ABSTRACT
Uncertainty is ubiquitous in almost all branches of science and their applications including physics, chemistry, biology, engineering, environmental science and social science. Following the seminal work of [88], uncertainty is often classified into two types. One is epistemic (or reducible) uncertainty, which can possibly be reduced by improved measurements or improvements in the modeling process. The other is aleatory (or irreducible) uncertainty, which is the result of intrinsic variability/stochasticity of the system. Uncertainty propagation through a dynamic system has enjoyed consider-
able research attention during the past decade due to the wide applications of mathematical models in studying the dynamical behavior of systems. In this chapter, we consider uncertainty propagation in a continuous time dynamical system through two types of differential equations, where the dynamical system is described by the system of ordinary differential equations
x˙(t) = g(t,x(t)), x(0) = x0, (7.1)
with x = (x1, x2, . . . , xn) T , g = (g1, g2, . . . , gn)
T being n-dimensional nonrandom functions of t and x, and x0 being an n-dimensional column vector. One type of differential equation is a stochastic differential equation (SDE), a classification reserved for differential equations driven by white noise (defined later). The other one is a random differential equation (RDE, a term popularized for some years since its early use in [104, 105]), a classification reserved for differential equations driven by other types of random inputs such as colored noise (defined later) and both colored noise and white noise. One of the goals of this chapter is to give a short and introductory review
of a number of theoretical results on SDEs and RDEs presented in different fields with the hope of making researchers in one field aware of the work in other fields and to explain the connections between them. Specifically, we focus on the equations describing the time evolution of the probability density functions of the associated stochastic processes. As we shall see below, these equations have their own applications in a wide range of fields, such as physics, chemistry, biological systems and engineering. A second goal of this chapter is to discuss the relationship between the stochastic processes resulting from SDEs and RDEs.
