2 The model
In this section, a formal description of the model is given. Time is discrete
and continues forever. In every period t ∈ N+ , a state is drawn by nature
from a set S = {1, ..., L}, where L is strictly greater than 1. Before defining
how nature draws the states, we first need to introduce some notations.
Denote by St (t ∈ N ∪ {∞}) the t-Cartesian product of S. For every
history st ∈ St (t ∈ N), a cylinder with base on st is defined to be the
set C(st) = {s ∈ S∞ s = (st,...)} of all infinite histories whose t initial
elements coincide with st. Define the set Γt (t ∈ N) to be the σ-algebra
which consists of all finite unions of cylinders with base on St.1 The sequence
(Γt)t∈N generates a filtration, and define Γ to be the σ -algebra generated
by ∪t∈NΓt. Given an arbitrary probability measure Q on (S∞ , Γ), we define
dQ0 ≡ 1 and dQt to be the Γt-measurable function defined for every st ∈ St
(t ∈ N+) as
dQt(s) = Q(C(st)) where s = (st, ...).
Given data up to and at period t - 1 (t ∈ N), the probability according
to Q of a state of nature at period t, denoted by Qt, is
Qt( s ) = dQt( s ∖for everys ∈ S ∞,
dQt-1(s)
with the convention that if dQt-1 (s)=0 then Qt(s) is defined arbitrarily.
In every period and for every finite history, nature draws a state of nature
according to an arbitrary probability distribution P on (S∞ , Γ). To simplify
1The set Γ0 is defined to be the trivial σ-algebra, and Γ-1 = Γ0 .