planemath上关于progressive stochastic process的解释:

A stochastic process is said to be adapted to a filtration on the measurable space if is an -measurable random variable for each . However, for continuous-time processes, where the time ranges over an arbitrary index set , the property of being adapted is too weak to be helpful in many situations. Instead, considering the process as a map

it is useful to consider the measurability of .

The process is progressive or progressively measurable if, for every , the stopped process is -measurable. In particular, every progressively measurable process will be adapted and jointly measurable. In discrete time, when is countable, the converse holds and every adapted process is progressive.

A set is said to be progressive if its characteristic function is progressive. Equivalently,

for every . The progressively measurable sets form a -algebra, and a stochastic process is progressive if and only if it is measurable with respect to this -algebra.