First order strong approximations of scalar SDEs defined in a domain

Abstract

We are interested in strong approximations of one-dimensional SDEs which have non-Lipschitz coefficients and which take values in a domain. Under a set of general assumptions we derive an implicit scheme that preserves the domain of the SDEs and is strongly convergent with rate one. Moreover, we show that this general result can be applied to many SDEs we encounter in mathematical finance and bio-mathematics. We will demonstrate flexibility of our approach by analyzing classical examples of SDEs with sublinear coefficients (CIR, CEV models and Wright–Fisher diffusion) and also with superlinear coefficients (3/2-volatility, Aït-Sahalia model). Our goal is to justify an efficient Multilevel Monte Carlo method for a rich family of SDEs, which relies on good strong convergence properties.

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

Fig. 1

References

  1. 1.

    Alfonsi, A.: On the discretization schemes for the CIR (and Bessel squared) processes. Monte Carlo Methods Appl. 11(4), 355–384 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  2. 2.

    Alfonsi, A.: Strong order one convergence of a drift implicit Euler scheme: application to the CIR process. Stat. Probab. Lett. 83(2), 602–607 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  3. 3.

    Andersen, L.B.G.: Simple and efficient simulation of the Heston stochastic volatility model. J. Comput. Finance 11(3), 1–42 (2008)

    Google Scholar 

  4. 4.

    Berkaoui, A., Bossy, M., Diop, A.: Euler scheme for SDEs with non-Lipschitz diffusion coefficient: strong convergence. ESAIM Probab. Stat. 12, 1–11 (2007)

    Article  MathSciNet  Google Scholar 

  5. 5.

    Cox, J.: Notes on option pricing I: constant elasticity of variance diffusions. Unpublished manuscript. Stanford University, Stanford (1975)

    Google Scholar 

  6. 6.

    Dangerfield, C.E., Kay, D., MacNamara, S., Burrage, K.: A boundary preserving numerical algorithm for the Wright–Fisher model with mutation. BIT Numer. Math. 52(2), 283–304 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  7. 7.

    Dereich, S., Neuenkirch, A., Szpruch, L.: An Euler-type method for the strong approximation of the Cox–Ingersoll–Ross process. Proc. R. Soc. A Math. Phys. Eng. Sci. 468(2140), 1105–1115 (2012)

    MathSciNet  Google Scholar 

  8. 8.

    Either, S., Kurtz, T.: Markov processes: characterization and convergences. John Wiley Sons, New York (1986)

    Book  Google Scholar 

  9. 9.

    Giles, M.: Monte Carlo and Quasi-Monte Carlo methods 2006. Conference Proceedings. In: Keller, A. (ed.) Improved multilevel Monte Carlo convergence using the Milstein scheme, pp. 343–358. Springer, New York (2008)

    Google Scholar 

  10. 10.

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

    Article  MATH  MathSciNet  Google Scholar 

  11. 11.

    Heston, S.L.: A simple new formula for options with stochastic volatility. Course notes. Washington University, St. Louis (1997)

    Google Scholar 

  12. 12.

    Higham, D.J., Mao, X.: Convergence of Monte Carlo simulations involving the mean-reverting square root process. J. Comput. Finance 8(3), 35–62 (2005)

    Google Scholar 

  13. 13.

    Higham, D.J., Mao, X., Stuart, A.M.: Strong convergence of Euler-type methods for nonlinear stochastic differential equations. SIAM J. Numer. Anal. 40(3), 1041–1063 (2003)

    Article  MathSciNet  Google Scholar 

  14. 14.

    Higham, D.J., Mao, X., Szpruch, L.: Convergence, non-negativity and stability of a new Milstein scheme with applications to finance. Discret. Contin. Dyn. Syst. Ser. B 18(8), 2083–2100 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  15. 15.

    Howison, S., Schwarz, D.: Risk-neutral pricing of financial instruments in emission markets: a structural approach. SIAM J. Math. Finance 3, 709–739 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  16. 16.

    Hurd, T.R., Kuznetsov, A.: Explicit formulas for Laplace transforms of stochastic integrals. Markov Process. Relat. Fields 14(2), 277–290 (2008)

    MATH  MathSciNet  Google Scholar 

  17. 17.

    Jentzen, A., Hutzenthaler, M., Kloeden, P.E.: Divergence of the multilevel Monte Carlo Euler method for nonlinear stochastic differential equations. Ann. Appl. Prob. 23(5), 1721–1760 (2013)

    Google Scholar 

  18. 18.

    Jentzen, A., Hutzenthaler, M., Kloeden, P.E.: Strong convergence of an explicit numerical method for SDEs with non-globally Lipschitz continuous coefficients. Ann. Appl. Probab. 22(4), 1611–1641 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  19. 19.

    Iacus, S.M.: Simulation and Inference for stochastic differential equations: with R examples. Springer, New York (2008)

    Book  Google Scholar 

  20. 20.

    Jentzen, A., Kloeden, P.E., Neuenkirch, A.: Pathwise approximation of stochastic differential equations on domains: higher order convergence rates without global Lipschitz coefficients. Numerische Mathematik 112(1), 41–64 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  21. 21.

    Kahl, C., Günther, M., Rosberg, T.: Structure preserving stochastic integration schemes in interest rate derivative modeling. Appl. Numer. Math. 58(3), 284–295 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  22. 22.

    Kahl, C., Jäckel, P.: Fast strong approximation Monte Carlo schemes for stochastic volatility models. Quant. Finance 6(6), 513–536 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  23. 23.

    Karatzas, I., Shreve, S.E.: Brownian motion and stochastic calculus. Springer, New York (1991)

    MATH  Google Scholar 

  24. 24.

    Karlin, S., Taylor, H.M.: A second course in stochastic processes. Academic Press, Dublin (1981)

    MATH  Google Scholar 

  25. 25.

    Kloeden, P.E., Neuenkirch, A.: Recent developments in computational finance. In: Gerstner, T., Kloeden, P.E. (eds.) Convergence of numerical methods for stochastic differential equations in mathematical finance, pp. 49–81. World Scientific, Singapore (2013)

    Google Scholar 

  26. 26.

    Kloeden, P.E., Platen, E.: Numerical solution of stochastic differential equations. Springer, New York (1992)

    Book  MATH  Google Scholar 

  27. 27.

    Larsen, K.S., Sørensen, M.: Diffusion models for exchange rates in a target zone. Math. Finance 17(2), 285–306 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  28. 28.

    Lord, R., Koekkoek, R., van Dijk, D.J.C.: A comparison of biased simulation schemes for stochastic volatility models. Quant. Finance 10(2), 177–194 (2009)

    Article  MathSciNet  Google Scholar 

  29. 29.

    Mao, X., Szpruch, L.: Strong convergence rates for backward Euler–Maruyama method for non-linear dissipative-type stochastic differential equations with super-linear diffusion coefficients. Stochastics 85(1), 144–171 (2013)

    MATH  MathSciNet  Google Scholar 

  30. 30.

    Szpruch, L., Mao, X., Higham, D.J., Pan, J.: Strongly nonlinear Aït-Sahalia-type interest rate model and its numerical approximation. BIT Numer. Math. 51(2), 405–425 (2010)

    Article  MathSciNet  Google Scholar 

  31. 31.

    Zeidler, E.: Nonlinear functional analysis and its applications. Springer, New York (1985)

    Book  MATH  Google Scholar 

  32. 32.

    Zhu, J.: Modular pricing of options: an application of Fourier analysis. Springer, New York (2009)

    Google Scholar 

Download references

Acknowledgments

We would like to thank the referees for their valuable and insightful comments and remarks. Moreover, we would like to thank Martin Altmayer for helpful comments on an earlier version of the manuscript.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Andreas Neuenkirch.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Neuenkirch, A., Szpruch, L. First order strong approximations of scalar SDEs defined in a domain. Numer. Math. 128, 103–136 (2014). https://doi.org/10.1007/s00211-014-0606-4

Download citation

Mathematics Subject Classification (2000)

  • 60H10
  • 65J15