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A positivity preserving numerical method for stochastic R&D model

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  • Zhang, Mengqing
  • Zhang, Qimin

Abstract

The stochastic research and development (R&D) model plays an important role in the growth rates of technological progress and capital accumulation. However, we can not obtain the explicit solution of stochastic R&D model. To explore a numerical approximate method preserving positivity for the R&D model with uncertainty from the population growth, we first construct an explicit Euler–Maruyama (EM) method and then verify it converges to the true solution with strong order 12(1−1p). But the EM method may lead to a negative approximation which is unrealistic. So, we establish the balanced explicit method (BEM) to overcome the defect of EM method and give sufficient conditions to preserve the positivity of BEM, then the convergence order of BEM is obtained. Finally, numerical simulations are carried out to verify our theoretical work.

Suggested Citation

  • Zhang, Mengqing & Zhang, Qimin, 2019. "A positivity preserving numerical method for stochastic R&D model," Applied Mathematics and Computation, Elsevier, vol. 351(C), pages 193-203.
    • Handle: RePEc:eee:apmaco:v:351:y:2019:i:c:p:193-203
    DOI: 10.1016/j.amc.2018.12.003
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    References listed on IDEAS

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    1. Tan, Jianguo & Men, Weiwei & Pei, Yongzhen & Guo, Yongfeng, 2017. "Construction of positivity preserving numerical method for stochastic age-dependent population equations," Applied Mathematics and Computation, Elsevier, vol. 293(C), pages 57-64.
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    6. 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.
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    Cited by:

    1. Tan, Jianguo & Chen, Yang & Men, Weiwei & Guo, Yongfeng, 2021. "Positivity and convergence of the balanced implicit method for the nonlinear jump-extended CIR model," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 182(C), pages 195-210.
    2. Gamboa, M. & López-García, M. & Lopez-Herrero, M.J., 2024. "On the exact and population bi-dimensional reproduction numbers in a stochastic SVIR model with imperfect vaccine," Applied Mathematics and Computation, Elsevier, vol. 468(C).

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