Abstract
Deterministic differential equations are useful tools for mathematical modelling. The consideration of uncertainty into their formulation leads to random differential equations. Solving a random differential equation means computing not only its solution stochastic process but also its main statistical functions such as the expectation and standard deviation. The determination of its first probability density function provides a more complete probabilistic description of the solution stochastic process in each time instant. In this paper, one presents a comprehensive study to determinate the first probability density function to the solution of linear random initial value problems taking advantage of the so-called random variable transformation method. For the sake of clarity, the study has been split into thirteen cases depending on the way that randomness enters into the linear model. In most cases, the analysis includes the specification of the domain of the first probability density function of the solution stochastic process whose determination is a delicate issue. A strong point of the study is the presentation of a wide range of examples, at least one of each of the thirteen casuistries, where both standard and nonstandard probabilistic distributions are considered.
1. Introduction and Motivation
Over the last few decades, random differential equations (RDEs) have been demonstrated to be powerful tools to model numerous problems appearing in many different areas such as physics, engineering, economics, epidemiology, and hydrology. The consideration of randomness into their formulation through initial/boundary conditions, source terms, and/or coefficients adapts better than their deterministic counterpart to model the uncertainty associate to the experimental measurement required to set the above inputs as well as the inherent complexity involved in many real modelling problems. This approach leads to face new and exciting questions different from the corresponding ones appearing in the deterministic scenario. Indeed, instead of obtaining just the solution stochastic process (SP) of RDEs, the theory is also concerned with its probabilistic properties, mainly the computations of the expectation and variance functions. The computation of the first probability density function (1-PDF) of the solution SP, say , is much more desirable since, from it, one can compute the previous statistical functions as simple particular cases and, in addition, it provides a comprehensive probabilistic description of the solution SP for each time instant . However, the computation of the 1-PDF constitutes a major challenge that can been achieved in only a few cases.
The aim of this paper is to determine the 1-PDF, , of the solution SP to the linear random initial value problem (IVP): where the data , and are assumed to be continuous random variables (continuous RVs) defined on a common probability space , whose domains are assumed to be respectively. Hereinafter, in order to avoid cumbersome notation, we will hide the sample dependence when writing domains of continuous RVs. In this way, for instance, the domain will be written as rather than the first expression in (2). The same can be said for , , and the domain of any other RV throughout this paper. We allow the left (right) endpoint of each interval of the domain takes the value (); that is, we also consider unbounded continuous RVs. Throughout the paper, we will denote by , , and the PDFs of the continuous RVs , , and , respectively. The case where , and are pairwise dependent continuous RVs will also be treated. In such case, , , and will denote the joint PDFs of the random vectors: , , and , respectively. Finally, we will also deal with the case where , , and are dependent continuous RVs, then will represent their joint PDF. Notice that the domains of these two- and three-dimensional PDFs often can be written directly as products of the sets , , and given by (2).
In order to compute the 1-PDF , random variable transformation (RVT) method will be applied. RVT is a probability technique that allows us to calculate the PDF of a RV resulting after the algebraic transformation of another RV, say , whose PDF, , is known. In its simplest scalar formulation, the method reads as follows: if is a continuous RV lying on the domain or support , whose PDF is and being a monotone mapping on , then where is the inverse function of on , which is assumed to have a continuous derivative on and denotes the modulus of the derivative of . In the particular case that increases (decreases) on , the domain of is determined by ().
Notice that we are interested in computing the 1-PDF to the solution of (1) which is a SP rather than a RV, whereas RVT technique is mainly designed to handle (transformations of) continuous RVs. In order to take advantage of RVT, we first will fix and then we will apply RVT to the (transformed) RV . Therefore, we can say that RVT technique provides a time-transversal description of the 1-PDF .
Some of the earliest applications of RVT method to RDEs can be found in [1, ch.6] where this technique is applied to study a linear oscillator assuming randomness just in the two initial conditions related to the position and velocity. Most of the subsequent contributions have focused on the study of particular equations assuming specific probabilistic distributions for the involved uncertainty which facilitates the analysis. Here, we point out some recent contributions that illustrate quite well the current trends of RVT method in dealing with RDEs. In [2], authors solve the radiative transfer equation in a semi-infinite continuous stochastic medium with Rayleigh scattering. RVT method is applied to obtain the 1-PDF of the solution when the optical depth space variable is assumed to be a RV belonging to the following particular distributions: exponential and Gaussian. Higher order statistical moments of the solution stochastic process are also computed. An analogous study on the stochastic transport equation of neutral particles with anisotropic scattering can be found in [3]. In [4, 5], authors apply RVT technique to develop a stochastic finite element method for solving some stochastic problems with random excitation.
The application of RVT technique to the exact determination of the 1-PDF of the solution SP of RDEs requires the previous computation of the exact solution of the RDE under study. However, in the outstanding contribution [6], author takes advantage of RVT method together with classical numerical techniques to illustrate, through a wide range of examples, the potentiality of this method to approximate the 1-PDF for the solution SP of some RDEs.
As it has been announced previously, in this paper, we will compute the 1-PDF of the IVP (1) whose exact solution is available. For the sake of clarity, in the presentation, we will divide the study in the three main IVPs (I)–(III) listed in Table 1. In Case (I), we consider the homogeneous (H) problem whereas Cases (II) and (III) deal with the nonhomogeneous (NH) cases. Within each case, we distinguish, in a systematic manner, the different possibilities regarding the randomness of each of the involved parameters , , and . These casuistries include the situations where the parameters are statistically dependent. In this context, and as it has been pointed out previously, if for example and are statistically dependent, then will denote the joint PDF of the random vector , and the same can be said for the rest of the possibilities. The IVPs (I) and (II) can be seen as particular cases of the IVP (III) when and , respectively. When uncertainty can only be attributted to and , and the parameter can be set in a deterministic way, it is more realistic and convenient to assume that the joint PDF is known rather than . Notice that the construction of the joint PDF of only two continuous RVs from measured data can become a difficult problem which accuracy can deteriorate severely if one includes a new and inappropriate RV into its formulation [7]. The most accuracy of the PDF of the random input parameter, the best approximation of the 1-PDF of the solution SP of the IVP (1). Therefore, the consideration of all the thirteen separate cases listed in Table 1 turns out more recommendable from a practical point of view.
For the sake of clarity in the development of all the cases listed in Table 1, throughout this paper, the input parameters, which are assumed to be continuous RVs, will be denoted by upper cases, while deterministic magnitudes will be written by lower cases. More precisely, for instance, in Case III.4, we will denote by and the random inputs, while the multiplicative coefficient in the RDE will be denoted by rather than .
The paper is organized as follows. In Section 2 we summarize the main results concerned with RVT that will be applied throughout the paper. We particularly state different versions of this useful technique including its application to different transformations that will facilitate the presentation of the results. Sections 3, 4, and 5 provide a detailed study where the 1-PDF of the solution SP, , corresponding to the IVPs (I), (II), and (III) listed in Table 1, respectively, is computed. For each one of the thirteen casuistries, the study shows, at least, an illustrative example. The choice of the PDFs considered in the examples has been made to show the ability of the method to deal with both standard and non-standard dependent probability distributions. In Section 6 we include some considerations related to the application and better understanding of RVT method that we found particularly useful. Conclusions are drawn in the closing section.
2. Preliminaries
Below, we state several versions of the RVT technique as well as some related results emerging from its applications that will play a relevant role in our subsequent developments. Most of these results can be found in [8, 9] or they are a direct consequence of them.
Theorem 1 (RVT technique: scalar version). Let be a continuous RV with PDF and domain . Let be a new RV generated by the map which is assumed to be continuously differentiable on and such that except at a finite number of points. Let one suppose that, for each , there exist points: such that Then
Although Theorem 1 unifies the treatment of the different cases that one can present to compute the PDF, , in practice, it is easier to determine by dividing the domain of RV into subintervals where the mapping is monotone and then applying formula (3) on each subinterval. The process to compute on the whole domain of is completed by adding the corresponding expressions calculated previously for each subinterval.
In the simplest but significant case where the map is linear, Theorem 1 reads as follows.
Proposition 2 (RVT technique: linear transformation). Let be a continuous RV with domain and PDF . Then, the PDF of the linear transformation , is given by
If , then w.p. 1 stands for with probability 1 and
where denotes the Dirac delta distribution.
The following result is a direct application of Theorem 1 in the case that .
Proposition 3 (RVT technique: exponential transformation). Let be a continuous RV with domain and PDF . Then the PDF of the exponential transformation , with , is given by
If or , then with probability 1 and
The computation of the joint PDF of two or more continuous RVs using the RVT method can also be performed by using the following generalization of formula (3).
Theorem 4 (RVT technique: multidimensional version). Let be a random vector of dimension with joint PDF . Let be a one-to-one deterministic map which is assumed to be continuous with respect to each one of its arguments and with continuous partial derivatives. Then, the joint PDF of the random vector is given by where is the inverse transformation of : and is the Jacobian of the transformation; that is, which is assumed to be different from zero.
As we will see later, the analysis of Cases I-3, II-3, and III-3-6 requires the computation of the PDF of the sum and product of two continuous RVs which turns out by the application of Theorem 4 in its two-dimensional version. Thus, for the sake of clarity in the exposition, we specialize Theorem 4 in this significant case.
Theorem 5 (RVT technique: two-dimensional version). Let be a two-dimensional RV with joint PDF . Let be a one-to-one deterministic map from to ; that is, there exists its inverse transformation: on the range of the map (12). Let one assume that both maps (12) and (13) are continuous. Let further assume that the following partial derivatives exist and are continuous and the Jacobian of the inverse map satisfies on the range of the transformation (12). Then, the joint PDF of the two-dimensional RV is given by
Next, we apply Theorem 5 in the particular case that transformation only depends on variable and only depends on variable . As it will be seen later, this result will be crucial in further applications.
Proposition 6. Let be a two-dimensional RV with joint PDF . Let be a one-to-one deterministic map from to ; that is, there exists its inverse transformation: on the range of the map (17). Let one assume that both maps (17) and (18) are continuous and the two following derivatives that exist are continuous and satisfy on the range of the transformation (17). Then, the joint PDF of the two-dimensional RV is given by
On the other hand, applying Theorem 5 to and (or ) and and (or ), we obtain the PDF of the sum and product of two continuous RVs, respectively. We state both results in the two following propositions.
Proposition 7 (RVT technique: sum of two continuous RVs). Let be a continuous random vector with joint PDF and respective domains and . Then the PDF of their sum is given by or, equivalent, by If and are independent continuous RVs, since , where denotes the PDF of , , (21) and (22) inform one that the PDF of the sum of two independent continuous RVs is just the convolution of their respective PDFs:
Proposition 8 (RVT technique: product of two continuous RVs). Let be a continuous random vector with joint PDF with respective domains and . Then the PDF of their product is given by Equivalently, if and , then If and are independent continuous RVs with PDF's and , respectively, then (24) and (25) become respectively.
As usual we have not specified the domain of variation of in (24) and (25) since it is cumbersome. However, later we will detail it in some illustrative cases where it appears (see for instance Case III.6).
We close this section by extending Proposition 7 for the case of three terms since it will be required to deal with Case III.7. This result comes directly from the application of Theorem 4.
Proposition 9 (RVT technique: sum of three continuous RVs). Let be a continuous random vector with joint PDF and respective domains , , and . Then the PDF of their sum is given by or, equivalent, by or
3. Case Study: Initial Value Problem (I)
This section is devoted to obtain the 1-PDF of the solution SP to the IVP (I) in each of the three cases listed in Table 1. Notice that has the following expression:
3.1. Case I.1: Is a Random Variable
As we pointed out previously for the sake of clarity in the presentation, we rewrite (30) by distinguishing the deterministic character of parameter (which is written with a lower case letter): Next, we first fix and denote . Then we apply Proposition 2 to Then, taking into account that the domain of RV is given by (2), one gets where We illustrate the previous development in the following example where is assumed to be a standard continuous RV, although further distributions, not necessarily standard, could be considered.
Example 10. Let us assume that has a Gaussian distribution, , , and . Therefore, according to (33)-(34), the 1-PDF of is given by For each , the domain of has been determined taking into account in (34) that in this case and . It can be checked that is a PDF for each . Figure 1 shows at different values of in the particular case that , and , so and .
3.2. Case I.2: Is a Random Variable
First, we rewrite (30) by highlighting the deterministic character of the initial condition : Next, we fix and denote . In the first analysis, we will assume that . Then we apply Proposition 3 to Then, taking into account that and the domain of RV is given by (2), one gets where According to (2), if , then For , from (36) , which is a deterministic initial condition. Then its 1-PDF can be written through the Dirac delta function as follows:
In the example below, we illustrate the previous development in the case where .
Example 11. Let us assume that has a beta distribution, , , and . Therefore, according to (38)-(39), the 1-PDF of is given by where denotes the beta deterministic special function. Since and , it is guaranteed that . As a consequence, given by (42) is well defined. For each , the domain of has been determined taking into account in (39) that in this case and . It can be checked that is a PDF for each . Figure 2 shows at different values of in the particular case that , , and . For , according to (41), .
3.3. Case I.3: Is a Random Vector
Let us denote by the joint PDF of the random vector . Now, we rewrite (30) in the following equivalent form: In order to apply RVT, we fix and denote , , and . To compute the PDF of , first we will determine the joint PDF of and by applying Proposition 6 to Then taking into account one gets Once the joint PDF of has been determined, the computation of the PDF of follows directly by applying Proposition 8. Indeed, as , we will apply formula (25) to , and . This yields or equivalently by using (46): where Notice that if , then and is just the PDF of RV , which is easily obtained from the datum as the following marginal distribution:
Example 12. Let be a two-dimensional RV whose PDF is defined by Notice that we are implicitly assuming independence between and since the joint PDF factorizes as the product of the marginal PDFs , , and , . Then, taking into account (48)–(50) and (52), the 1-PDF of is given by For the sake of clarity in the graphical representation of , Figure 3 shows two equivalent plots of in the case that . Notice that the plot on the left side is the resultant surface where some 1-PDF have been highlighted for different fixed times, whereas these 1-PDFs have been represented in these fixed times on the right side. As the two integrals appearing into the expression (53) can be computed explicitly, an equivalent expression to with is given by For ,
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4. Case Study: Initial Value Problem (II)
This section is addressed to determine the 1-PDF of the solution SP to the IVP (II) in each of the three cases listed in Table 1. Notice that has the following expression:
4.1. Case II.1: Is a Random Variable
As we did in Case I.1, for the sake of clarity in the presentation, we rewrite (56) by emphasizing the deterministic character of parameter (which is written with a lower case letter): Next, we fix and denote . Then we apply Proposition 2 to Taking into account that the domain of RV is given by (2), one gets where
Example 13. Let us assume that has a gamma distribution, , . Therefore, according to (59)-(60), the 1-PDF of is given by: where denotes the deterministic gamma special function. For each , the domain of has been determined taking into account in (60) that, in this case, and . It can be checked that is a PDF for each . Figure 4 shows at different values of in the particular case that , , and .
4.2. Case II.2: Is a Random Variable
First, we rewrite (56) by highlighting the deterministic character of the initial condition : Next, we fix and denote . Then we apply Proposition 2 to This yields where For , as it also occurred in Case I.2, and therefore
Example 14. Let us assume that has a -distribution with degrees of freedom, , and . Therefore, according to (64)-(65), the 1-PDF of is given by For each , the domain of has been determined taking into account in (65) that in this case and . It can be checked that is a PDF for each . Figure 5 shows at different values of in the particular case that (3), , and . For , according to (66), .
4.3. Case II.3: Is a Random Vector
We will denote by the joint PDF of continuous RVs and . Let us rewrite (56) in the following equivalent form: In order to compute the 1-PDF of , , we first fix and consider the continuous RVs , , and . Then, we will apply Propositions 6 and 7. Indeed, in a first step we compute the joint PDF by applying Proposition 6 to Taking into account that one gets where and lie on Finally, we apply Proposition 7 using the following identification: , , and . This yields or more explicitly, using that and (71), where If , then and, similar to Case I.3, is just the PDF of RV , which is easily obtained from the datum as the following marginal distribution:
Example 15. Let be a two-dimensional vector whose joint PDF is given by
Then, carrying out the involved computations according to (74)-(75), the 1-PDF of is given by
Taking into account that , the value of this integral is
where
As we did in the example of Case (see Figure 3), for the sake of clarity, Figure 6 shows two equivalent plots of given by (79)-(80).
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5. Case Study: Initial Value Problem (III)
This section deals with the computation of the 1-PDF of the solution SP to the IVP (III) in each of the seven cases listed in Table 1. Notice that has the following expression: Throughout the sequent analysis depending on whether is considered to be a RV (Cases III.3, III.5, III.6, and III.7) or a deterministic constant (Cases III.1, III.2, and III.4), see Table 1; we will assume that or , respectively.
5.1. Case III.1: Is a Random Variable
In order to take advantage of the RVT method and to study this case, it is convenient to rewrite (81) in the equivalent form: Next, we first fix and denote . Then we apply Proposition 2 to Then, taking into account that the domain of RV is given by (2), one gets where
Example 16. Let be an exponential RV, , and let . Then, according to (84)-(85), the 1-PDF of is given by where the domain of has been determined taking into account in (85) that in this case and . Again, it can be checked that is a PDF for each . Figure 7 shows at different values of in the case that , , , and .
Before closing this case, we provide an example where the usefulness of the 1-PDF to determine the statistical moments of the solution and to compute the probability of sets of interest is shown. To illustrate these applications, we will choose the context of Example 16, although it could be applied to any example throughout this paper.
Example 17. Let us consider the context of Example 16 where the 1-PDF of , , has been computed (see expression (86)). Then, the statistical moment of order of with respect to the origin can be computed directly in terms of as follows: As a consequence, the following expressions for the mean and the variance of are obtained: Figure 8 shows the expectation and variance of for the same values of , , , and considered in Example 16.
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The computation of probabilities also can be performed directly through the 1-PDF. For instance, it may be of interest to compute the probability that the solution lies between two fixed values, say, and :
5.2. Case III.2: Is a Random Variable
Now we assume that only is a RV in the IVP (1) and denotes its PDF. For convenience, in this case, we rewrite (81) as follows: First, we fixed and denote . Then we again apply Proposition 2 to determine the 1-PDF of , taking into account that Notice that independently of the sign of the deterministic parameter . Then, where For , as it also occurred in Cases I.2 and II.2, and therefore
Example 18. Let us assume that follows a gamma distribution of parameters ; that is, . Therefore, according to (92)-(93), the 1-PDF of is given by For each , the domain of has been determined taking into account in (93) that in this case and . It can be checked that is a PDF for each . Figure 9 shows at different values of in the particular case that , , , and . For , according to (94), .
5.3. Case III.3: Is a Random Variable
So far we have observed that the application of RVT method to determine the PDF of a RV, say , generated by a transformation , relies strongly on the feasibility of computing the inverse of the map . Fortunately, this has been done exactly in the previous cases, but, in general, it is not possible in the current case where just is assumed to be a RV. In fact, once has been fixed, we must isolate in the equation: which is not possible to perform in an exact manner. To circumvent this drawback, we will apply the Lagrange-Bürmann formula which gives the Taylor series expansion of the inverse of an analytic function.
Theorem 19 (Lagrange-Bürmann formula, see [10]). Suppose that is defined as a function of the variable by an equation of the form: where is analytic about the point and . Then, it is possible to invert (or to solve) the equation for on a neighbourhood , of :
Although this result permits obtaining, from a theoretical stand point, the inverse function of , in practice, often this can only be achieved in an approximate manner since the infinite series (97) must be truncated to be kept computationally feasible. Moreover, the representation (97) of the inverse is only valid in a certain neighbourhood , of , whose diameter must be determined carefully in each case study.
Let us apply Theorem 19 to determine the 1-PDF of . Fixing and denoting , we consider the map (96) which is analytic about any numerical value , on the assumed domain to the RV . As the map (96), in general, is not monotone, we have to apply Theorem 1. In a first step, we divide the domain of the map (or equivalently, the domain of the RV ) into subintervals: where is monotone. Then, we will fix a such subinterval , where the contribution to the total 1-PDF is going to be calculated. For this, we select a point where condition is met. By applying the Lagrange-Bürmann formula, we construct the inverse of the map on that, in the sequel, will be denoted by : In practice, this infinite series could only converge on a subset of (if increases on or (if decreases on . In such case, the function can be completed on the whole interval by taking another (or other, if necessary) appropriate point(s), say , and then repeating the above process. In this manner, a piecewise inverse function on will be constructed.
In accordance with RVT method, besides constructing the inverse , one requires to compute its derivative. We again take advantage of Lagrange-Bürmann formula to complete this computation. In fact, notice that once has been constructed, from (98), one gets Notice that if has been defined by means of a piecewise function, then will also be defined in the same way.
The process will be culminated by repeating again the previous argument on all the subintervals . Following the previous development, the 1-PDF of can be computed as follows: where and are defined by (98) and (99), respectively.
Often, the infinite series (98) has to be truncated at the term to control computational burden. In this way, we obtain an approximation of the inverse: Thus, an approximation of its derivative is The infinite series (98) and (99) have the same convergence radius; however, often in practice, we need to handle their corresponding truncations (101) and (102), respectively. In this case, the selection of the appropriate depends also on the quality of the approximations provided by both truncated series.
Repeating the foregoing process on each interval , , one gets the corresponding approximation of given by
In the following example, we illustrate the previous development.
Example 20. Let us assume that has a beta distribution of parameters , , , , and . Figure 10 shows the approximation of at different values of . The approximation has been performed by (103), (101), and (102) with being because of monotony of . In order to carry out the computations, has been split into 7 subintervals in accordance with the process described previously. In each subinterval, an approximation of degree has been used.
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5.4. Case III.4: Is a Random Vector
Let us consider the IVP (1) and suppose that both and are continuous RVs with joint PDF . We rewrite (81) in the equivalent form: In order to compute the 1-PDF of , , we first fix and consider the continuous RVs , , and . Then, we will apply Propositions 6 and 7. Indeed, in a first step, we compute the joint PDF by applying Proposition 6 to and taking into account that This leads to where Notice that to determine the variation of , we have used that for each and . Finally, we apply Proposition 7 using the following identification: , , and . This yields or more explicitly, using (107), where , , , and are defined by (108).
If , then and, similar to Cases I.3 and II.3, is the PDF of RV . It is obtained as the following marginal distribution of the :
Example 21. Let be a two-dimensional Gaussian vector, , where
and denotes the correlation coefficient between and . Then, according to (110) and (108), the 1-PDF of is given by
where
Notice that to determine the integration limits in (113), we have considered in (108) that and . Figure 11 shows at different values of in the case that , , , , , , and .
For , as the marginal distribution of a bivariate Gaussian distribution is also a Gaussian distribution with mean and variance of the corresponding component of the random vector, one gets
5.5. Case III.5: Is a Random Vector
Let us consider the IVP (1) and now we assume that both and are continuous RVs with joint PDF . We rewrite (81) in the following equivalent form: In order to apply RVT, we fix and denote , , and . To compute the PDF of , first we will determine the joint PDF of and by applying Theorem 5 to Then taking into account , the involved Jacobian simplifies to Therefore, whereIn the case that , the computation of the domain of variation of and in (119) is more complicated. To express it, we first introduce the numbers: Then, after elaborated computations, one gets where Notice that and/or can become . In this case, the extremes of the intervals above that depend on and/or become , depending on the sign of parameter .
Once the joint PDF of has been determined, the computation of the PDF of follows directly by applying Proposition 7. Indeed, as (since into Case III, ; otherwise we would be in Case I), we will apply formula (22) to , , and . This yields where denotes the domain of variation of which, according to (120)–(124), depends on both the sign of the products and . Using (119), it is equivalent to The range of variation of , denoted by , can be straightforwardly computed taking into account and the domains of and which have been determined previously in (120)–(124). As its practical determination in specific examples is simple from the previous exposition, in order to avoid an unwieldy notation, we do not rewrite the final general expression.
Example 22. Let be a two-dimensional Gaussian vector, , where
And let denotes the correlation coefficient between and . Then, according to (126), (122), and (124) to determine the domain of , the 1-PDF of is given by
where
Hereinafter, we will assume that to facilitate our discussion through the example. We are in the case that , , and ; ; hence, , , and ; thus, and , which, according to (126), (122), and (124), determines the limits of integration of (128). Specifically,
Following an analogous reasoning, it is easy to check from (121) and (123) that and . Therefore, the domain of variation to is . As a consequence, in (128), lies in .
Figure 12 shows at different values of in the case that , , , , , , and .
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5.6. Case III.6: Is a Random Vector
Let us consider the IVP (1) and let us assume that and are continuous RVs with joint PDF . For convenience, we rewrite (81) in the following equivalent form: In order to apply RVT method, we fix and denote , , and . We first consider the case where . To compute the PDF of , first we will determine the joint PDF of and by applying Theorem 5 to As , in order to compute the Jacobian , it is enough to calculate the two following partial derivatives: which are well defined since , , and due to by hypothesis and . Also notice that . Moreover, taking into account that , one gets The values , , , and that determine the domain of variation of and can be computed taking into account that the function is positive and increasing on the whole real line in the current case where . This yieldsAs it happened in the foregoing Case III.5, for the determination of the domains of and , we will use the notations and introduced in (121). After non difficult but elaborated computations, one gets where
Finally, we apply Proposition 7 with the following identifications: , , and . As the variation of given by (136), (138) and (139) is easily controlled in terms of the data than of , in order to facilitate in practice the determination of the limits of integration of the integral which define the 1-PDF , we will use formula (22) rather than (21). This yields where is defined by (136) and (138) or (139) depending on the signs of and the product . As in Case III.5, we do not explicit the range of variation of , denoted by , since its writing is cumbersome but not difficult from previous exposition.
Now, we deal with the case where and we keep the assumption . For convenience, we rewrite (81) in the following equivalent form: To apply RVT method, we fix and denote , , and . To compute the PDF of , first we will determine the joint PDF of and by applying Theorem 5 to As , in order to compute the Jacobian , it is enough to calculate the two following partial derivatives: which are well defined since and , due to by hypothesis . Then, one gets Again, the values , , , and determining the domain of variation of and can be computed taking into account that the function is increasing on the whole real line in the current case where . After non difficult but elaborated computations, this yields, where
Next, we apply Proposition 8 with the following identifications: , , and . As the variation of given by (145) and (147), which depends on the sign of , , and , is easily controlled in terms of the data than of , in order to facilitate in practice the limits of integration of the integral defining the 1-PDF , we will use formula (25) rather than (24). This yields where denotes the domain of variation of . Again, we do not explicit the range of variation of , denoted by , since its writing is cumbersome but simple from previous exposition.
Finally, we consider the case that implies . Therefore,
Example 23. With the aim of showing the generality of the obtained results to deal with different classes of continuous RVs, in this example, we will consider that the joint PDF of the input continuous RVs and is constructed by a copula function. Let and be two uniform continuous RVs on the interval ; that is, . Using the Farlie-Gumbel-Morgenstern copula [11], we construct the random vector with joint PDF hence, and are statistically dependent. As a characteristic of copula functions, notice that the two marginal distributions of are just the individual distributions of and . In the following, we will consider the case previously developed for and we will take . First, notice that according to (141) and taking into account that , the domains of variation of and are and , respectively; thus, lies on . However, we must refine the domain of integration to of (148), in such a way that the two arguments of the PDF lie inside the interval , where the input continuous RVs and are defined. Thus, fixing and , we must determine such as hold. By (141), notice that in our context , with ; hence, second inequality in (151) is guaranteed. As , the first inequality in (151) is equivalent to . Therefore, in (148) is given by , where is the solution of the equation . At this point, notice that the function is decreasing for and this justifies the consideration of the previous term: . To summarize, the 1-PDF of is given by where . In Figure 13, we have plotted at different values of .
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5.7. Case III.7: Is a Random Vector
Finally, we consider the IVP (1) and assume that every input is randomized; that is, is a random vector with joint PDF . For convenience, we rewrite (81) in the following equivalent form: In order to apply RVT method, we fix and denote , , , and . To compute the PDF of , first we will determine the joint PDF of , , and by applying Theorem 4 to Notice that so and are well defined as and are continuous RVs such as and , respectively. After some computations, it is easy to check that the absolute value of the Jacobian required to apply Theorem 4 is given by Notice that we have used . Therefore, Now, we will apply Proposition 9 with the following identifications: , , , and . As the variation of and is easily controlled in terms of the data than of , in order to facilitate in practice the specification of the limits of integration of the integral defining the 1-PDF , we will use formula (27). This yields for and . As in the previous case, we do not detail the range of variation of , , since it is very involved.
Finally, we consider the case . From (1), one gets . Therefore,
Example 24. Let be a three-dimensional Gaussian vector, , where is a vector in which represents the mean and is a symmetric positive definite real matrix of size usually referred to as the variance-covariance matrix. Then, according to (157), the 1-PDF of is given by where In the following, we will take : Now, we do not provide an explicit expression of since it is very sophisticated. Figure 14 shows at different values of ().
(a)
(b)
(c)
(d)
6. Some Final Remarks
In this section, we will point out some considerations related to the practical application of RVT method in dealing with the computation of the 1-PDF.
Remark 25. Throughout this paper, we have determined the 1-PDF of the solution SP to IVP (1) in all the cases listed in Table 1. It must be pointed out that it has depended heavily on doing an appropriate choice of the involved variables when applying the RVT method. To illustrate this statement, let us consider the foregoing Case III.5 when and are the only input continuous RVs. When we dealt with this case, we first decomposed the solution SP (81) in the form given by (116) and then we applied Theorem 5 as is shown in (117). This decomposition was carefully chosen to guarantee the successful application of Theorem 5. In fact, alternative decompositions of (81) are possible; however, they could not be adequate to achieve our goal. For example, if we rewrite (81) in the following form: then, the application of Theorem 5 to does not lead to fruitful results since we cannot isolate and and this would ruin our goal. In other cases, we can find out that two or more choices do not have the previous drawback and then the best selection will be the one which facilitates the involved computations (the easiest expression for the Jacobian, the simplest determination of the domains, etc.).
Remark 26. Although in this paper we have concentrated on the determination of the 1-PDF , which describes the probabilistic behaviour of the solution SP at each time instant , it must be pointed out that the computation of higher PDFs is also possible and desirable. In fact, for instance, the -PDF, say, provides a whole probabilistic information of in two time instants and . Therefore, from it, we can calculate relevant probabilistic properties such as the correlation function As it can be guested, from the previous development, the computation of follows in broad outline that . In Example 27, we compute the -PDF within the framework of foregoing Case II.3 including the correlation function.
Example 27. Let us fix such as and denote and . We want to determine the joint PDF of continuous RVs and . For it, we will apply Theorem 5 to where the expression of the solution SP given by (56) has been considered. Now, taking into account that then, the involved Jacobian is given by This leads to where
In spite of having computed the 2-PDF in the previous example, it must be noticed that, in general, its determination to problem (1) becomes very difficult even prohibitive, particularly the specification of the associated domains.
Remark 28. In some situations, the determination of the 1-PDF provides the full probabilistic information of the solution SP; hence, the computation of higher PDFs is not necessary. This is illustrated in the following example.
Example 29. Let the IVP be considered in Case III.1 where the only random input is the initial condition . We know that Its 1-PDF is given by (84). From it, we can compute any probabilistic information for every time instant, say . Let us consider another time instant, say . Notice that, in this case, the solution SP at can be represented as follows: From this expression, we see that the behaviour of the solution at the time instant is deterministically given by (a linear transformation of) . Therefore, the computation of the 2-PDF is not required. This can be confirmed from another point of view. Let us assume without loss of generality that the expectation of the initial condition is zero: and its variance is . From (170), it is easy to check that Then, the correlation coefficient function is given by
7. Conclusions
In this paper, we have determined the first probability density function (1-PDF) of the solution stochastic process of the linear random differential equation taking advantage of random variable transformation (RVT) method. The study has been made in a systematic way in order to facilitate and clarify the development and exposition of the results as well as to facilitate their practical use. The wide range of the exhibited examples, that include both standard and non-standard probabilistic distributions, show the ability of RVT technique to deal with the computation of the 1-PDF to models based on linear random differential equations. Notice that throughout the paper no independence among the random parameters has been assumed in order to achieve general results.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This work has been partially supported by the Ministerio de Economía y Competitividad under Grant no. DPI2010-20891-C02-01 and Universitat Politècnica de València under Grant no. PAID06-11-2070.