Strong convergence rates for markovian representations of fractional processes

Philipp Harms

doi:10.3934/dcdsb.2020367

Previous Article
Limit cycle bifurcations of piecewise smooth near-Hamiltonian systems with a switching curve
DCDS-B Home
This Issue
Next Article
The effect of surface pattern property on the advancing motion of three-dimensional droplets

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 and 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.
[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.
[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.
[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.
[5]	C. D. Aliprantis and K. C. Border, Infinite Dimensional Analysis. A Hitchhiker's Guide, 3rd edition
[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.
[7]	C. Bayer, P. Friz and J. Gatheral, Pricing under rough volatility, Quantitative Finance, 16 (2016), 887-904. doi: 10.1080/14697688.2015.1099717.
[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.
[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.
[10]	M. Beiglböck and P. Siorpaes, Pathwise versions of the Burkholder–Davis–Gundy inequality, Bernoulli, 21 (2015), 360-373. doi: 10.3150/13-BEJ570.
[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.
[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.
[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.
[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.
[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.
[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.
[17]	T. Dieker, Simulation of Fractional Brownian Motion, Master's thesis, University of Twente, 2004.
[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.
[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.
[20]	M. Fukasawa, Asymptotic analysis for stochastic volatility: Martingale expansion, Finance and Stochastics, 15 (2011), 635-654. doi: 10.1007/s00780-010-0136-6.
[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.
[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.
[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.
[24]	J. Gatheral, T. Jaisson and M. Rosenbaum, Volatility is rough, Quantitative Finance, 18 (2018), 933-949. doi: 10.1080/14697688.2017.1393551.
[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.
[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.
[27]	B. Horvath, A. Jacquier and A. Muguruza, Functional central limit theorems for rough volatility, arXiv: 1711.03078, 2017. doi: 10.2139/ssrn.3078743.
[28]	J. R. Hosking, Modeling persistence in hydrological time series using fractional differencing, Water Resources Research, 20 (1984), 1898-1908. doi: 10.1029/WR020i012p01898.
[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.
[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.
[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.
[32]	L. Mytnik and T. S. Salisbury, Uniqueness for Volterra-type stochastic integral equations, arXiv: 1502.05513, 2015.
[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.
[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.

show all references

References:

[1]	E. Abi Jaber, Lifting the Heston model, Quantitative Finance, 19 (2019), 1995-2013. doi: 10.1080/14697688.2019.1615113.
[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.
[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.
[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.
[5]	C. D. Aliprantis and K. C. Border, Infinite Dimensional Analysis. A Hitchhiker's Guide, 3rd edition
[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.
[7]	C. Bayer, P. Friz and J. Gatheral, Pricing under rough volatility, Quantitative Finance, 16 (2016), 887-904. doi: 10.1080/14697688.2015.1099717.
[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.
[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.
[10]	M. Beiglböck and P. Siorpaes, Pathwise versions of the Burkholder–Davis–Gundy inequality, Bernoulli, 21 (2015), 360-373. doi: 10.3150/13-BEJ570.
[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.
[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.
[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.
[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.
[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.
[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.
[17]	T. Dieker, Simulation of Fractional Brownian Motion, Master's thesis, University of Twente, 2004.
[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.
[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.
[20]	M. Fukasawa, Asymptotic analysis for stochastic volatility: Martingale expansion, Finance and Stochastics, 15 (2011), 635-654. doi: 10.1007/s00780-010-0136-6.
[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.
[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.
[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.
[24]	J. Gatheral, T. Jaisson and M. Rosenbaum, Volatility is rough, Quantitative Finance, 18 (2018), 933-949. doi: 10.1080/14697688.2017.1393551.
[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.
[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.
[27]	B. Horvath, A. Jacquier and A. Muguruza, Functional central limit theorems for rough volatility, arXiv: 1711.03078, 2017. doi: 10.2139/ssrn.3078743.
[28]	J. R. Hosking, Modeling persistence in hydrological time series using fractional differencing, Water Resources Research, 20 (1984), 1898-1908. doi: 10.1029/WR020i012p01898.
[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.
[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.
[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.
[32]	L. Mytnik and T. S. Salisbury, Uniqueness for Volterra-type stochastic integral equations, arXiv: 1502.05513, 2015.
[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.
[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.

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

Figure Options

Download full-size image

Download as PowerPoint slide

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 $

Figure Options

Download full-size image

Download as PowerPoint slide

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)

Figure Options

Download full-size image

Download as PowerPoint slide

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) $

Table Options

Download as excel

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) $

[1]	Ajay Jasra, Kody J. H. Law, Yaxian Xu. Markov chain simulation for multilevel Monte Carlo. Foundations of Data Science, 2021, 3 (1) : 27-47. doi: 10.3934/fods.2021004
[2]	Michael B. Giles, Kristian Debrabant, Andreas Rössler. Analysis of multilevel Monte Carlo path simulation using the Milstein discretisation. Discrete and Continuous Dynamical Systems - B, 2019, 24 (8) : 3881-3903. doi: 10.3934/dcdsb.2018335
[3]	Jiakou Wang, Margaret J. Slattery, Meghan Henty Hoskins, Shile Liang, Cheng Dong, Qiang Du. Monte carlo simulation of heterotypic cell aggregation in nonlinear shear flow. Mathematical Biosciences & Engineering, 2006, 3 (4) : 683-696. doi: 10.3934/mbe.2006.3.683
[4]	James Broda, Alexander Grigo, Nikola P. Petrov. Convergence rates for semistochastic processes. Discrete and Continuous Dynamical Systems - B, 2019, 24 (1) : 109-125. doi: 10.3934/dcdsb.2019001
[5]	Zhiyan Ding, Qin Li. Constrained Ensemble Langevin Monte Carlo. Foundations of Data Science, 2022, 4 (1) : 37-70. doi: 10.3934/fods.2021034
[6]	Rolf Rannacher. A short course on numerical simulation of viscous flow: Discretization, optimization and stability analysis. Discrete and Continuous Dynamical Systems - S, 2012, 5 (6) : 1147-1194. doi: 10.3934/dcdss.2012.5.1147
[7]	Giacomo Dimarco. The moment guided Monte Carlo method for the Boltzmann equation. Kinetic and Related Models, 2013, 6 (2) : 291-315. doi: 10.3934/krm.2013.6.291
[8]	Guillaume Bal, Ian Langmore, Youssef Marzouk. Bayesian inverse problems with Monte Carlo forward models. Inverse Problems and Imaging, 2013, 7 (1) : 81-105. doi: 10.3934/ipi.2013.7.81
[9]	Theodore Papamarkou, Alexey Lindo, Eric B. Ford. Geometric adaptive Monte Carlo in random environment. Foundations of Data Science, 2021, 3 (2) : 201-224. doi: 10.3934/fods.2021014
[10]	Nicolas Vauchelet. Numerical simulation of a kinetic model for chemotaxis. Kinetic and Related Models, 2010, 3 (3) : 501-528. doi: 10.3934/krm.2010.3.501
[11]	Omid Nikan, Seyedeh Mahboubeh Molavi-Arabshai, Hossein Jafari. Numerical simulation of the nonlinear fractional regularized long-wave model arising in ion acoustic plasma waves. Discrete and Continuous Dynamical Systems - S, 2021, 14 (10) : 3685-3701. doi: 10.3934/dcdss.2020466
[12]	A. Settati, A. Lahrouz, Mohamed El Fatini, A. El Haitami, M. El Jarroudi, M. Erriani. A Markovian switching diffusion for an SIS model incorporating Lévy processes. Discrete and Continuous Dynamical Systems - B, 2022 doi: 10.3934/dcdsb.2022072
[13]	Ana I. Muñoz, José Ignacio Tello. Mathematical analysis and numerical simulation of a model of morphogenesis. Mathematical Biosciences & Engineering, 2011, 8 (4) : 1035-1059. doi: 10.3934/mbe.2011.8.1035
[14]	Kolade M. Owolabi, Edson Pindza. Numerical simulation of multidimensional nonlinear fractional Ginzburg-Landau equations. Discrete and Continuous Dynamical Systems - S, 2020, 13 (3) : 835-851. doi: 10.3934/dcdss.2020048
[15]	Qun Liu, Daqing Jiang, Tasawar Hayat, Ahmed Alsaedi. Dynamical behavior of a multigroup SIRS epidemic model with standard incidence rates and Markovian switching. Discrete and Continuous Dynamical Systems, 2019, 39 (10) : 5683-5706. doi: 10.3934/dcds.2019249
[16]	Eleonora Messina. Numerical simulation of a SIS epidemic model based on a nonlinear Volterra integral equation. Conference Publications, 2015, 2015 (special) : 826-834. doi: 10.3934/proc.2015.0826
[17]	Alberto Zingaro, Ivan Fumagalli, Luca Dede, Marco Fedele, Pasquale C. Africa, Antonio F. Corno, Alfio Quarteroni. A geometric multiscale model for the numerical simulation of blood flow in the human left heart. Discrete and Continuous Dynamical Systems - S, 2022, 15 (8) : 2391-2427. doi: 10.3934/dcdss.2022052
[18]	Chjan C. Lim, Joseph Nebus, Syed M. Assad. Monte-Carlo and polyhedron-based simulations I: extremal states of the logarithmic N-body problem on a sphere. Discrete and Continuous Dynamical Systems - B, 2003, 3 (3) : 313-342. doi: 10.3934/dcdsb.2003.3.313
[19]	Joseph Nebus. The Dirichlet quotient of point vortex interactions on the surface of the sphere examined by Monte Carlo experiments. Discrete and Continuous Dynamical Systems - B, 2005, 5 (1) : 125-136. doi: 10.3934/dcdsb.2005.5.125
[20]	Olli-Pekka Tossavainen, Daniel B. Work. Markov Chain Monte Carlo based inverse modeling of traffic flows using GPS data. Networks and Heterogeneous Media, 2013, 8 (3) : 803-824. doi: 10.3934/nhm.2013.8.803

2021 Impact Factor: 1.497

Tools

Metrics

Tools

Article outline

Figures and Tables

2021 Impact Factor: 1.497

Tools

Figures and Tables

2021 Impact Factor: 1.497

American Institute of Mathematical Sciences

Strong convergence rates for markovian representations of fractional processes

References:

References:

Tools

Metrics

Other articles
by authors

Tools

Tools

Tools

Metrics

Other articles
by authors

American Institute of Mathematical Sciences

Strong convergence rates for markovian representations of fractional processes

References:

References:

Tools

Metrics

Other articlesby authors

Tools

Tools

Tools

Metrics

Other articlesby authors

Other articles
by authors

Other articles
by authors