Psychological Aspects of Market Crashes

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 S^t (t ∈ N ∪ {∞}) the t-Cartesian product of S. For every
history st ∈ S^t (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 S^t.¹ The sequence
(Γt)t∈N generates a filtration, and define Γ to be the σ -algebra generated
by ∪_t∈NΓ_t. 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 ∈ S^t
(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 every^s ∈ 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 .

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