Numerical solution for anti-persistent process based stochastic integral equations

Abstract

In this article, we propose the shifted Legendre polynomial solutions for anti-persistent process based stochastic integral equations. The operational matrices for stochastic integration and fractional stochastic integration are efficiently generated using the properties of shifted Legendre polynomials. In addition, the original problem can be reduced to a system of simultaneous equations with (N + 1) unknowns in the function approximation. By solving the given stochastic integral equations, we obtain numerical solutions. The proposed method’s convergence is derived in terms of the error function’s expectation, and the upper bound of the error in L² norm is also discussed in detail. The applicability of this methodology is demonstrated using numerical examples and the solution’s quality is statistically validated by comparing it with the exact solution.