of Advances Studies
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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.
Keywords: Fractional processes, Markovian representation, Monte Carlo simulation, numerical discretization, strong convergence rates, rough Bergomi model.
Mathematics Subject Classification: Primary: 60G22; Secondary: 60G15, 65C05, 91G60.
Citation: Philipp Harms. Strong convergence rates for markovian representations of fractional processes. Discrete & Continuous Dynamical Systems - B, 2021, 26 (10) : 5567-5579. doi: 10.3934/dcdsb.2020367
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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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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 $">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 $
with $ H = 0.1 $. Right: slopes of the lines in the left plot (dots) and predicted convergence rate (line)">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)
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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