The Mathematics of Financial Modelingand Investment Management

11-FinEcon-Time Series Page 284 Wednesday, February 4, 2004 12:58 PM

284 The Mathematics of Financial Modeling and Investment Management

cations. Spacing of time points might also be irregular but deterministic.

For instance, week-ends introduce irregular spacing in a sequence of

daily closing prices. These questions can be easily handled within the

context of discrete time series.

The diffusion of electronic transactions has made available high-fre-

quency data related to individual transactions. These data are randomly

spaced as the intervals between two transactions are random variables. If

one wants to consider randomly spaced time intervals, discrete-time

models will not suffice; one must use either marked point processes (dis-

cussed briefly in Chapter 13) or continuous-time processes through the

use of master equations. In this chapter and the next we discuss only

time series at discrete and fixed intervals of time. Here we introduce con-

cepts, representations, and models of time series. In the next chapter we

will discuss model selection and estimation.

CONCEPTS OF TIME SERIES

A time series is a collection of random variables Xt indexed with a dis-

crete time index t = ...–2,–1,0,1,2,.... The variables Xt are defined over a

probability space (Ω,P,ℑ), where Ω is the set of states, P is a probability

measure, and ℑ is the σ-algebra of events, equipped with a discrete fil-

tration {ℑt} that determines the propagation of information (see Chapter

6). A realization of a time series is a countable sequence of real num-

bers, one for each time point.

The variables Xt are characterized by finite-dimensional distributions

(see the section on stochastic processes in Chapter 6) as well as by condi-

tional distributions, Fs(xs/ℑt), s > t. The latter are the distributions of the

variable x at time s given the σ-algebra {ℑt} at time t. Note that condition-

ing is always conditioning with respect to a σ-algebra though (see Chap-

ter 6) we will not always strictly use this notation and will condition with

respect to the value of variables, for instance:

Fs(xs/xt), s > t

If the series starts from a given point, initial conditions must be fixed.

Initial conditions might be a set of fixed values or a set of random vari-

ables. If the initial conditions are not fixed values but random variables,

one has to consider the correlation between the initial values and the ran-

dom shocks of the series. A usual assumption is that the initial conditions

and the random shocks of the series are statistically independent.