Abstract
We adopt the multilevel Monte Carlo method introduced by M. Giles (Multilevel Monte Carlo path simulation, Oper. Res. 56(3):607–617, 2008) to SDEs with additive fractional noise of Hurst parameter H>1/2. For the approximation of a Lipschitz functional of the terminal state of the SDE we construct a multilevel estimator based on the Euler scheme. This estimator achieves a prescribed root mean square error of order ε with a computational effort of order ε −2.
Similar content being viewed by others
References
Abramowitz, M., & Stegun, I. A. (1964). Handbook of mathematical functions with formulas, graphs and mathematical tables. U.S. Department of Commerce: Washington.
Baudoin, F., & Coutin, L. (2007). Operators associated with a stochastic differential equation driven by fractional Brownian motions. Stochastic Processes and Their Applications, 117, 550–574.
Benth, F. E. (2003). On arbitrage-free pricing of weather derivatives based on fractional Brownian motion. Applied Mathematical Finance, 10, 303–324.
Brody, D., Syroka, J., & Zervos, M. (2002). Dynamical pricing of weather derivatives. Quantitative Finance, 2, 189–198.
Coeurjolly, J. F. (2000). Simulation and identification of the fractional Brownian motion: a bibliographical and comparative study. Journal of Statistical Software, 5, 1–53.
Craigmile, P. F. (2003). Simulating a class of stationary Gaussian processes using the Davies-Harte algorithm, with application to long memory processes. Journal of Time Series Analysis, 24, 505–511.
Denk, G., Meintrup, D., & Schäffler, S. (2001). Transient noise simulation: modeling and simulation of 1/f-noise. In K. Antreich et al. (Eds.), Int. ser. numer. math. : Vol. 146. Modeling, simulation, and optimization of integrated circuits (pp. 251–267). Basel: Birkhäuser.
Denk, G., & Winkler, R. (2007). Modelling and simulation of transient noise in circuit simulation. Mathematical and Computer Modelling of Dynamical Systems, 13(4), 383–394.
Duffie, D., & Glynn, P. (1995). Efficient Monte Carlo simulation of security prices. Annals of Applied Probability, 5(4), 897–905.
Garrido Atienza, M. J., Kloeden, P. E., & Neuenkirch, A. (2009). Discretization of the attractor of a system driven by fractional Brownian motion. Applied Mathematics & Optimization, 60(2), 151–172.
Giles, M. (2008). Multilevel Monte Carlo path simulation. Operations Research, 56(3), 607–617.
Giles, M. (2007). Improved multilevel Monte Carlo convergence using the Milstein scheme. In A. Keller et al. (Eds.), Monte Carlo and quasi-Monte Carlo methods 2006. Proceedings (pp. 343–354). Berlin: Springer.
Gradinaru, M., & Nourdin, I. (2009, to appear). Milstein’s type scheme for fractional SDEs. Annales de l’Institut Henri Poincaré (B): Probability and Statistics.
Guasoni, P. (2006). No arbitrage with transaction costs, with fractional Brownian motion and beyond. Mathematical Finance, 16, 569–582.
Imkeller, P. (2008). IMPAN lecture notes : Vol. 1. Malliavin’s calculus and applications in stochastic control and finance. Warsaw: Polish Academy of Science, Institute of Mathematics.
Jolis, M. (2007). On the Wiener integral with respect to the fractional Brownian motion on an interval. Journal of Mathematical Analysis and Applications, 330, 1115–1127.
Kloeden, P. E., & Platen, E. (1999). Numerical solution of stochastic differential equations, 3rd ed. Berlin: Springer.
Kou, S. C. (2008). Stochastic modeling in nanoscale biophysics: subdiffusion within proteins. Annals of Applied Statistics, 2(2), 501–535.
Milstein, G. N. (1995). Numerical integration of stochastic differential equations. Dordrecht: Kluwer Academic.
Mishura, Y., & Shevchenko, G. (2008). The rate of convergence for Euler approximations of solutions of stochastic differential equations driven by fractional Brownian motion. Stochastics, 80(5), 489–511.
Neuenkirch, A. (2006). Optimal approximation of SDE’s with additive fractional noise. Journal of Complexity, 22, 459–474.
Neuenkirch, A. (2008). Optimal pointwise approximation of stochastic differential equations driven by fractional Brownian motion. Stochastic Processes and Their Applications, 118(12), 2294–2333.
Neuenkirch, A., & Nourdin, I. (2007). Exact rate of convergence of some approximation schemes associated to SDEs driven by a fBm. Journal of Theoretical Probability, 20(4), 871–899.
Nourdin, I. (2005). Schémas d’approximation associés à une équation différentielle dirigée par une fonction höldérienne; cas du mouvement Brownien fractionnaire. Comptes Rendus de l’Académie des Sciences. Série I. Mathématique, 340(8), 611–614.
Nualart, D. (2006). The Malliavin calculus and related topics, 2nd ed. Berlin: Springer.
Nualart, D., & Saussereau, B. (2009). Malliavin calculus for stochastic differential equations driven by fractional Brownian motion. Stochastic Processes and Their Applications, 119(2), 391–409.
Young, L. C. (1936). An inequality of Hölder type connected with Stieltjes integration. Acta Mathematica, 67, 251–282.
Zähle, M. (2005). Stochastic differential equations with fractal noise. Mathematische Nachrichten, 278(9), 1097–1106.
Author information
Authors and Affiliations
Corresponding author
Additional information
Partially supported by the DFG project “Pathwise numerical analysis of stochastic evolution equations”.
Rights and permissions
About this article
Cite this article
Kloeden, P.E., Neuenkirch, A. & Pavani, R. Multilevel Monte Carlo for stochastic differential equations with additive fractional noise. Ann Oper Res 189, 255–276 (2011). https://doi.org/10.1007/s10479-009-0663-8
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10479-009-0663-8