We give a very simple and elementary proof of the existence of a weakly compact family of probability measures {
P
θ
:
θ ∈ gJ} representing an important sublinear expectation—
G-expectation
$
\mathbb{E}
$
\mathbb{E}
[·]. We also give a concrete approximation of a bounded continuous function
X(
ω) by an increasing sequence of cylinder functions
L
ip
(Ω) in order to prove that
C
b
(Ω) belongs to the completion of
L
ip
(Ω) under the natural norm
$
\mathbb{E}
$
\mathbb{E}
[| · |].
Keywords Probability and distribution uncertainty -
G-normal distribution -
G-Brownian motion - continuous paths
2000 MR Subject Classification 60H10 - 60H05 - 60H30 - 60E05 - 60E07 - 62C05 - 62D05
The author thanks the partial support from The National Basic Research Program of China (973 Program) grant No. 2007CB814900
(Financial Risk).