Theory of Statistics - George Mason University

126 1 Probability Theory
If {Xt} is adapted to the filtration {Ft}, we often write Xt ∈ Ft. We also
call the process {Xt} nonanticipating, for obvious reasons.
Definition 1.59 (filtered probability space)
Given a probability space (Ω, F, P) and a filtration {Ft} of sub-σ-fields of F,
we form the filtered probability space (Ω, F, {Ft : t ∈ [0, ∞[}, P).
1.6.2 Continuous Time Processes
For a stochastic process over a continuous index set I we must be concerned
about the continuity of the process in time. The problem arises because the
countably-additive property of a measure (equation (0.1.8)) does not carry
over to uncountable unions. For a process X(t, ω) where t is in uncountable
index set, say, for example, an interval, we will be faced with the necessity
to evaluate probabilities of sets of the form ∪t≥0At. Such unions are not
necessarily in the underlying σ-field.
** continuation motivation
We can define continuity of X(t, ω) on I in the usual way at a given point
ω0 ∈ Ω. Next, we consider continuity of a stochastic process over Ω.
Definition 1.60 (sample continuous)
Given a probability space (Ω, F, P) and a function
X : I × Ω ↦→ IR,
we say X is sample continuous if X(ω) : I ↦→ IR is continuous for almost all
ω (with respect to P).
The phrase almost surely continuous, or just continuous, is often used instead
of sample continuous.
*** add more ... examples
The path of a stochastic process may be continuous, but many useful
stochastic processes are mixtures of continuous distributions and discrete
jumps. In such cases, in order to assign any reasonable value to the path
at the point of discontinuity, we naturally assume that time is unidirectional
and the discontinuity occurs at the time of the jump, and then the path evolves
continuously from that point; that is, after the fact, the path is continuous
from the right. The last value from the left is a limit of a continuous function.
In French, we would describe this as continu à droite, limité à gauche; that is
cadlag. Most models of stochastic processes are assumed to be cadlag.
1.6.3 Markov Chains
The simplest stochastic process is a sequence of exchangeable random variables;
that is, a sequence with no structure. A simple structure can be imposed
by substituting conditioning for independence. A sequence of random
Theory of Statistics c○2000–2013 James E. Gentle