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An efficient spectral method for the numerical solution to some classes of stochastic differential equations
Author(s) -
Chauvière Cédric,
Djellout Hacène
Publication year - 2021
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.7157
Subject(s) - mathematics , stochastic differential equation , spectral method , stochastic partial differential equation , collocation method , convergence (economics) , smoothness , differential equation , collocation (remote sensing) , order of accuracy , ordinary differential equation , numerical analysis , polynomial , mathematical analysis , numerical partial differential equations , computer science , machine learning , economic growth , economics
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.