Regression-Based Variance Reduction Approach for Strong Approximation Schemes

  • Denis BelomestnyEmail author
  • Stefan Häfner
  • Mikhail Urusov
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 208)


In this paper, we present a novel approach towards variance reduction for discretised diffusion processes. The proposed approach involves specially constructed control variates and allows for a significant reduction in the variance for the terminal functionals. In this way, the complexity order of the standard Monte Carlo algorithm (\(\varepsilon ^{-3}\)) can be reduced down to \(\varepsilon ^{-2}\sqrt{\left| \log (\varepsilon )\right| }\) in case of the Euler scheme with \(\varepsilon \) being the precision to be achieved. These theoretical results are illustrated by several numerical examples.


Monte Carlo methods Regression methods Control variates Stochastic differential equations Strong schemes 



Stefan Häfner thanks the Faculty of Mathematics of the University of Duisburg-Essen, where this work was carried out.


  1. 1.
    Akahori, J., Amaba, T., Okuma, K.: A discrete-time Clark-Ocone formula and its application to an error analysis (2013). arXiv:1307.0673v2
  2. 2.
    Belomestny, D., Häfner, S., Nagapetyan, T., Urusov, M.: Variance reduction for discretised diffusions via regression (2016). arXiv:1510.03141v3
  3. 3.
    Giles, M.B.: Multilevel Monte Carlo path simulation. Op. Res. 56(3), 607–617 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Györfi, L., Kohler, M., Krzyżak, A., Walk, H.: A distribution-free theory of nonparametric regression. Springer Series in Statistics. Springer, New York (2002)Google Scholar
  5. 5.
    Karatzas, I., Shreve, S.: Brownian Motion and Stochastic Calculus, vol. 113. Springer Science & Business Media, Berlin (2012)Google Scholar
  6. 6.
    Kloeden, P., Platen, E.: Numerical Solution of Stochastic Differential Equations, vol. 23. Springer, Berlin (1992)Google Scholar
  7. 7.
    Milstein, G.N., Tretyakov, M.V.: Stochastic numerics for mathematical physics. Scientific Computation. Springer, Berlin (2004)Google Scholar
  8. 8.
    Milstein, G.N., Tretyakov, M.V.: Practical variance reduction via regression for simulating diffusions. SIAM J. Numer. Anal. 47(2), 887–910 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Müller-Gronbach, T., Ritter, K., Yaroslavtseva, L.: On the complexity of computing quadrature formulas for marginal distributions of SDEs. J. Complex. 31(1), 110–145 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Newton, N.J.: Variance reduction for simulated diffusions. SIAM J. Appl. Math. 54(6), 1780–1805 (1994)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Denis Belomestny
    • 1
    • 2
    Email author
  • Stefan Häfner
    • 3
  • Mikhail Urusov
    • 1
  1. 1.Duisburg-Essen UniversityEssenGermany
  2. 2.National Research University Higher School of EconomicsMoscowRussia
  3. 3.PricewaterhouseCoopers GmbHFrankfurtGermany

Personalised recommendations