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A Posteriori Error Control and Adaptivity for the IMEX BDF2 Method for PIDEs with Application to Options Pricing Models

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Abstract

In this paper, we study a posteriori error estimates of the IMEX BDF2 scheme for time discretizations of solving parabolic partial integro-differential equations, which describe the jump-diffusion option pricing model in finance. Because of the initial singularities of the solution which is due to the nonsmoothness of the payoff function, a posteriori error control and adaptivity will be crucial in solving numerically this type of equations. To derive optimal order a posteriori error estimates, quadratic reconstructions for the IMEX BDF2 method are introduced. By using these continuous, piecewise time reconstructions, the upper and lower error bounds depending only on the discretization parameters and the data of the problems are derived. Based on these a posteriori error estimates, we further develop a time adaptive algorithm. The numerical implementations are performed with both nonuniform partitions and adaptivity in time. The adaptive algorithm reduce the computational cost substantially and provides efficient error control for the solution.

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All data generated or analyzed during this study are included in this published article.

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Acknowledgements

The authors would like to thank the referee for comments and suggestions that led to improvements in the presentation of this paper.

Funding

This work was supported by grants from the National Natural Science Foundation of China (Grant Nos. 12271367, 11771060) and Shanghai Science and Technology Planning Projects (Grant No. 20JC1414200), and sponsored by Natural Science Foundation of Shanghai (Grant No. 20ZR1441200).

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  1. Department of Mathematics, Shanghai Normal University, Shanghai, 200234, China

    Wansheng Wang, Mengli Mao & Yi Huang

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  1. Wansheng Wang
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Wang, W., Mao, M. & Huang, Y. A Posteriori Error Control and Adaptivity for the IMEX BDF2 Method for PIDEs with Application to Options Pricing Models. J Sci Comput 93, 55 (2022). https://doi.org/10.1007/s10915-022-02013-4

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