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Estimation of 1-dimensional nonlinear stochastic differential equations based on higher-order partial differential equation numerical scheme and its application

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Abstract

A method based on higher-order partial differential equation (PDE) numerical scheme are proposed to obtain the transition cumulative distribution function (CDF) of the diffusion process (numerical differentiation of the transition CDF follows the transition probability density function (PDF)), where a transformation is applied to the Kolmogorov PDEs first, then a new type of PDEs with step function initial conditions and 0, 1 boundary conditions can be obtained. The new PDEs are solved by a fourth-order compact difference scheme and a compact difference scheme with extrapolation algorithm. After extrapolation, the compact difference scheme is extended to a scheme with sixth-order accuracy in space, where the convergence is proved. The results of the numerical tests show that the CDF approach based on the compact difference scheme to be more accurate than the other estimation methods considered; however, the CDF approach is not time-consuming. Moreover, the CDF approach is used to fit monthly data of the Federal funds rate between 1983 and 2000 by CKLS model.

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Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant No. 11401591), the Project Sponsored by the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry (No. 2013693), and the Self-determined Research Funds of CCNU from the Colleges (No. CCNU14A05040).

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Authors and Affiliations

  1. School of Mathematics and Statistics, Central China Normal University, Wuhan, 430079, China

    Peiyan Li

  2. School of Statistics and Mathematics, Zhongnan University of Economics and Law, Wuhan, 430073, China

    Wei Gu

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  1. Peiyan Li
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  2. Wei Gu
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Correspondence to Wei Gu.

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Li, P., Gu, W. Estimation of 1-dimensional nonlinear stochastic differential equations based on higher-order partial differential equation numerical scheme and its application. Front. Math. China 12, 1441–1455 (2017). https://doi.org/10.1007/s11464-017-0663-y

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  • DOI: https://doi.org/10.1007/s11464-017-0663-y

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