Abstract and Applied Analysis

Volume 2016 (2016), Article ID 6372108, 22 pages

http://dx.doi.org/10.1155/2016/6372108

## Random First-Order Linear Discrete Models and Their Probabilistic Solution: A Comprehensive Study

Instituto Universitario de Matemática Multidisciplinar, Universitat Politècnica de València, Camino de Vera s/n, Building 8G, 2nd Floor, 46022 Valencia, Spain

Received 3 October 2015; Accepted 1 February 2016

Academic Editor: Patricia J. Y. Wong

Copyright © 2016 M.-C. Casabán et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This paper presents a complete stochastic solution represented by the first probability density function for random first-order linear difference equations. The study is based on Random Variable Transformation method. The obtained results are given in terms of the probability density functions of the data, namely, initial condition, forcing term, and diffusion coefficient. To conduct the study, all possible cases regarding statistical dependence of the random input parameters are considered. A complete collection of illustrative examples covering all the possible scenarios is provided.

#### 1. Introduction and Motivation

The birth/death rates of species in biology, the volatility of assets in finance, the transmission rates of the spread of epidemics or social addictions in epidemiology, the diffusion and advection coefficients of mass transport processes in physics, and so forth are quantities that, in practice, involve uncertainty. Thus, their deterministic modelling is clearly limited. This motivates the search of mathematical models that consider randomness in their formulation. Deterministic differential and difference equations have been demonstrated to be useful mathematical representations for modelling numerous real problems. The consideration of randomness in these types of equations is a relatively recent research area whose main goal is to extend classical deterministic results to the random scenario. Regarding continuous models, most of the contributions have focussed on Itô-type stochastic differential equations. In this class of differential equations, uncertainty is considered through a Gaussian and stationary stochastic process (SP) called white noise, which is the derivative of the Wiener SP [1–3]. Autoregressive (AR) models can be interpreted as their discrete counterpart. These types of models are extensively used in time series analysis in statistics [4]. Some recent interesting models based on Itô-type stochastic differential equations include [5–7], for instance. Complementary to these approaches, uncertainty can be directly introduced in differential and difference equations by assuming that coefficients, source term, and/or initial/boundary conditions are random variables and/or stochastic processes. Under this approach, the probability distributions associated with RVs and SPs are not required to be Gaussian. This approach leads to the area usually referred to as random differential/difference equations [8], [9, p. 66]. Intensive studies on random differential and difference equations have been undertaken only over the last few decades. Currently, they are exerting a profound influence on the analysis of many problems in engineering and science [10, 11]. Most of these contributions are based on mean square calculus [8, 12–14].

The solution of a random difference equation is a discrete SP, say . In dealing with random difference equations, the main goals are computing the solution SP and its statistical characteristics, such as the mean function, , and the variance function, . Despite being more complicated, the computation of the first probability density function (1-PDF), , associated with solution is more convenient since from 1-PDF, besides determining the mean and the variance functions, one can also compute higher-order statistical moments of :

In the context of random differential equations, a number of contributions have dealt with the computation of the 1-PDF in specific problems appearing in physics [15–17] or in mathematics [18, 19]. A comprehensive study for the random first-order linear differential equation under general hypotheses has been recently published by the authors of [20]. The unifying element of all these contributions is the Random Variable Transformation (RVT) method. This technique permits, under certain hypotheses that will be specified later, the computation of the PDF of a random variable (RV) resulting after mapping another RV whose PDF is known [21–23]. Although RVT technique is a classical probability result, it must be pointed out that its application to study differential equations with randomness is likely due to Professor M. El-Tawil. He was the first author who had used RVT technique to present some approximate solutions of random differential equations. In [24], the RVT method together with different numerical schemes (finite difference and Runge-Kutta) is implemented to get the 1-PDF of the solution SP in solving both partial and/or ordinary (first- and/or second-order) random differential equations. His contribution includes the definition of probabilistic error and a formula for its computation as well.

As continuation of the study initiated in [20], the aim of this paper is to develop a comprehensive study to determine the -PDF of the solution discrete SP of the following random initial value problem:taking advantage of RVT method. All the input parameters, , , and , are assumed to be continuous RVs defined on a common probability space . Although RVT method is the unifying technique used to conduct the analysis of model (2) and also the one studied in [20], it is important to point out that both problems have distinctly different nature. Indeed, model (2) is discrete whereas problem faced in [20] is continuous counterpart. As we will see later, Proposition 1 constitutes the key result we have had to establish to conduct the analysis of problem (2). Its formulation is based on RVT method. Throughout the paper, significant differences between the new analytical expressions and graphical behaviour of the -PDF of the solution of (2) and its continuous counterpart will be also exhibited.

As it also happens in the deterministic framework, in general, the study of random difference equations has been less prolific than of random differential equations. In [25], the authors study the mean square exponential stability of impulsive stochastic difference equations. In [26], one studies random matrix linear difference equations assuming that diffusion coefficient in (2) is a deterministic matrix rather than a RV. In [26], the authors focus on the computation of the mean vector and variance-covariance matrix of the solution discrete SP instead of the 1-PDF. In this paper, we present a comprehensive study of model (2) assuming randomness in all the inputs, , , and . This includes the general case where inputs are statistically dependent. From a statistical point of view, (2) is a generalization of autoregressive model of order 1, , where uncertainty is enclosed in the term through white noise.

For the sake of clarity in the presentation and in order to facilitate the comparison of the results obtained in this paper against the ones achieved in [20], we will keep the notation used in both contributions identical. Hence, the domain of the random inputs , , and will be denoted by respectively. From this point forward, we will omit the -notation when writing RVs. In this manner, for instance, we will write rather than . The same can be said for the notation of the PDFs that appear throughout this paper. For example, will denote the PDF of RV ; will denote the joint PDF of RVs and ; we will write for the joint PDF of the random vector , and so on. As usual, we will assume that any PDF is null outside its domain.

Based on the same arguments exhibited in [20, Section 1], we will distinguish the thirteen cases listed in Table 1 to conduct our study. These casuistries consider whether the random difference equation (2) is homogeneous or nonhomogeneous as well as all possible cases regarding the random or deterministic nature of the input parameters , , and . Note that, by splitting the study in all these cases, the comparison of the results concerning the discrete problem (2) against its continuous counterpart is facilitated. Examples have also been devised with the same aim. Even more, in the majority of the examples, the same statistical distributions have been taken as in [20] to highlight better analogies and differences between both models.