Top

Published in:

2024 | OriginalPaper | Chapter

The Order Barrier for the \(L^1\)-approximation of the Log-Heston SDE at a Single Point

Authors : Annalena Mickel, Andreas Neuenkirch

Published in: Monte Carlo and Quasi-Monte Carlo Methods

Publisher: Springer International Publishing

Activate our intelligent search to find suitable subject content or patents.

search-config

AI-assisted search

Off

Abstract

We study the \(L^1\)-approximation of the log-Heston SDE at the terminal time point by arbitrary methods that use an equidistant discretization of the driving Brownian motion. We show that such methods can achieve at most order \( \min \{ \nu , \tfrac{1}{2} \}\), where \(\nu \) is the Feller index of the underlying CIR process. As a consequence Euler-type schemes are optimal for \(\nu \ge 1\), since they have convergence order \(\tfrac{1}{2}-\epsilon \) for \(\epsilon >0\) arbitrarily small in this regime.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

über 102.000 Bücher
über 537 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Finance + Banking
Management + Führung
Marketing + Vertrieb
Maschinenbau + Werkstoffe
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 390 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Maschinenbau + Werkstoffe

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 340 Zeitschriften

aus folgenden Fachgebieten:

Bauwesen + Immobilien
Business IT + Informatik
Finance + Banking
Management + Führung
Marketing + Vertrieb
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

previous chapter Large Scale Gaussian Processes with Matheron’s Update Rule and Karhunen-Loève Expansion

next chapter A Randomised Lattice Rule Algorithm with Pre-determined Generating Vector and Random Number of Points for Korobov Spaces with

Alfonsi, A.: Strong order one convergence of a drift implicit Euler scheme: application to the CIR process. Stat. Probab. Lett. 83, 602–607 (2013)MathSciNetCrossRef

Cambanis, S., Hu, Y.: Exact convergence rate of the Euler-Maruyama scheme, with application to sampling design. Stochastics Rep. 59, 211–240 (1996)MathSciNetCrossRef

Castell, F., Gaines, J.: The ordinary differential equation approach to asymptotically efficient schemes for solution of stochastic differential equations. Ann. Inst. Henri Poincaré, Probab. Stat. 32, 231–250 (1996)

Clark, J., Cameron, R.: The maximum rate of convergence of discrete approximations for stochastic differential equations. In: Grigelionis, B. (eds.), Stochastic Differential Systems (Proc. IFIP-WG 7/1 Working Conf., Vilnius, 1978) Lecture Notes in Control and Information Sci. vol. 25, pp. 162–171. Springer, Berlin, Heidelberg (1980)

Dereich, S., Neuenkirch, A., Szpruch, L.: An Euler-type method for the strong approximation of the Cox-Ingersoll-Ross process. Proc. R. Soc. A. 468, 1105–1115 (2012)MathSciNetCrossRef

Feller, W.: Two singular diffusion problems. Ann. Math. 54, 173–182 (1951)MathSciNetCrossRef

Hefter, M., Herzwurm, A.: Strong convergence rates for Cox-Ingersoll-Ross processes—full parameter range. J. Math. Anal. Appl. 459, 1079–1101 (2016)MathSciNetCrossRef

Hefter, M., Herzwurm, A.: Optimal strong approximation of the one-dimensional squared Bessel process. Commun. Math. Sci. 15, 2121–2141 (2017)MathSciNetCrossRef

Hefter, M., Herzwurm, A., Müller-Gronbach, T.: Lower error bounds for strong approximation of scalar SDEs with non-Lipschitzian coefficients. Ann. Appl. Probab. 29, 178–216 (2019)MathSciNetCrossRef

10.

Hefter, M., Jentzen, A.: On arbitrarily slow convergence rates for strong numerical approximations of Cox-Ingersoll-Ross processes and squared Bessel processes. Finance Stoch. 23, 139–172 (2019)MathSciNetCrossRef

11.

Hofmann, N., Müller-Gronbach, T., Ritter, K.: Optimal approximation of stochastic differential equations by adaptive step-size control. Math. Comput. 69, 1017–1034 (2000)MathSciNetCrossRef

12.

Hofmann, N., Müller-Gronbach, T., Ritter, K.: Step size control for the uniform approximation of systems of stochastic differential equations with additive noise. Ann. Appl. Probab. 10, 616–633 (2000)MathSciNetCrossRef

13.

Hofmann, N., Müller-Gronbach, T., Ritter, K.: The optimal discretization of stochastic differential equations. J. Compl. 17, 117–153 (2001)MathSciNetCrossRef

14.

Hofmann, N., Müller-Gronbach, T., Ritter, K.: Linear vs standard information for scalar stochastic differential equations. J. Compl. 18, 394–414 (2002)MathSciNetCrossRef

15.

Hurd, T.R., Kuznetsov, A.: Explicit formulas for Laplace transforms of stochastic integrals. Markov Process. Relat. Fields 14, 277–290 (2008)MathSciNet

16.

Hutzenthaler, M., Jentzen, A., Noll, M.: Strong convergence rates and temporal regularity for Cox-Ingersoll-Ross processes and Bessel processes with accessible boundaries (2014). https://doi.org/10.48550/arXiv.1403.6385

17.

Jentzen, A., Müller-Gronbach, T., Yaroslavtseva, L.: On stochastic differential equations with arbitrary slow convergence rates for strong approximation. Commun. Math. Sci. 14, 1477–1500 (2016)MathSciNetCrossRef

18.

Müller-Gronbach, T., Ritter, K.: Minimal errors for strong and weak approximation of stochastic differential equations. In: Keller, A., Heinrich, S., Niederreiter, H. (eds.) Monte Carlo and Quasi-Monte Carlo Methods 2006, pp. 53–82. Springer, Berlin (2008)CrossRef

19.

Mickel, A.: Weak and strong approximation of the log-Heston Model by Euler-type methods and related topics. Dissertation, University of Mannheim (2023)

20.

Mickel, A., Neuenkirch, A.: Sharp \({L}^1\)-approximation of the log-Heston SDE by Euler-type methods. J. Comput. Fin. 26, 67–100 (2023)

21.

Müller-Gronbach, T.: The optimal uniform approximation of systems of stochastic differential equations. Ann. Appl. Probab. 12, 664–690 (2002)MathSciNetCrossRef

22.

Müller-Gronbach, T.: Strong approximation of systems of stochastic differential equations. Habilitation thesis, Technical University of Darmstadt (2002)

23.

Müller-Gronbach, T.: Optimal pointwise approximation of SDEs based on Brownian motion at discrete points. Ann. Appl. Probab. 14, 1605–1642 (2004)MathSciNetCrossRef

24.

Müller-Gronbach, T., Yaroslavtseva, L.: A note on strong approximation of SDEs with smooth coefficients that have at most linearly growing derivatives. J. Math. Anal. Appl. 467, 1013–1031 (2018)MathSciNetCrossRef

25.

Newton, N.J.: An asymptotically efficient difference formula for solving stochastic differential equations. Stochastics 19, 175–206 (1986)MathSciNetCrossRef

26.

Newton, N.J.: An efficient approximation for stochastic differential equations on the partition of symmetrical first passage times. Stochastics Stochastics Rep. 29, 227–258 (1990)MathSciNetCrossRef

27.

Newton, N.J.: Asymptotically efficient Runge-Kutta methods for a class of Itô and Stratonovich equations. SIAM J. Appl. Math. 51, 542–567 (1991)MathSciNetCrossRef

28.

Traub, J., Wasilkowski, G., Woźniakowski, H.: Information-Based Complexity. Academic Press Inc, Boston, MA (1988)

29.

Yaroslavtseva, L.: On non-polynomial lower error bounds for adaptive strong approximation of SDEs. J. Compl. 42, 1–18 (2017)MathSciNetCrossRef

Title: The Order Barrier for the -approximation of the Log-Heston SDE at a Single Point
Authors: Annalena Mickel
Andreas Neuenkirch
Publisher: Springer International Publishing
Book: Monte Carlo and Quasi-Monte Carlo Methods
Print ISBN: 978-3-031-59761-9

Electronic ISBN: 978-3-031-59762-6

Copyright Year: 2024
DOI: https://doi.org/10.1007/978-3-031-59762-6_24

Springer Professional

Abstract

Please log in to get access to your license.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Springer Professional "Technik"

Springer Professional "Wirtschaft"