In mathematics, the Kolmogorov continuity theorem is a theorem that guarantees that a stochastic process that satisfies certain constraints on the moments of its increments will be continuous (or, more precisely, have a "continuous version"). It is credited to the Soviet mathematician Andrey Nikolaevich Kolmogorov.
Let
(
S
,
d
)
be some metric space, and let
X
:
[
0
,
+
∞
)
×
Ω
→
S
be a stochastic process. Suppose that for all times
T
>
0
, there exist positive constants
α
,
β
,
K
such that
E
[
d
(
X
t
,
X
s
)
α
]
≤
K
|
t
−
s
|
1
+
β
for all
0
≤
s
,
t
≤
T
. Then there exists a modification of
X
that is a continuous process, i.e. a process
X
~
:
[
0
,
+
∞
)
×
Ω
→
S
such that
X
~
is sample-continuous;
for every time
t
≥
0
,
P
(
X
t
=
X
~
t
)
=
1.
Furthermore, the paths of
X
~
are almost surely locally
γ
-Hölder-continuous for every
0
<
γ
<
β
α
.
In the case of Brownian motion on
R
n
, the choice of constants
α
=
4
,
β
=
1
,
K
=
n
(
n
+
2
)
will work in the Kolmogorov continuity theorem. Moreover for any positive integer
m
, the constants
α
=
2
m
,
β
=
m
−
1
will work, for some positive value of
K
that depends on
n
and
m
.
