## Abstract

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*.

## Keywords

Correlation Function Brownian Motion Poisson Process Covariance Function Compound Poisson Process## Preview

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