Skip to main content Accessibility help
×
Home

A Variable Step Size Riemannian Sum for an Itô Integral

  • E. Rapoo (a1)

Abstract

We investigate the problem of using a Riemannian sum with random subintervals to approximate the iterated Itô integral ∫wdw - or, equivalently, solving the corresponding stochastic differential equation by Euler's method with variable step sizes. In the past this task has been used as a counterexample to illustrate that variable step sizes must be used with extreme caution in stochastic numerical analysis. This article establishes a class of variable step size schemes which do work.

    • Send article to Kindle

      To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

      Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

      Find out more about the Kindle Personal Document Service.

      A Variable Step Size Riemannian Sum for an Itô Integral
      Available formats
      ×

      Send article to Dropbox

      To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

      A Variable Step Size Riemannian Sum for an Itô Integral
      Available formats
      ×

      Send article to Google Drive

      To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

      A Variable Step Size Riemannian Sum for an Itô Integral
      Available formats
      ×

Copyright

Corresponding author

Postal address: Department of Mathematical Sciences, University of South Africa, PO Box 392, UNISA 0003, Republic of South Africa. Email address: rapooe@unisa.ac.za

References

Hide All
[1] Bichteler, K. (1981). Stochastic integration and L{p} theory of semimartingales. Ann. Prob. 9, 4989.
[2] Bouleau, N. and Lépingle, D. (1994). Numerical Methods for Stochastic Processes. John Wiley, New York.
[3] Burrage, K., Burrage, P.M. and Tian, T. (2004). Numerical methods for strong solutions of stochastic differential equations: an overview. Proc. R. Soc. London 460, 373402.
[4] Burrage, P. M. and Burrage, K. (2002). A variable stepsize implementation for stochastic differential equations. SIAM J. Sci. Comput. 24, 848864.
[5] Burrage, P.M., Herdiana, R. and Burrage, K. (2004). Adaptive stepsize based on control theory for stochastic differential equations. J. Comput. Appl. Math. 170, 317336.
[6] Gaines, J. G. and Lyons, T. L. (1994). Random generation of stochastic area integrals. SIAM J. Appl. Math. 54, 11321146.
[7] Gaines, J. G. and Lyons, T. L. (1997). Variable step size control in the numerical solution of stochatic differential equations. SIAM J. Appl. Math. 57, 14551484.
[8] Hofmann, N., Müller-Gronbach, T. and Ritter, K. (2000). Optimal approximation of stochastic differential equations by adaptive step-size control. Math. Comput. 69, 10171034.
[9] Hofmann, N., Müller-Gronbach, T. and Ritter, K. (2001). The optimal discretization of stochastic differential equations. J. Complexity 17, 117153.
[10] Ikeda, N. and Watanabe, S. (1981). Stochastic Differential Equations and Diffusion processes. North-Holland, Amsterdam.
[11] Kloeden, P. E. and Platen, E. (1992). Numerical Solution of Stochastic Differential Equations. Springer, Berlin.
[12] Lamba, H., Mattingly, J. C. and Stuart, A. M. (2007). An adaptive Euler–Maruyama scheme for SDEs: convergence and stability. IMA J. Numerical Anal. 27, 479506.
[13] Lehn, J., Rössler, A. and Schein, O. (2002). Adaptive schemes for the numerical solution of SDEs – a comparison. J. Comput. Appl. Math. 138, 297308.
[14] Mauthner, S. (1998). Step size control in the numerical solution of stochastic differential equations. J. Comput. Appl. Math. 100, 93109.
[15] Milstein, G. N. and Tretyakov, M. V. (1997). Mean-square numerical methods for stochastic differential equations with small noises. SIAM J. Sci. Comput. 18, 10671087.
[16] Newton, N. J. (1990). An efficient approximation for stochastic differential equations on the partition of symmetrical first passage times. Stoch. Stoch Reports 29, 227258.
[17] Revuz, D. and Yor, M. (1994). Continuous Martingales and Brownian Motion, 2nd edn. Springer, Berlin.
[18] Römisch, W. and Winkler, R. (2006). Stepsize control for mean-square numerical methods for stochastic differential equations with small noise. SIAM J. Sci. Comput. 28, 604625.
[19] Szepessy, A., Tempone, R. and Zouraris, G. E. (2001). Adaptive weak approximation of stochastic differential equations. Commun. Pure Appl. Math. 54, 11691214.
[20] Talay, D. (1995). Simulation of stochastic differential systems. In Probabilistic Methods in Applied Physics (Lecture Notes Physics 451), eds Kree, P. and Wedig, W., Springer, Berlin, pp. 5496.

Keywords

MSC classification

Related content

Powered by UNSILO

A Variable Step Size Riemannian Sum for an Itô Integral

  • E. Rapoo (a1)

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed.