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.
Highlights
Introduction and MotivationOver 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
We state several versions of the random variable transformation (RVT) technique as well as some related results emerging from its applications that will play a relevant role in our subsequent developments
As the variation of Z2 given by (145) and (147), which depends on the sign of a1, a2, and a1a2, is controlled in terms of the data than of Z1, in order to facilitate in practice the limits of integration of the integral defining the 1-PDF f1(z, t), we will use formula (25) rather than (24)
Summary
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 case where Z0, B and A are pairwise dependent continuous RVs will be treated In such case, fZ0,B(z0, b), fZ0,A(z0, a), and fB,A(b, a) will denote the joint PDFs of the random vectors: (Z0, B), (Z0, A), and (B, A), respectively. In order to compute the 1-PDF f1(z, t), random variable transformation (RVT) method will be applied. 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.
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