Taylor approximation of the solution of age-dependent stochastic delay population equations with Ornstein-Uhlenbeck process and Poisson jumps

Wenrui Li; Qimin Zhang; Meyer-Baese Anke; Ming Ye; Yan Li; Wenrui Li; Qimin Zhang; Meyer-Baese Anke; Ming Ye; Yan Li

doi:10.3934/mbe.2020145

Mathematical Biosciences and Engineering

2020, Volume 17, Issue 3: 2650-2675. doi: 10.3934/mbe.2020145

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Research article

Taylor approximation of the solution of age-dependent stochastic delay population equations with Ornstein-Uhlenbeck process and Poisson jumps

1.
School of Mathematics and Statistics, Ningxia University, Yinchuan 750021, China
2.
Department of Scientific Computing, Florida State University, Tallahassee FL 32306-4120, USA
3.
Department of Earth, Ocean, and Atmospheric Science and Department of Scientific Computing, Florida State University, Tallahassee FL 32306, USA

Received: 04 December 2019 Accepted: 18 February 2020 Published: 09 March 2020

Numerical approximation is a vital method to investigate the properties of stochastic age-dependent population systems, since most stochastic age-dependent population systems cannot be solved explicitly. In this paper, a Taylor approximation scheme for a class of age-dependent stochastic delay population equations with mean-reverting Ornstein-Uhlenbeck (OU) process and Poisson jumps is presented. In case that the coefficients of drift and diffusion are Taylor approximations, we prove that the numerical solutions converge to the exact solutions for these equations. Moreover, the convergence order of the numerical scheme is given. Finally, some numerical simulations are discussed to illustrate the theoretical results.
- stochastic age-dependent population equations,
- time delay,
- ornstein-uhlenbeck process,
- taylor approximation
Citation: Wenrui Li, Qimin Zhang, Meyer-Baese Anke, Ming Ye, Yan Li. Taylor approximation of the solution of age-dependent stochastic delay population equations with Ornstein-Uhlenbeck process and Poisson jumps[J]. Mathematical Biosciences and Engineering, 2020, 17(3): 2650-2675. doi: 10.3934/mbe.2020145

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Abstract

Numerical approximation is a vital method to investigate the properties of stochastic age-dependent population systems, since most stochastic age-dependent population systems cannot be solved explicitly. In this paper, a Taylor approximation scheme for a class of age-dependent stochastic delay population equations with mean-reverting Ornstein-Uhlenbeck (OU) process and Poisson jumps is presented. In case that the coefficients of drift and diffusion are Taylor approximations, we prove that the numerical solutions converge to the exact solutions for these equations. Moreover, the convergence order of the numerical scheme is given. Finally, some numerical simulations are discussed to illustrate the theoretical results.

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