This paper presents a complete stochastic solution represented by the first probability density function for random firstorder 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.
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 Itôtype 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 [
The solution of a random difference equation is a discrete SP, say
In the context of random differential equations, a number of contributions have dealt with the computation of the 1PDF in specific problems appearing in physics [
As continuation of the study initiated in [
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 [
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 [
Based on the same arguments exhibited in [
List of the thirteen different cases considered to conduct the full study. This classification is made regarding whether the discrete initial value problem is homogeneous (H) or nonhomogeneous (NH) and the way that uncertainty is considered (one random variable or a random vector in two or three dimensions).
Type  Discrete initial value problem  Case 

H 

I.1 


NH 

II.1 

III.1 
The paper is organized as follows. Section
As it has been pointed out in the previous section, the goal of this paper is to compute the
Next, we will establish the following result concerning the PDF of a RV which is obtained after mapping another RV via a power transformation. This result will play a relevant role in the analysis of Case I.2. listed in Table
Let
(i) If
(ii) If
(iii) If
(iv) If
(i) If
(ii) If
(iii) Let us assume
To complete the computation of PDF
We complete the computation of PDF
To summarize, from (
(iv) Let us assume that
This section is addressed to compute the
For the sake of clarity in the presentation, we rewrite solution (
Let
Note that if
In order to facilitate the comparison of the
Let us assume
In order to emphasize the deterministic nature of the initial condition
Let
Let us assume that
This example exhibits a different behaviour of the
Throughout this case, the joint PDF of the random vector
Let
In this section, we deal with the computation of the
In this case, solution (
Let
In this case, the solution discrete stochastic process (
Let
In accordance with the notation previously introduced,
Let us consider the random vector
where
Figure
This section is addressed to determine the
Analogously to the previous sections, for the sake of clarity in the presentation, we will rewrite each one of the involved parameters in (
In this case, if
Notice that expression (
Let
Below, we show an example with the aim of illustrating that once the
Within the context of Example
For example, taking
In Figure
In addition to computing the mean and the variance, further significant information related to the solution SP can be computed from the
Mean (a) and variance (b) of
Let us assume
If
Then, for
Notice that expression (
Let
2D and 3D plots of
So far, we have taken advantage of RVT method to compute the
Throughout this example, we will use the notation introduced in [
2D and 3D plots of
Let us denote by
If
Let us fix
For the trivial case
Notice that expression (
Let us assume that the random vector
In this case, solution (
Let us denote by
Let us consider the twodimensional Gaussian vector
Figure
Solution (
Let us denote by
So far, we have considered standard distributions in one or more dimensions to illustrate the obtained theoretical results. Now, we will assume that the joint PDF of the input parameters
2D and 3D plots of
In this last case, solution (
Let us denote by
Let us assume that
In this paper, we have provided general explicit formulae to compute the first probability density function (1PDF) of the solution stochastic process to random firstorder linear difference equations. It has been done in the general case where the involved random inputs are statistically dependent. The study has been based on the Random Variable Transformation technique. When solving random difference equations, most of the available studies focus on the computation of the solution stochastic process and its expectation and variance functions. However, the computation of explicit formulae to determine the 1PDF is more advisable since it permits the computation of other higherorder moments and the probability of certain sets of interest as well. We have shown, through the theoretical development, that the study here presented generalizes its deterministic counterpart. In addition, all the theoretical results have been illustrated by a comprehensive list of examples. Finally, note that our analysis can be extended to determine the 1PDF of the solution to random nonlinear firstorder difference equations in future studies.
In order to facilitate the handling of all the results obtained throughout the paper in practice, in the following cases, Cases I–III, we sum up all the expressions of the 1PDF of the solution stochastic process of problem (
Notice that the domains are defined in expression (
where
where
The authors declare that there is no conflict of interests regarding the publication of this paper.
This work has been partially supported by the Ministerio de Economía y Competitividad Grant MTM201341765P.