Research Article | Open Access
M.-C. Casabán, J.-C. Cortés, J.-V. Romero, M.-D. Roselló, "Random First-Order Linear Discrete Models and Their Probabilistic Solution: A Comprehensive Study", Abstract and Applied Analysis, vol. 2016, Article ID 6372108, 22 pages, 2016. https://doi.org/10.1155/2016/6372108
Random First-Order Linear Discrete Models and Their Probabilistic Solution: A Comprehensive Study
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 . 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 , [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 . 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 , 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 , 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 , it is important to point out that both problems have distinctly different nature. Indeed, model (2) is discrete whereas problem faced in  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 , the authors study the mean square exponential stability of impulsive stochastic difference equations. In , one studies random matrix linear difference equations assuming that diffusion coefficient in (2) is a deterministic matrix rather than a RV. In , 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 , 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  to highlight better analogies and differences between both models.
The paper is organized as follows. Section 2 is addressed to introduce the preliminaries related to RVT technique required to conduct our study. In this section, we establish a key result related to the PDF of the power transformation of RVs which will be crucial to deal with Case I.2 of Table 1, where uncertainty just enters in model (2) through the RV . Section 3 is divided into three subsections where the 1-PDF of the discrete solution SP of (2) is determined for each one of Cases I, II, and III listed in Table 1. Illustrative examples covering the thirteen cases are provided throughout the paper. In the last section, we present our conclusions. Finally, we present an appendix where the main obtained results are collected in order to facilitate their practical use.
As it has been pointed out in the previous section, the goal of this paper is to compute the -PDF of the solution SP of problem (2) in each one of the cases listed in Table 1. The key result to achieve this goal is the RVT method. This is a probabilistic technique that allows us to calculate the PDF of a random variable/vector which is obtained after mapping another random variable/vector whose PDF is known. Depending on the type of mapping as well as its dimension, several versions of RVT method can be established. Throughout this paper, the general scalar version and its specialization to the linear case, as well as the general multidimensional version, will be required. These results are stated in [20, Theorem 1, Proposition 2 and Theorem 4], respectively.
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 1. It is important to underline the notion that power transformation is a distinctive feature to describe the solution SP of the discrete model (2) against the exponential transformation which appears when its continuous counterpart is dealt with. In this sense, the next result plays the same role as [20, Proposition 3] performed there.
Proposition 1 (RVT technique: power transformation). Let be a continuous RV with domain and PDF . Let one denote by the Dirac delta function. Then, the PDF of the power transformation , with and , is given by the following:
(i) If , (ii) If ,(iii) If is even, then one has the following:
Case 1 ():Case 2 ():Case 3 ().
Case 3.1 (): where on the domainCase 3.2 (): where on the domain(iv) If and is odd, then one has the following:
Case 1 ( or ):Case 2 ():If , for ,
Proof. (i) If , then w.p. and its PDF is given by(ii) If , the mapping is monotone on the whole domain of RV ; then the inverse function of takes the formwhose derivative is given byThen, applying expression [20, Eq. (3)] and taking into account (18)-(19), one gets(iii) Let us assume is even. We distinguish three cases depending on the domain of the RV . In order to avoid any confusion, it is important to note that these cases are mutually exclusive.
Case 1 (). Assuming , the mapping is monotone in . Hence, the inverse function of , denoted by , takes the formand its derivative, , is given byThen, applying [20, Eq. (3)] and taking into account (21)-(22), one getsWhen , the reasoning is analogous. Then, considering expression (23) when , both cases for the sign of can be expressed as follows:on the domainCase 2 (). In this case, the mapping is monotone in . Using an analogous development as we did in Case 1, one obtains on the domainCase 3 (). We will consider two subcases: and . We split each subcase into appropriate subintervals in order to apply [20, Theorem 1] in order to compute the PDF.
Case 3.1 (). Let us consider the piece . On the subinterval , the mapping (denoted by , for the sake of clarity) is monotone and its inverse iswhose derivative, , for , is given byOn the other hand, on the piece , its corresponding mapping is monotone and its inverse isand, for ,Notice that and if . Then, applying [23, Theorem 2.1.8] and taking into account (28)–(31), one getson the domainAs usual, we assume outside domain (33).
To complete the computation of PDF on the whole domain, finally we consider the subinterval , where the RV is positive. Hence, we are in Case 1 and according to (24) it follows thaton the domainAgain, as usual, we assume outside domain (35). Notice that one satisfies To summarize, from (32)–(35), the complete PDF of in this case is given by Case 3.2 (). Let us consider the piece . Following analogous reasoning as in Case 3.1, according to (32), one obtains the piece of the PDF of the power transformation :on the domainWe assume outside domain (39).
We complete the computation of PDF on the whole domain considering the subinterval . In this subinterval, is negative. As it was shown in Case 2, the PDF is given byon the domainWe assume outside domain (41).
To summarize, from (38)–(41), the complete PDF of in this case is(iv) Let us assume that and is odd. The mapping is monotone on the whole domain of RV ; then the inverse function of takes the formwhose derivative, for , is given byNotice that if . Therefore, we distinguish two cases depending on the domain of the RV .
Case 1 ( or ). Applying [20, Theorem 1] and taking into account (43)-(44), one getsCase 2 (). Applying [20, Theorem 1] and taking into account (43)-(44), one getsFinally, if and , then with probability (w.p. ) and its PDF is given by
3. Case Study: Homogeneous Discrete Initial Value Problem (I)
This section is addressed to compute the -PDF of the solution discrete stochastic process of the homogeneous discrete initial value problem (I) in all different cases collected in Table 1. In this case, the solution can be expressed as follows:
3.1. Case I.1: Is a Random Variable
For the sake of clarity in the presentation, we rewrite solution (48) by using the lowercase letter in order to indicate the deterministic character of parameter :
Let be an arbitrary and fixed integer. Let us assume and denote . By applying [20, Proposition 2] to one gets the 1-PDF
Note that if , from (49), it follows that w.p. for each and . So, the 1-PDF for the trivial case, , can be written as
In order to facilitate the comparison of the -PDF of the solution of problem (49) against its continuous counterpart provided in , in the following example, is assumed to be a Gaussian RV. Note that we are going to consider a standard distribution, although the method is also able to be applied to nonstandard distributions.
Example 1. Let us assume and consider a Gaussian distribution, . Hence, applying (51), the -PDF of is given by It can be checked that is a PDF for each . Figure 1 shows for , in the particular case that : (a) and (b). Note the different behavior of 1-PDF depending on the modulus of the parameter . This is in agreement with the expectation and variance of the solution which are given, respectively, by Indeed, in Figure 1, we observe that, for each , the 1-PDF is symmetric about , whereas, in the case that (), it becomes flat (sharp) as increases. This means that its variability around zero, that is, the variance, tends to infinity (zero).
3.2. Case I.2: Is a Random Variable
In order to emphasize the deterministic nature of the initial condition , we recast (48) by using the lowercase letter :
Let be an arbitrary and fixed integer and denote . The RV represents power transformation of RV ; that is, can be written as . By applying Proposition 1 to , , and , one obtains the 1-PDF . For the sake of clarity, we do not provide the corresponding explicit expression for since it just consists of substituting the previous identification. Below, we show an illustrative example.
Example 2. Let us assume that has a uniform distribution on the interval , , and . Therefore, according to Proposition 1, the -PDF of is given by It can be checked that is a PDF for each . Figure 2 shows at different values of .
This example exhibits a different behaviour of the -PDF of the solution of (55) depending on whether is odd or even.
3.3. Case I.3: Is a Random Vector
Throughout this case, the joint PDF of the random vector will be denoted by . Let be an arbitrary and fixed integer and denote . To compute the PDF of , first we will determine the joint PDF of the RVs and by applying [20, Theorem 4] to the two-dimensional RV withFrom (57), the inverse transformation of , , takes the form Taking into account the fact that , the involved Jacobian simplifies to . Therefore, the joint PDF is given by Going back to the original RVs, that is, and , one getsFinally, considering the marginal density function of in (60), the 1-PDF of is given by
Example 3. Let be a random vector and let us assume that its PDF is given byBy (61)-(62), the following -PDF of is obtained: Figure 3 shows for . From , we observe that the density of probability accumulates around , which is in agreement with the asymptotic behaviour of the solution which tends to zero as .
4. Case Study: Nonhomogeneous Discrete Initial Value Problem (II)
In this section, we deal with the computation of the -PDF of the solution discrete SP of the nonhomogeneous discrete initial value problem (II). This will be done for every one of the cases considered in Table 1. Now, the solution has the following form:As we did in Section 3 and for the sake of clarity in the presentation, we will recast the input parameters in (64) by lowercase letters when they indicate deterministic quantities.
4.1. Case II.1: Is a Random Variable
Example 4. Let be a gamma RV of parameters , . Then, by (67), the -PDF of reads where means the classical gamma function. Notice that, for each , the domain of follows from the corresponding domain of a gamma distribution. In Figure 4, the -PDF is plotted at different values of considering and . In this case, from the plot of , one observes that the expectation increases as does. It is straightforward to check that the expectation lies on the straight line , whereas the variance takes the constant value 2.
4.2. Case II.2: Is a Random Variable
In this case, the solution discrete stochastic process (64) takes the form For , and the PDF is . If is an arbitrary and fixed integer, denoting , the PDF is obtained by applying [20, Proposition 2] toThis leads to the 1-PDF
Example 5. Let be a RV with -distribution with degrees of freedom, , . Then, by (71), the -PDF of writesFor each , the domain of has been determined considering the domain of -distribution with degrees of freedom. For the sake of clarity, Figure 5 shows a 2D plot (a) and a 3D plot (b) of at different values of in the particular case that and .
4.3. Case II.3: Is a Random Vector
In accordance with the notation previously introduced, stands for the joint PDF of the random vector . Let be an arbitrary and fixed integer and denote . To compute the PDF of , first we will determine the joint PDF of the RVs and by applying [20, Theorem 1] to the two-dimensional RV withFrom (73), the inverse transformation of : takes the form By [20, Theorem 4] and taking into account the fact that , the joint PDF is given by Going back to the original RVs, that is, and , one gets Finally, considering the marginal density function of , one gets the 1-PDF of :
Example 6. Let us consider the random vector whose joint PDF is defined byBy (77), the -PDF of is Computing the above integral, one getswhereFigure 6 shows two equivalent plots of given by (80)-(81). It is straightforward to check that the expectation and variance of the solution are given by respectively. Notice that the values of the expectation and variance obtained from expressions (82) agree with the plots shown in Figure 6, where one observes that the 1-PDF is, for each , symmetric about and its support increases as tends to infinity in such a way that the -PDF’s shape becomes flattened. Then, the variability about zero, in this case the variance, increases as does.
5. Case Study: Nonhomogeneous Discrete Initial Value Problem (III)
This section is addressed to determine the -PDFs of the solution SPs of problem (III) in Cases III.1–III.7 collected in Table 1. Now, the solution has the following form:which is well defined due to the hypothesis .
Analogously to the previous sections, for the sake of clarity in the presentation, we will rewrite each one of the involved parameters in (83) by lowercase letters when it denotes a deterministic quantity.
5.1. Case III.1: Is a Random Variable
In this case, if , solution (83) takes the formLet be an arbitrary and fixed integer and denote . The application of [20, Proposition 2] topermits computing the 1-PDF. This yieldsFor the trivial case , solution (83) takes the form and hence the 1-PDF is given by
Example 7. Let be an exponential RV of parameter , . Then, by (86), the -PDF of writes where the domain has been determined taking into account the domain of an exponential RV. Figure 7 shows at different values of depending on whether is odd (a) or even (b) for , , and .