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First-Order Weak Balanced Schemes for Stochastic Differential Equations

Author

Listed:
  • H. A. Mardones

    (Universidad de Concepción)

  • C. M. Mora

    (Universidad de Concepción)

Abstract

We address the numerical solution of stochastic differential equations with multiplicative noise (SDEs) by means of balanced schemes. In particular, we study the design of balanced schemes that achieve the first order of weak convergence without involve the simulation of multiple stochastic integrals. We start by using the linear scalar SDE as a test problem to show that it is possible to construct almost sure stable first-order weak schemes based on the addition of stabilizing functions to the drift terms. Then, we consider multidimensional linear SDEs. In this case, we propose to find appropriate stabilizing weights through an optimization procedure. Finally, we illustrate the potential of the new methodology by solving the stochastic Duffing-Van Der Pol equation, which is a classical test non-linear SDE. Numerical experiments show a good performance of the numerical methods introduced in this paper.

Suggested Citation

  • H. A. Mardones & C. M. Mora, 2020. "First-Order Weak Balanced Schemes for Stochastic Differential Equations," Methodology and Computing in Applied Probability, Springer, vol. 22(2), pages 833-852, June.
    • Handle: RePEc:spr:metcap:v:22:y:2020:i:2:d:10.1007_s11009-019-09733-5
    DOI: 10.1007/s11009-019-09733-5
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    References listed on IDEAS

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    3. G. N. Milstein & Eckhard Platen & H. Schurz, 1998. "Balanced Implicit Methods for Stiff Stochastic Systems," Published Paper Series 1998-1, Finance Discipline Group, UTS Business School, University of Technology, Sydney.
    4. Kahl Christian & Schurz Henri, 2006. "Balanced Milstein Methods for Ordinary SDEs," Monte Carlo Methods and Applications, De Gruyter, vol. 12(2), pages 143-170, April.
    5. Baxendale, Peter H., 2004. "Stochastic averaging and asymptotic behavior of the stochastic Duffing-van der Pol equation," Stochastic Processes and their Applications, Elsevier, vol. 113(2), pages 235-272, October.
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