
# 15. Stochastic Processes

#### Summary

A stochastic process is a collection of random variables $$\bs{X} = \{X_t: t \in T\}$$ defined on a common probability space, taking values in a common set $$S$$ (the state space), and indexed by a set $$T$$, often either $$\N$$ or $$[0, \infty)$$ and thought of as time (discrete or continuous respectively). In the first section, we discuss questions of existence and construction, and various types of equivalence of random processes. In the second section we discuss filtrations, which are collections of $$\sigma$$-algebras, also indexed by $$T$$, that represent our state of knowledge as the process evolves. We also discuss stopping times which are random elements of $$T$$ that do not involve having to look into the future, in a sense. The subsequent section explore some broad categories of random processes. In part, this chapter serves as a foundation for the study of other special random processes, particularly Markov chains and Brownian motion.

#### Quote

• When in disgrace with Fortune and men's eyes
I all alone beweep my outcast state ...
—Shakespeare, Sonnet 29