- Source: Kolmogorov continuity theorem
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.
Statement
Let
(
S
,
d
)
{\displaystyle (S,d)}
be some complete separable metric space, and let
X
:
[
0
,
+
∞
)
×
Ω
→
S
{\displaystyle X\colon [0,+\infty )\times \Omega \to S}
be a stochastic process. Suppose that for all times
T
>
0
{\displaystyle T>0}
, there exist positive constants
α
,
β
,
K
{\displaystyle \alpha ,\beta ,K}
such that
E
[
d
(
X
t
,
X
s
)
α
]
≤
K
|
t
−
s
|
1
+
β
{\displaystyle \mathbb {E} [d(X_{t},X_{s})^{\alpha }]\leq K|t-s|^{1+\beta }}
for all
0
≤
s
,
t
≤
T
{\displaystyle 0\leq s,t\leq T}
. Then there exists a modification
X
~
{\displaystyle {\tilde {X}}}
of
X
{\displaystyle X}
that is a continuous process, i.e. a process
X
~
:
[
0
,
+
∞
)
×
Ω
→
S
{\displaystyle {\tilde {X}}\colon [0,+\infty )\times \Omega \to S}
such that
X
~
{\displaystyle {\tilde {X}}}
is sample-continuous;
for every time
t
≥
0
{\displaystyle t\geq 0}
,
P
(
X
t
=
X
~
t
)
=
1.
{\displaystyle \mathbb {P} (X_{t}={\tilde {X}}_{t})=1.}
Furthermore, the paths of
X
~
{\displaystyle {\tilde {X}}}
are locally
γ
{\displaystyle \gamma }
-Hölder-continuous for every
0
<
γ
<
β
α
{\displaystyle 0<\gamma <{\tfrac {\beta }{\alpha }}}
.
Example
In the case of Brownian motion on
R
n
{\displaystyle \mathbb {R} ^{n}}
, the choice of constants
α
=
4
{\displaystyle \alpha =4}
,
β
=
1
{\displaystyle \beta =1}
,
K
=
n
(
n
+
2
)
{\displaystyle K=n(n+2)}
will work in the Kolmogorov continuity theorem. Moreover, for any positive integer
m
{\displaystyle m}
, the constants
α
=
2
m
{\displaystyle \alpha =2m}
,
β
=
m
−
1
{\displaystyle \beta =m-1}
will work, for some positive value of
K
{\displaystyle K}
that depends on
n
{\displaystyle n}
and
m
{\displaystyle m}
.
See also
Kolmogorov extension theorem
References
Daniel W. Stroock, S. R. Srinivasa Varadhan (1997). Multidimensional Diffusion Processes. Springer, Berlin. ISBN 978-3-662-22201-0. p. 51
Kata Kunci Pencarian:
- Kolmogorov continuity theorem
- Kolmogorov's theorem
- Continuity
- Kolmogorov extension theorem
- Continuity theorem
- Kolmogorov–Arnold representation theorem
- Andrey Kolmogorov
- Kolmogorov–Smirnov test
- Fréchet–Kolmogorov theorem
- Diffusion process
