# 4.6 Stochastic Processes

Measure time *t* in appropriate units—days, months, years. A **time series** is a series {^{–α}** x**, … ,

^{–1}

**,**

*x*^{0}

**} of**

*x**n*-dimensional vectors observed iteratively over a period of time [–α, 0]. The natural number

*n*is called the

**dimensionality**of the time series. If

*n*= 1, the time series is

**univariate**; otherwise it is

**multivariate**. Vectors

*are recorded with a frequency corresponding to a single unit of time. These concepts are illustrated in Exhibits 4.5 and 4.6 with a two-dimensional time series of daily high and low temperatures at the summit of Mt. Washington during the month of January 2015.*

^{t}**x***t*= –30 corresponds to January 1, 2015. Source: Mt. Washington Observatory.

Often, each value * ^{t}x* of a time series

**corresponds to an observation made precisely at time**

*x**t*, but this is not always the case. In our Mt. Washington example, each day’s high and low temperatures

^{t}x_{1}and

^{t}x_{2}are realized at different times during a given day, but we associate them both with the specific integer point in time

*t*.

Presumably, the process that generated a time series will continue into the future. We are interested in future values, which we treat as random. To model these, we specify a model called a stochastic process based upon the time series. A** stochastic process**—or **process**—is a sequence of random vectors ^{t}**X** with *t* taking on integer values.3 Values *t* extend back to –∞ and forward to ∞. Modeling all these terms may seem excessive, especially for practical work. We do so as a mathematical convenience. It saves us having to artificially model some initial or terminal behavior for the process.

Given a time series {^{–α}** x**, … ,

^{–1}

**,**

*x*^{0}

**}, we construct a stochastic process {… ,**

*x*^{–1}

**,**

*X*^{0}

**,**

*X*^{1}

**, …} by treating the time series as a single realization of the corresponding segment {**

*X*^{–α}

**, … ,**

*X*^{–1}

**,**

*X*^{0}

**} of the stochastic process. We apply statistical techniques to specify a stochastic process that is consistent with such a realization. The undertaking is called**

*X***time series analysis**.

###### Exercises

What is the difference between a time series and a stochastic process?

Solution

In our Mt. Washington example, each recorded high temperature ^{t}x_{1} is associated with the specific integer points in time *t*, but might have been realized at any point during a 24-hour period. Give two examples of financial time series:

- one for which values
are actually realized at time^{t}x*t*, and - another for which values
may be realized at any time in an interval that we associate with time^{t}x*t*.