Representations and classifications of stochastic processes
Authors: Johnson Dudley Paul 
Published: July 2010

We show that to every stochastic process X one can associate a unique collection
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis normal upper Phi comma normal upper Phi Subscript plus Baseline comma upper T left-parenthesis t right-parenthesis comma upper E left-parenthesis upper U right-parenthesis comma p Superscript asterisk Baseline right-parenthesis">

(
Φ
,

Φ
+

,
T
(
t
)
,
E
(
U
)
,

p
∗

)

(\Phi ,{\Phi _ + },T(t),E(U),{p^\ast })

consisting of a linear space

Φ
\Phi

, on which is defined a linear functional

p
∗

{p^ \ast }

, together with a convex subset

Φ
+

{\Phi _ + }

which is invariant under the semigroup of operators

T
(
t
)

T(t)

and the resolution of the identity

E
(
U
)

E(U)

. The joint distributions of X, there being one version for each

ϕ
∈

Φ
+

\phi \in {\Phi _ + }

, are then given by
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper P Subscript phi Baseline left-parenthesis upper X left-parenthesis t 1 right-parenthesis element-of upper U 1 comma midline-horizontal-ellipsis comma upper X left-parenthesis t 1 plus midline-horizontal-ellipsis plus t Subscript n Baseline right-parenthesis element-of upper U Subscript n Baseline right-parenthesis equals p Superscript asterisk Baseline upper E left-parenthesis upper U Subscript n Baseline right-parenthesis upper T left-parenthesis t Subscript n Baseline right-parenthesis midline-horizontal-ellipsis upper E left-parenthesis upper U 1 right-parenthesis upper T left-parenthesis t 1 right-parenthesis phi period">

P
ϕ

(
X
(

t
1

)
∈

U
1

,
⋯
,
X
(

t
1

+
⋯
+

t
n

)
∈

U
n

)
=

p
∗

E
(

U
n

)
T
(

t
n

)
⋯
E
(

U
1

)
T
(

t
1

)
ϕ
.

{P_\phi }(X({t_1}) \in {U_1}, \cdots ,X({t_1} + \cdots + {t_n}) \in {U_n}) = {p^ \ast }E({U_n})T({t_n}) \cdots E({U_1})T({t_1})\phi .

\]
To each

ϕ
\phi

contained in the extreme points

Φ

+
+

{\Phi _{ + + }}

of

Φ
+

{\Phi _ + }

and each time t we find a probability measure

P
t
∗

(
ϕ
,
⋅
)

P_t^ \ast (\phi , \cdot )

on

Φ

+
+

{\Phi _{ + + }}

such that
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T left-parenthesis t right-parenthesis phi equals integral Underscript normal upper Phi Subscript plus plus Baseline Endscripts psi upper P Subscript t Superscript asterisk Baseline left-parenthesis phi comma d psi right-parenthesis">

T
(
t
)
ϕ
=

∫

Φ

+
+

ψ

P
t
∗

(
ϕ
,
d
ψ
)

T(t)\phi = {\smallint _{{\Phi _{ + + }}}}\psi P_t^ \ast (\phi ,d\psi )

.

P
t
∗

P_t^ \ast

is the transition probability function of a temporally homogeneous Markov process

X
∗

{X^ \ast }

on

Φ

+
+

{\Phi _{ + + }}

for which there exists a function f such that

X
=
f
(

X
∗

)

X = f({X^ \ast })

. We show that in a certain sense

X
∗

{X^ \ast }

is the smallest of all Markov processes Y for which there exists a function g with

X
=
g
(
Y
)

X = g(Y)

. We then apply these results to a class of stochastic process in which future and past are independent given the present and the conditional distribution, on the past, of a collection of random variables in the future.