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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

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
References:
[1]

E. Abi Jaber, Lifting the Heston model, Quantitative Finance, 19 (2019), 1995-2013.  doi: 10.1080/14697688.2019.1615113.  Google Scholar

[2]

E. Abi Jaber and O. El Euch, Markovian structure of the Volterra Heston model, Statistics & Probability Letters, 149 (2019), 63-72.  doi: 10.1016/j.spl.2019.01.024.  Google Scholar

[3]

E. Abi Jaber and O. El Euch, Multifactor approximation of rough volatility models, SIAM Journal on Financial Mathematics, 10 (2019), 309-349.  doi: 10.1137/18M1170236.  Google Scholar

[4]

E. Abi Jaber, M. Larsson and S. Pulido, Affine Volterra processes, The Annals of Applied Probability, 29 (2019), 3155-3200.  doi: 10.1214/19-AAP1477.  Google Scholar

[5]

C. D. Aliprantis and K. C. Border, Infinite Dimensional Analysis. A Hitchhiker's Guide, 3rd edition  Google Scholar

[6]

E. Alòs, J. A. León and J. Vives, On the short-time behavior of the implied volatility for jump-diffusion models with stochastic volatility, Finance and Stochastics, 11 (2007), 571-589.  doi: 10.1007/s00780-007-0049-1.  Google Scholar

[7]

C. Bayer, P. Friz and J. Gatheral, Pricing under rough volatility, Quantitative Finance, 16 (2016), 887-904.  doi: 10.1080/14697688.2015.1099717.  Google Scholar

[8]

C. Bayer, P. K. Friz, P. Gassiat, J. Martin and B. Stemper, A regularity structure for rough volatility, Mathematical Finance, 30 (2020), 782-832.  doi: 10.1111/mafi.12233.  Google Scholar

[9]

C. Bayer, P. K. Friz, A. Gulisashvili, B. Horvath and B. Stemper, Short-time near-the-money skew in rough fractional volatility models, Quantitative Finance, 19 (2019), 779-798.  doi: 10.1080/14697688.2018.1529420.  Google Scholar

[10]

M. Beiglböck and P. Siorpaes, Pathwise versions of the Burkholder–Davis–Gundy inequality, Bernoulli, 21 (2015), 360-373.  doi: 10.3150/13-BEJ570.  Google Scholar

[11]

M. Bennedsen, A. Lunde and M. S. Pakkanen, Decoupling the short-and long-term behavior of stochastic volatility, arXiv: 1610.00332, 2016. doi: 10.2139/ssrn.2846756.  Google Scholar

[12]

M. Bennedsen, A. Lunde and M. S. Pakkanen, Hybrid scheme for Brownian semistationary processes, Finance and Stochastics, 21 (2017), 931-965.  doi: 10.1007/s00780-017-0335-5.  Google Scholar

[13]

H. Brass and K. Petras, Quadrature Theory, vol. 178 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 2011. doi: 10.1090/surv/178.  Google Scholar

[14]

P. Carmona and L. Coutin, Fractional Brownian motion and the Markov property, Electronic Communications in Probability, 3 (1998), 95-107.  doi: 10.1214/ECP.v3-998.  Google Scholar

[15]

P. Carmona, L. Coutin and G. Montseny, Approximation of some Gaussian processes, Statistical Inference for Stochastic Processes, 3 (2000), 161-171.  doi: 10.1023/A:1009999518898.  Google Scholar

[16]

C. Cuchiero and J. Teichmann, Generalized Feller processes and Markovian lifts of stochastic Volterra processes: The affine case, Journal of Evolution Equations, (2020), 1–48. doi: 10.1007/s00028-020-00557-2.  Google Scholar

[17]

T. Dieker, Simulation of Fractional Brownian Motion, Master's thesis, University of Twente, 2004. Google Scholar

[18]

C. R. Dietrich and G. N. Newsam, Fast and exact simulation of stationary Gaussian processes through circulant embedding of the covariance matrix, SIAM Journal on Scientific Computing, 18 (1997), 1088-1107.  doi: 10.1137/S1064827592240555.  Google Scholar

[19]

M. Forde and H. Zhang, Asymptotics for rough stochastic volatility models, SIAM Journal on Financial Mathematics, 8 (2017), 114-145.  doi: 10.1137/15M1009330.  Google Scholar

[20]

M. Fukasawa, Asymptotic analysis for stochastic volatility: Martingale expansion, Finance and Stochastics, 15 (2011), 635-654.  doi: 10.1007/s00780-010-0136-6.  Google Scholar

[21]

M. Gaß, K. Glau, M. Mahlstedt and M. Mair, Chebyshev interpolation for parametric option pricing, Finance and Stochastics, 22 (2018), 701-731.  doi: 10.1007/s00780-018-0361-y.  Google Scholar

[22]

M. Gaß, K. Glau and M. Mair, Magic points in finance: Empirical integration for parametric option pricing, SIAM Journal on Financial Mathematics, 8 (2017), 766-803.  doi: 10.1137/16M1101301.  Google Scholar

[23]

P. Gassiat, On the martingale property in the rough Bergomi model, Electronic Communications in Probability, 24 (2019), Paper No. 33, 9 pp. doi: 10.1214/19-ECP239.  Google Scholar

[24]

J. Gatheral, T. Jaisson and M. Rosenbaum, Volatility is rough, Quantitative Finance, 18 (2018), 933-949.  doi: 10.1080/14697688.2017.1393551.  Google Scholar

[25]

S. E. Graversen and G. Peskir, Maximal inequalities for the Ornstein–Uhlenbeck process, Proceedings of the American Mathematical Society, 128 (2000), 3035-3041.  doi: 10.1090/S0002-9939-00-05345-4.  Google Scholar

[26]

P. Harms and D. Stefanovits, Affine representations of fractional processes with applications in mathematical finance, Stochastic Processes and their Applications, 129 (2019), 1185-1228.  doi: 10.1016/j.spa.2018.04.010.  Google Scholar

[27]

B. Horvath, A. Jacquier and A. Muguruza, Functional central limit theorems for rough volatility, arXiv: 1711.03078, 2017. doi: 10.2139/ssrn.3078743.  Google Scholar

[28]

J. R. Hosking, Modeling persistence in hydrological time series using fractional differencing, Water Resources Research, 20 (1984), 1898-1908.  doi: 10.1029/WR020i012p01898.  Google Scholar

[29]

T. Hytönen, J. van Neerven, M. Veraar and L. Weis, Analysis in Banach Spaces, vol. 67, Springer, Cham, 2017. doi: 10.1007/978-3-319-69808-3.  Google Scholar

[30]

R. McCrickerd and M. S. Pakkanen, Turbocharging Monte Carlo pricing for the rough Bergomi model, Quantitative Finance, 18 (2018), 1877–1886. doi: 10.1080/14697688.2018.1459812.  Google Scholar

[31]

A. A. Muravlev, Representation of a fractional Brownian motion in terms of an infinite-dimensional Ornstein–Uhlenbeck process, Russian Mathematical Surveys, 66 (2011), 439-441.  doi: 10.1070/RM2011v066n02ABEH004746.  Google Scholar

[32]

L. Mytnik and T. S. Salisbury, Uniqueness for Volterra-type stochastic integral equations, arXiv: 1502.05513, 2015. Google Scholar

[33]

N. N. Vakhania, V. I. Tarieladze and S. A.Chobanyan, Probability distributions on Banach spaces, vol. 14, Springer Science & Business Media, 1987. doi: 10.1007/978-94-009-3873-1.  Google Scholar

[34]

M. Veraar, The stochastic Fubini theorem revisited, Stochastics An International Journal of Probability and Stochastic Processes, 84 (2012), 543-551.  doi: 10.1080/17442508.2011.618883.  Google Scholar

show all references

References:
[1]

E. Abi Jaber, Lifting the Heston model, Quantitative Finance, 19 (2019), 1995-2013.  doi: 10.1080/14697688.2019.1615113.  Google Scholar

[2]

E. Abi Jaber and O. El Euch, Markovian structure of the Volterra Heston model, Statistics & Probability Letters, 149 (2019), 63-72.  doi: 10.1016/j.spl.2019.01.024.  Google Scholar

[3]

E. Abi Jaber and O. El Euch, Multifactor approximation of rough volatility models, SIAM Journal on Financial Mathematics, 10 (2019), 309-349.  doi: 10.1137/18M1170236.  Google Scholar

[4]

E. Abi Jaber, M. Larsson and S. Pulido, Affine Volterra processes, The Annals of Applied Probability, 29 (2019), 3155-3200.  doi: 10.1214/19-AAP1477.  Google Scholar

[5]

C. D. Aliprantis and K. C. Border, Infinite Dimensional Analysis. A Hitchhiker's Guide, 3rd edition  Google Scholar

[6]

E. Alòs, J. A. León and J. Vives, On the short-time behavior of the implied volatility for jump-diffusion models with stochastic volatility, Finance and Stochastics, 11 (2007), 571-589.  doi: 10.1007/s00780-007-0049-1.  Google Scholar

[7]

C. Bayer, P. Friz and J. Gatheral, Pricing under rough volatility, Quantitative Finance, 16 (2016), 887-904.  doi: 10.1080/14697688.2015.1099717.  Google Scholar

[8]

C. Bayer, P. K. Friz, P. Gassiat, J. Martin and B. Stemper, A regularity structure for rough volatility, Mathematical Finance, 30 (2020), 782-832.  doi: 10.1111/mafi.12233.  Google Scholar

[9]

C. Bayer, P. K. Friz, A. Gulisashvili, B. Horvath and B. Stemper, Short-time near-the-money skew in rough fractional volatility models, Quantitative Finance, 19 (2019), 779-798.  doi: 10.1080/14697688.2018.1529420.  Google Scholar

[10]

M. Beiglböck and P. Siorpaes, Pathwise versions of the Burkholder–Davis–Gundy inequality, Bernoulli, 21 (2015), 360-373.  doi: 10.3150/13-BEJ570.  Google Scholar

[11]

M. Bennedsen, A. Lunde and M. S. Pakkanen, Decoupling the short-and long-term behavior of stochastic volatility, arXiv: 1610.00332, 2016. doi: 10.2139/ssrn.2846756.  Google Scholar

[12]

M. Bennedsen, A. Lunde and M. S. Pakkanen, Hybrid scheme for Brownian semistationary processes, Finance and Stochastics, 21 (2017), 931-965.  doi: 10.1007/s00780-017-0335-5.  Google Scholar

[13]

H. Brass and K. Petras, Quadrature Theory, vol. 178 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 2011. doi: 10.1090/surv/178.  Google Scholar

[14]

P. Carmona and L. Coutin, Fractional Brownian motion and the Markov property, Electronic Communications in Probability, 3 (1998), 95-107.  doi: 10.1214/ECP.v3-998.  Google Scholar

[15]

P. Carmona, L. Coutin and G. Montseny, Approximation of some Gaussian processes, Statistical Inference for Stochastic Processes, 3 (2000), 161-171.  doi: 10.1023/A:1009999518898.  Google Scholar

[16]

C. Cuchiero and J. Teichmann, Generalized Feller processes and Markovian lifts of stochastic Volterra processes: The affine case, Journal of Evolution Equations, (2020), 1–48. doi: 10.1007/s00028-020-00557-2.  Google Scholar

[17]

T. Dieker, Simulation of Fractional Brownian Motion, Master's thesis, University of Twente, 2004. Google Scholar

[18]

C. R. Dietrich and G. N. Newsam, Fast and exact simulation of stationary Gaussian processes through circulant embedding of the covariance matrix, SIAM Journal on Scientific Computing, 18 (1997), 1088-1107.  doi: 10.1137/S1064827592240555.  Google Scholar

[19]

M. Forde and H. Zhang, Asymptotics for rough stochastic volatility models, SIAM Journal on Financial Mathematics, 8 (2017), 114-145.  doi: 10.1137/15M1009330.  Google Scholar

[20]

M. Fukasawa, Asymptotic analysis for stochastic volatility: Martingale expansion, Finance and Stochastics, 15 (2011), 635-654.  doi: 10.1007/s00780-010-0136-6.  Google Scholar

[21]

M. Gaß, K. Glau, M. Mahlstedt and M. Mair, Chebyshev interpolation for parametric option pricing, Finance and Stochastics, 22 (2018), 701-731.  doi: 10.1007/s00780-018-0361-y.  Google Scholar

[22]

M. Gaß, K. Glau and M. Mair, Magic points in finance: Empirical integration for parametric option pricing, SIAM Journal on Financial Mathematics, 8 (2017), 766-803.  doi: 10.1137/16M1101301.  Google Scholar

[23]

P. Gassiat, On the martingale property in the rough Bergomi model, Electronic Communications in Probability, 24 (2019), Paper No. 33, 9 pp. doi: 10.1214/19-ECP239.  Google Scholar

[24]

J. Gatheral, T. Jaisson and M. Rosenbaum, Volatility is rough, Quantitative Finance, 18 (2018), 933-949.  doi: 10.1080/14697688.2017.1393551.  Google Scholar

[25]

S. E. Graversen and G. Peskir, Maximal inequalities for the Ornstein–Uhlenbeck process, Proceedings of the American Mathematical Society, 128 (2000), 3035-3041.  doi: 10.1090/S0002-9939-00-05345-4.  Google Scholar

[26]

P. Harms and D. Stefanovits, Affine representations of fractional processes with applications in mathematical finance, Stochastic Processes and their Applications, 129 (2019), 1185-1228.  doi: 10.1016/j.spa.2018.04.010.  Google Scholar

[27]

B. Horvath, A. Jacquier and A. Muguruza, Functional central limit theorems for rough volatility, arXiv: 1711.03078, 2017. doi: 10.2139/ssrn.3078743.  Google Scholar

[28]

J. R. Hosking, Modeling persistence in hydrological time series using fractional differencing, Water Resources Research, 20 (1984), 1898-1908.  doi: 10.1029/WR020i012p01898.  Google Scholar

[29]

T. Hytönen, J. van Neerven, M. Veraar and L. Weis, Analysis in Banach Spaces, vol. 67, Springer, Cham, 2017. doi: 10.1007/978-3-319-69808-3.  Google Scholar

[30]

R. McCrickerd and M. S. Pakkanen, Turbocharging Monte Carlo pricing for the rough Bergomi model, Quantitative Finance, 18 (2018), 1877–1886. doi: 10.1080/14697688.2018.1459812.  Google Scholar

[31]

A. A. Muravlev, Representation of a fractional Brownian motion in terms of an infinite-dimensional Ornstein–Uhlenbeck process, Russian Mathematical Surveys, 66 (2011), 439-441.  doi: 10.1070/RM2011v066n02ABEH004746.  Google Scholar

[32]

L. Mytnik and T. S. Salisbury, Uniqueness for Volterra-type stochastic integral equations, arXiv: 1502.05513, 2015. Google Scholar

[33]

N. N. Vakhania, V. I. Tarieladze and S. A.Chobanyan, Probability distributions on Banach spaces, vol. 14, Springer Science & Business Media, 1987. doi: 10.1007/978-94-009-3873-1.  Google Scholar

[34]

M. Veraar, The stochastic Fubini theorem revisited, Stochastics An International Journal of Probability and Stochastic Processes, 84 (2012), 543-551.  doi: 10.1080/17442508.2011.618883.  Google Scholar

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 $
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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)
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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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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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