An efficient spectral method for the numerical solution to some classes of stochastic differential equations

Abstract

We consider a new approach for the numerical approximation to some classes of stochastic differential equations driven by white noise. The proposed method shares some features with the stochastic collocation techniques, and in particular, it takes advantage of the assumption of smoothness of the functional to be approximated, to achieve fast convergence. The solution to the stochastic differential equation (SDE) is represented by means of Lagrange polynomials. The coefficients of the polynomial basis are functions of time, and they can be computed by solving a system of deterministic ordinary differential equations. Another motivation of this work lies in the novelty of the numerical scheme that does not belong to classical techniques to solve SDEs. Numerical examples are presented to illustrate the accuracy and the efficiency of the proposed method. In particular, we observe a spectral convergence for the mean and the variance of the solution.

A periodic averaging method for impulsive stochastic age‐structured population model in a polluted environment

“…Since model () is nonautonomous and cannot be solved explicitly, numerical approximation methods are necessary. In the past several years, a number of numerical methods have been developed with comprehensive numerical analysis 21–24 . For example, Li et al 21 studied the strong convergence properties of partially truncated EM method for the stochastic age‐structured Susceptible‐Infected‐Removed epidemic model with a local Lipschtiz condition.…”

“…In the past several years, a number of numerical methods have been developed with comprehensive numerical analysis. [21][22][23][24] For example, Li et al 21 studied the strong convergence properties of partially truncated EM method for the stochastic age-structured Susceptible-Infected-Removed epidemic model with a local Lipschtiz condition. Liu et al 23 constructed a stopped EM method for nonlinear stochastic differential equations (SDEs) to discuss the strong convergence properties.…”