In mathematics, progressive measurability is a property in the theory of stochastic processes. A progressively measurable process, while defined quite technically, is important because it implies the stopped process is measurable. Being progressively measurable is a strictly stronger property than the notion of being an adapted process. Progressively measurable processes are important in the theory of Itō integrals.
Let
(
Ω
,
F
,
P
)
be a probability space;
(
X
,
A
)
be a measurable space, the state space;
{
F
t
∣
t
≥
0
}
be a filtration of the sigma algebra
F
;
X
:
[
0
,
∞
)
×
Ω
→
X
be a stochastic process (the index set could be
[
0
,
T
]
or
N
0
instead of
[
0
,
∞
)
).
The process
X
is said to be progressively measurable (or simply progressive) if, for every time
t
, the map
[
0
,
t
]
×
Ω
→
X
defined by
(
s
,
ω
)
↦
X
s
(
ω
)
is
B
o
r
e
l
(
[
0
,
t
]
)
⊗
F
t
-measurable. This implies that
X
is
F
t
-adapted.
A subset
P
⊆
[
0
,
∞
)
×
Ω
is said to be progressively measurable if the process
X
s
(
ω
)
:=
χ
P
(
s
,
ω
)
is progressively measurable in the sense defined above, where
χ
P
is the indicator function of
P
. The set of all such subsets
P
form a sigma algebra on
[
0
,
∞
)
×
Ω
, denoted by
P
r
o
g
, and a process
X
is progressively measurable in the sense of the previous paragraph if, and only if, it is
P
r
o
g
-measurable.
It can be shown that
L
2
(
B
)
, the space of stochastic processes
X
:
[
0
,
T
]
×
Ω
→
R
n
for which the Ito integral
with respect to Brownian motion
B
is defined, is the set of equivalence classes of
P
r
o
g
-measurable processes in
L
2
(
[
0
,
T
]
×
Ω
;
R
n
)
.
Every adapted process with left- or right-continuous paths is progressively measurable. Consequently, every adapted process with càdlàg paths is progressively measurable.
Every measurable and adapted process has a progressively measurable modification.
