Stochastic Processes

In the previous chapter we defined a time series or a discrete time stochastic process as a countable family of random variables X = (X 1, X 2,…). Time series provide adequate models in many applications. For example, X n may denote the damage of a physical system after n loading cycles or the value of a stock at the end of day n. However, there are situations in which discrete time models are too coarse. For example, consider the design of an engineering system subjected to wind, wave, and other random forces over a time interval I. To calculate the system dynamic response, we need to know these forces at each time t ∈ I. The required collection of force values is an uncountable set of random variables indexed by t ∈ I, referred to as a continuous time stochastic process or just a stochastic process. We use upper case letters for all random quantities. A real-valued stochastic process is denoted by {X(t), t ∈ I} or X. If the process takes on values in ℝ d , d > 1, we use the notation {X(t), t ∈ I} or X.