Random non-autonomous second order linear differential equations: mean square analytic solutions and their statistical properties

J Calatayud,J.-C Cortés,L Villafuerte,M Jornet

doi:10.1186/s13662-018-1848-8

Abstract

In this paper we study random non-autonomous second order linear differential equations by taking advantage of the powerful theory of random difference equations. The coefficients are assumed to be stochastic processes, and the initial conditions are random variables both defined in a common underlying complete probability space. Under appropriate assumptions established on the data stochastic processes and on the random initial conditions, and using key results on difference equations, we prove the existence of an analytic stochastic process solution in the random mean square sense. Truncating the random series that defines the solution process, we are able to approximate the main statistical properties of the solution, such as the expectation and the variance. We also obtain error a priori bounds to construct reliable approximations of both statistical moments. We include a set of numerical examples to illustrate the main theoretical results established throughout the paper. We finish with an example where our findings are combined with Monte Carlo simulations to model uncertainty using real data.

Highlights

In this paper, we take advantage of the powerful theory of random difference equations to conduct a full probabilistic study to the random second order linear differential equation⎧ ⎪⎪⎨X (t) + A(t)X (t) + B(t)X(t) = 0, t ∈ R, ⎪⎪⎩XX (t0 (t0 ) ) = = Y0, Y1. (1)The data coefficients A(t) and B(t) are stochastic processes and the initial conditions Y0 and Y1 are random variables on an underlying complete probability space (, F, P)
The aim of this paper is, in the first step, to specify the meaning of random differential equation (1) via the Lp( ) random calculus or, more concretely, using the so-called mean square calculus that corresponds to p = 2; secondly, to find a proper stochastic process solution to (1); and thirdly, to compute its main statistical information under mild conditions
X(t) = Xn(t – t0)n, n=0 for t ∈ (t0 – r, t0 + r), where the sum is in L2( ). This stochastic process will be a solution to the random problem (1) in the sense of L2( )

Summary

Introduction

Important deterministic models appearing in the area of mathematical physics like Airy, Hermite, Legendre, Laguerre and Bessel differential equations have been randomized and rigorously studied in [3, 4, 7, 8, 10], respectively. In these contributions, approximate solution stochastic processes together with their main statistical moments (mean and variance) are constructed by taking advantage of the random mean square calculus. Additional studies dealing with random differential equations via random mean square calculus include [12, 21,22,23,24, 26,27,28, 31, 36], for instance

Objectives

Results

Conclusion