Optional Process

Suppose we are given a filtration on a measurable space . A stochastic process is said to be adapted if is -measurable for every time in the index set . For an arbitrary, uncountable, index set , this property is too restrictive to be useful. Instead, we can impose measurability conditions on considered as a map from to . For instance, we could require to be progressively measurable, but that is still too weak a condition for many purposes. A stronger condition is for to be optional. The index set is assumed to be a closed subset of in the following definition.

The class of optional processes forms the smallest set containing all adapted and right-continuous processes, and which is closed under taking limits of sequences of processes.

The -algebra, , on generated by the right-continuous and adapted processes is called the optional -algebra. Then, a process is optional if and only if it is -measurable.

Alternatively, the optional -algebra may be defined as

Here, is a stochastic interval, consisting of the pairs such that . In continuous-time, the equivalence of these two definitions for does require mild conditions on the filtration -- it is enough for to be universally complete.

In the discrete-time case where the index set countable, then the definitions above imply that a process is optional if and only if it is adapted.