Random Differential Inequalities (Enhanced Edition‪)‬

This chapter is essentially introductory in nature. Its main purpose is to
introduce some basic probabilistic concepts that are essential to the study
of random differential equations. We list some known results from standard
textbooks and sketch some useful results that are not so well known.
Sections 1.1 and 1.2 are concerned with events, probability measure, random
variables, expectations, and distribution functions. Section 1.3 is devoted to
convergence of random sequences depending on various modes of convergence.
Several important and well-known theorems about various kinds
of convergence are stated. Furthermore, some results concerning the weak
convergence of measures are also included. A brief discussion about conditional
probabilities and expectations is given in Section 1.4. Section 1.5
surveys certain fundamental notions and results in the general theory of
stochastic processes. We sketch an important concept due to Doob, namely,
the concept of separability of random processes in Section 1.6. Finally, we
collect in Section 1.7 some basic deterministic comparison results.