Probabilistic analysis of a class of impulsive linear random differential equations forced by stochastic processes admitting Karhunen-Loève expansions

Abstract

<p style='text-indent:20px;'>We study a full randomization of the complete linear differential equation subject to an infinite train of Dirac's delta functions applied at different time instants. The initial condition and coefficients of the differential equation are assumed to be absolutely continuous random variables, while the external or forcing term is a stochastic process. We first approximate the forcing term using the Karhunen-Loève expansion, and then we take advantage of the Random Variable Transformation method to construct a formal approximation of the first probability density function (1-p.d.f.) of the solution. By imposing mild conditions on the model parameters, we prove the convergence of the aforementioned approximation to the exact 1-p.d.f. of the solution. All the theoretical findings are illustrated by means of two examples, where different types of probability distributions are assumed to model parameters.</p>

Probabilistic analysis of a class of compartmental models formulated by random differential equations

“…As we pointed out after (18), expression (21) could be useful to compute the 1-PDF in some practical cases when the calculation of the (k − 1)-dimensional integral in (20) is complicated. In Section 3, we shall apply this general result to some specific source terms that depend parametrically on a finite number of random variables.…”

“…So far, we have considered in Examples 1 and 2 that the source terms, g m (t) ≡ g m (t; p m (𝜔)), 𝜔 ∈ Ω, m = 1, … , n, are defined by stochastic processes that depend parametrically on a finite set of random variables p m (𝜔) = (p m 1 (𝜔), … , p m M m (𝜔)). In both cases, we have used the RVT method to calculate the 1-PDF of the solution of the compartmental model (3) using the expression (21). However, non-parametric processes are also interesting and might be more accurate for some modeling cases.…”

“…It is worth pointing out that the Laplace transform provides a powerful tool to analyze these types of linear compartmental models formulated by RDEs. Recently, some contributions have dealt with the computation of the 1‐PDF of some classes of RDEs representing simple compartmental models by combining the random Laplace transform with the RVT technique, see for instance [21].…”