Mean Square Convergent Non-Standard Numerical Schemes for Linear Random Differential Equations with Delay

Julia Calatayud, Juan Carlos Cortés, Marc Jornet and Francisco Rodríguez
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Julia Calatayud: Instituto Universitario de Matemática Multidisciplinar, Building 8G, Access C, 2nd Floor, Universitat Politècnica de València, Camino de Vera s/n, 46022 Valencia, Spain
Juan Carlos Cortés: Instituto Universitario de Matemática Multidisciplinar, Building 8G, Access C, 2nd Floor, Universitat Politècnica de València, Camino de Vera s/n, 46022 Valencia, Spain
Marc Jornet: Instituto Universitario de Matemática Multidisciplinar, Building 8G, Access C, 2nd Floor, Universitat Politècnica de València, Camino de Vera s/n, 46022 Valencia, Spain
Francisco Rodríguez: Department of Applied Mathematics, University of Alicante, Apdo. 99, 03080 Alicante, Spain

Mathematics, 2020, vol. 8, issue 9, 1-17

Abstract: In this paper, we are concerned with the construction of numerical schemes for linear random differential equations with discrete delay. For the linear deterministic differential equation with discrete delay, a recent contribution proposed a family of non-standard finite difference (NSFD) methods from an exact numerical scheme on the whole domain. The family of NSFD schemes had increasing order of accuracy, was dynamically consistent, and possessed simple computational properties compared to the exact scheme. In the random setting, when the two equation coefficients are bounded random variables and the initial condition is a regular stochastic process, we prove that the randomized NSFD schemes converge in the mean square (m.s.) sense. M.s. convergence allows for approximating the expectation and the variance of the solution stochastic process. In practice, the NSFD scheme is applied with symbolic inputs, and afterward the statistics are explicitly computed by using the linearity of the expectation. This procedure permits retaining the increasing order of accuracy of the deterministic counterpart. Some numerical examples illustrate the approach. The theoretical m.s. convergence rate is supported numerically, even when the two equation coefficients are unbounded random variables. M.s. dynamic consistency is assessed numerically. A comparison with Euler’s method is performed. Finally, an example dealing with the time evolution of a photosynthetic bacterial population is presented.

Keywords: delay random differential equation; non-standard finite difference method; mean square convergence (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2020
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