Skip to main content

Multilevel Monte Carlo Implementation for SDEs Driven by Truncated Stable Processes

Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS,volume 163)

Abstract

In this article we present an implementation of a multilevel Monte Carlo scheme for Lévy-driven SDEs introduced and analysed in (Dereich and Li, Multilevel Monte Carlo for Lévy-driven SDEs: central limit theorems for adaptive Euler schemes, Ann. Appl. Probab. 26, No. 1, 136–185, 2016 [12]). The scheme is based on direct simulation of Lévy increments. We give an efficient implementation of the algorithm. In particular, we explain direct simulation techniques for Lévy increments. Further, we optimise over the involved parameters and, in particular, the refinement multiplier. This article complements the theoretical considerations of the above reference. We stress that we focus on the case where the frequency of small jumps is particularly high, meaning that the Blumenthal–Getoor index is larger than one.

Keywords

  • Multilevel Monte Carlo
  • Lévy-driven stochastic differential equation
  • Truncated stable distributions
  • Computation of expectations

This is a preview of subscription content, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (USA)
  • DOI: 10.1007/978-3-319-33507-0_1
  • Chapter length: 25 pages
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
eBook
USD   169.00
Price excludes VAT (USA)
  • ISBN: 978-3-319-33507-0
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
Softcover Book
USD   219.99
Price excludes VAT (USA)
Hardcover Book
USD   219.99
Price excludes VAT (USA)
Fig. 1
Fig. 2
Fig. 3
Fig. 4

References

  1. Applebaum, D.: Lévy processes and stochastic calculus. Cambridge Studies in Advanced Mathematics, vol. 116. Cambridge University Press, Cambridge (2009)

    Google Scholar 

  2. Asmussen, S., Rosiński, J.: Approximations of small jumps of Lévy processes with a view towards simulation. J. Appl. Probab. 38(2), 482–493 (2001)

    MathSciNet  CrossRef  MATH  Google Scholar 

  3. Bally, V., Talay, D.: The law of the Euler scheme for stochastic differential equations. I. Convergence rate of the distribution function. Probab. Theory Relat. Fields 104(1), 43–60 (1996)

    MathSciNet  CrossRef  MATH  Google Scholar 

  4. Becker, M.: Exact simulation of final, minimal and maximal values of Brownian motion and jump-diffusions with applications to option pricing. Comput. Manag. Sci. 7(1), 1–17 (2010)

    MathSciNet  CrossRef  MATH  Google Scholar 

  5. Ben Alaya, M., Kebaier, A.: Central limit theorem for the multilevel Monte Carlo Euler method. Ann. Appl. Probab. 25(1), 211–234 (2015)

    MathSciNet  CrossRef  MATH  Google Scholar 

  6. Bertoin, J.: Lévy Processes. Cambridge University Press, Cambridge (1996)

    Google Scholar 

  7. Bruti-Liberati, N., Nikitopoulos-Sklibosios, C., Platen, E.: First order strong approximations of jump diffusions. Monte Carlo Methods Appl. 12(3–4), 191–209 (2006)

    MathSciNet  CrossRef  MATH  Google Scholar 

  8. Chen, Z.S., Feng, L.M., Lin, X.: Simulating Lévy processes from their characteristic functions and financial applications. ACM Trans. Model. Comput. Simul. 22(3), 14 (2012)

    MathSciNet  CrossRef  Google Scholar 

  9. Dereich, S.: The coding complexity of diffusion processes under supremum norm distortion. Stoch. Process. Appl. 118(6), 917–937 (2008)

    MathSciNet  CrossRef  MATH  Google Scholar 

  10. Dereich, S.: Multilevel Monte Carlo algorithms for Lévy-driven SDEs with Gaussian correction. Ann. Appl. Probab. 21(1), 283–311 (2011)

    MathSciNet  CrossRef  MATH  Google Scholar 

  11. Dereich, S., Heidenreich, F.: A multilevel Monte Carlo algorithm for Lévy-driven stochastic differential equations. Stoch. Process. Appl. 121(7), 1565–1587 (2011)

    MathSciNet  CrossRef  MATH  Google Scholar 

  12. Dereich, S., Li, S.: Multilevel Monte Carlo for Lévy-driven SDEs: central limit theorems for adaptive Euler schemes. Ann. Appl. Probab. 26(1), 136–185 (2016)

    MathSciNet  CrossRef  MATH  Google Scholar 

  13. Dobrushin, R.L.: Prescribing a system of random variables by conditional distributions. Theory Probab. Appl. 15(3), 458–486 (1970)

    MathSciNet  CrossRef  MATH  Google Scholar 

  14. Giles, M.B.: Multilevel Monte Carlo path simulation. Oper. Res. 56(3), 607–617 (2008)

    MathSciNet  CrossRef  MATH  Google Scholar 

  15. Glasserman, P.: Monte Carlo methods in financial engineering. Applications of Mathematics (New York). Stochastic Modelling and Applied Probability, vol. 53. Springer, New York (2004)

    Google Scholar 

  16. Gradshteyn, I.S., Ryzhik, I.M.: Table of Integrals, Series, and Products. Academic, New York (1980)

    MATH  Google Scholar 

  17. Heinrich, S.: Multilevel Monte Carlo methods. Lect. Notes Comput. Sci. 2179, 58–67 (2001)

    CrossRef  MATH  Google Scholar 

  18. Jacod, J., Kurtz, T.G., Méléard, S., Protter, P.: The approximate Euler method for Lévy driven stochastic differential equations. Ann. Inst. H. Poincaré Probab. Statist. 41(3), 523–558 (2005). doi:10.1016/j.anihpb.2004.01.007

    CrossRef  MATH  Google Scholar 

  19. Kloeden, P.E., Platen, E.: Numerical Solution of Stochastic Differential Equations, Applications of Mathematics (New York), vol. 23. Springer, Berlin (1992)

    CrossRef  MATH  Google Scholar 

  20. Kohatsu-Higa, A., Tankov, P.: Jump-adapted discretization schemes for Lévy-driven SDEs. Stoch. Process. Appl. 120(11), 2258–2285 (2010)

    MathSciNet  CrossRef  MATH  Google Scholar 

  21. Li, S.: Multilevel Monte Carlo simulation for stochastic differential equations driven by Lévy processes. Ph.D. dissertation, Westfälische Wilhelms-Universität (2015)

    Google Scholar 

  22. Maghsoodi, Y.: Mean square efficient numerical solution of jump-diffusion stochastic differential equations. Sankhyā Ser. A 58(1), 25–47 (1996)

    MathSciNet  MATH  Google Scholar 

  23. Menn, C., Rachev, S.T.: Smoothly truncated stable distributions, GARCH-models, and option pricing. Math. Methods Oper. Res. 69(3), 411–438 (2009)

    MathSciNet  CrossRef  MATH  Google Scholar 

  24. Mordecki, E., Szepessy, A., Tempone, R., Zouraris, G.E.: Adaptive weak approximation of diffusions with jumps. SIAM J. Numer. Anal. 46(4), 1732–1768 (2008)

    MathSciNet  CrossRef  MATH  Google Scholar 

  25. Platen, E.: An approximation method for a class of Itô processes with jump component. Litovsk. Mat. Sb. 22(2), 124–136 (1982)

    MathSciNet  MATH  Google Scholar 

  26. Quek, T., De La Roche, G., Güvenç, I., Kountouris, M.: Small Cell Networks: Deployment, PHY Techniques, and Resource Management. Cambridge University Press, Cambridge (2013)

    CrossRef  Google Scholar 

  27. Rubenthaler, S.: Numerical simulation of the solution of a stochastic differential equation driven by a Lévy process. Stoch. Process. Appl. 103(2), 311–349 (2003)

    MathSciNet  CrossRef  MATH  Google Scholar 

  28. Sato, K.: Lévy processes and infinitely divisible distributions. Cambridge Studies in Advanced Mathematics, vol. 68. Cambridge University Press, Cambridge (1999)

    Google Scholar 

  29. Vasershtein, L.N.: Markov processes over denumerable products of spaces describing large system of automata. Problemy Peredači Informacii 5(3), 64–72 (1969)

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Steffen Dereich .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 2016 Springer International Publishing Switzerland

About this paper

Cite this paper

Dereich, S., Li, S. (2016). Multilevel Monte Carlo Implementation for SDEs Driven by Truncated Stable Processes. In: Cools, R., Nuyens, D. (eds) Monte Carlo and Quasi-Monte Carlo Methods. Springer Proceedings in Mathematics & Statistics, vol 163. Springer, Cham. https://doi.org/10.1007/978-3-319-33507-0_1

Download citation