Existence, uniqueness, and numerical analysis of fractional M-derivative stochastic integrodifferential models within the shifted Legendre Gauss pseudospectral algorithm

In this paper, we are concerned with improving theoretical and numerical justifications of fractional local M-derivative stochastic integrodifferential models of order [Formula: see text] with [Formula: see text] driven by Brownian motion, concerning the Itô type stochastic integration. Based on linear growth and global Lipschitz constraints presumed on the given drift and diffusion coefficients of the stochastic model, we employ Picard’s iteration idea, Arzela–Ascoli’s theorems, Itô’s isometry, and Gronwall’s inequality to assert the existence and uniqueness of the suggested solution. Once this assertion is provided, we establish a Gauss pseudospectral technique considering the shifted Legendre polynomials to obtain numerical approximations of the desired realizations. As required, we construct a reasonable procedure for Brownian motion simulations and employ the software package Mathematica to execute the steps. The Gauss pseudospectral algorithm transforms our model into an unpretentious system that involves algebraic equations that are no longer complicated and produce accurate numerical solutions. To emphasize the reality of this algorithm, we analyze the convergence utilizing some stochastic concepts. Finally, we perform a variety of stochastic applications with various coefficients and provide evidential results in figures, tables, and discussions to ensure the efficiency of the obtained consequences. Suggestions depending on this work are also notarized.