progressively measurable process
A stochastic process is said to be adapted to a filtration (http://planetmath.org/FiltrationOfSigmaAlgebras) 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.
|Title||progressively measurable process|
|Date of creation||2013-03-22 18:37:31|
|Last modified on||2013-03-22 18:37:31|
|Last modified by||gel (22282)|