In the mathematical theory of probability, a Borel right process, named after Émile Borel, is a particular kind of continuous-time random process.
Let
E
lE
E
\Omega
[0,infty)
E
E
t\in[0,infty)
Xt
t
\omega\in\Omega
Xt(\omega)\inE
\omega
t
lE
lE*
t\in[0,infty)
lFt=\sigma\left\{
-1 | |
X | |
s |
(B):s\in[0,t],B\inlE\right\},
* | |
lF | |
t |
=\sigma\left\{
-1 | |
X | |
s |
(B):s\in[0,t],B\inlE*\right\},
and then, let
lFinfty=\sigma\left\{
-1 | |
X | |
s |
(B):s\in[0,infty),B\inlE\right\},
* | |
lF | |
infty |
=\sigma\left\{
-1 | |
X | |
s |
(B):s\in[0,infty),B\inlE*\right\}.
For each Borel measurable function
f
E
x\inE
U\alphaf(x)=Ex\left[
infty | |
\int | |
0 |
e-\alphaf(Xt)dt\right].
Since
Ptf(x)=
x\left[f(X | |
E | |
t)\right] |
t → Xt
f
t → Ptf(x)
Therefore, together with the monotone class theorem, for any universally measurable function
f
(t,x) → Ptf(x)
lB([0,infty)) ⊗ lE*
\left(lB([0,infty)) ⊗ lE*\right)λ ⊗
λ
lB([0,infty))
\mu
lE*
\left(lB([0,infty)) ⊗ lE*\right)λ ⊗
lB([0,infty)) ⊗ lE*
λ ⊗ \mu
f
E
t → Ptf(x)
\alpha\in[0,infty)
U\alphaf(x)=
infty | |
\int | |
0 |
e-\alphaPtf(x)dt.
There is enough joint measurability to check that
\{U\alpha:\alpha\in(0,infty) \}
(E,lE*)
\{Pt:t\in[0,infty)\}
U\alphaf(x)=Ex\left[
infty | |
\int | |
0 |
e-\alphaf(Xt)dt\right].
The following are the defining properties of Borel right processes:
For each probability measure
\mu
(E,lE)
P\mu
(\Omega,lF*)
(Xt,
*, | |
lF | |
t |
P\mu)
\mu
\{Pt:t\in[0,infty)\}
Let
f
\alpha
(E,lE*)
\mu
(E,lE)
t → f(Xt)
P\mu
[0,infty)