1 predictable processes in discrete time
Suppose we have a filtration (http://planetmath.org/FiltrationOfSigmaAlgebras) on a measurable space . Then a stochastic process is predictable if is -measurable (http://planetmath.org/MeasurableFunctions) for every and is -measurable. So, the value of is known at the previous time step. Compare with the definition of adapted processes for which is -measurable.
2 predictable processes in continuous time
In continuous time, the definition of predictable processes is a little more subtle. Given a filtration with time index ranging over the non-negative real numbers, the class of predictable processes forms the smallest set of real valued stochastic processes containing all left-continuous -adapted processes and which is closed under taking limits of a sequence of processes.
Equivalently, a real-valued stochastic process
is predictable if it is measurable with respect to the predictable sigma algebra . This is defined as the smallest -algebra on making all left-continuous and adapted processes measurable.
Alternatively, is generated by either of the following collections of subsets of
Note that in these definitions, the sets and are stochastic intervals, and subsets of .
3 general predictable processes
The definition of predictable process given above can be extended to a filtration with time index lying in an arbitrary subset of the extended real numbers. In this case, the predictable sets form a -algebra on . If has a minimum element then let be the collection of sets of the form for , otherwise let be the empty set.Then, the predictable -algebra is defined by
Here, and are understood to be intervals containing only times in the index set . If is an interval of the real numbers then can be equivalently defined as the -algebra generated by the class of left-continuous and adapted processes with time index ranging over .
A stochastic process is predictable if it is -measurable. It can be verified that in the cases where or then this definition agrees with the ones given above.
|Date of creation||2013-03-22 18:36:30|
|Last modified on||2013-03-22 18:36:30|
|Last modified by||gel (22282)|