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Positivity-preserving numerical schemes for stochastic differential equations

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Abstract

The solutions of stochastic differential equations arising in biology, finance and so on often have positivity. However, numerical solutions by the standard schemes often fail to satisfy this property. In this paper, we propose positivity-preserving numerical schemes for stochastic differential equations by virtue of Itô’s formula. We also show the convergence result of the proposed scheme and demonstrate their effectiveness by numerical examples.

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Notes

  1. Theoretically, we can also apply any implicit schemes. However, an iterative method is sometimes required when solving the implicit scheme, in which case the iterations must be terminated for numerical computation. As a result, the range-preserving property proposed here cannot be guaranteed.

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Authors and Affiliations

  1. Graduate School of Systems Engineering and Science, Shibaura Institute of Technology, Saitama, Japan

    Keisuke Abiko

  2. Department of Mathematical Sciences, Shibaura Institute of Technology, Saitama, Japan

    Tetsuya Ishiwata

Authors
  1. Keisuke Abiko
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  2. Tetsuya Ishiwata
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Correspondence to Tetsuya Ishiwata.

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This work was supported by KAKENHI No. 19H05599.

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Abiko, K., Ishiwata, T. Positivity-preserving numerical schemes for stochastic differential equations. Japan J. Indust. Appl. Math. 39, 1095–1108 (2022). https://doi.org/10.1007/s13160-022-00554-7

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  • DOI: https://doi.org/10.1007/s13160-022-00554-7

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