A mean square chain rule and its applications in solving the random Chebyschev differential equation

TL;DR

This paper introduces a new mean square chain rule for stochastic processes, enabling the construction of solutions to the random Chebyshev differential equation using power series and providing methods for approximating mean and standard deviation functions.

Contribution

It develops a novel mean square chain rule and applies it to solve the random Chebyshev differential equation with rigorous mean square solutions.

Findings

Established a new mean square chain rule for stochastic processes.

Constructed mean square solutions for the random Chebyshev differential equation.

Provided approximation methods for mean and standard deviation functions.

Abstract

In this paper a new version of the chain rule for calculating the mean square derivative of a second-order stochastic process is proven. This random operational calculus rule is applied to construct a rigorous mean square solution of the random Chebyshev differential equation (r.C.d.e.) assuming mild moment hypotheses on the random variables that appear as coefficients and initial conditions of the corresponding initial value problem. Such solution is represented through a mean square random power series. Moreover, reliable approximations for the mean and standard deviation functions to the solution stochastic process of the r.C.d.e. are given. Several examples, that illustrate the theoretical results, are included.