Definition
Let
be a probability space;
be a measurable space, the state space;
be a filtration of the sigma algebra
;
be a stochastic process (the index set could be
or
instead of
);
be the Borel sigma algebra on
.
The process
is said to be progressively measurable [2] (or simply progressive) if, for every time
, the map
defined by
is
-measurable. This implies that
is
-adapted. [1]
A subset
is said to be progressively measurable if the process
is progressively measurable in the sense defined above, where
is the indicator function of
. The set of all such subsets
form a sigma algebra on
, denoted by
, and a process
is progressively measurable in the sense of the previous paragraph if, and only if, it is
-measurable.
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