The optional sigma-algebra, , was defined earlier in these notes as the sigma-algebra generated by the adapted and right-continuous processes. Then, a stochastic process is optional if it is -measurable. However, beyond the definition, very little use was made of this concept. While right-continuous adapted processes are optional by construction, and were used throughout the development of stochastic calculus, there was no need to make use of the general definition. On the other hand, optional processes are central to the theory of optional section and projection. So, I will now look at such processes in more detail, starting with the following alternative, but equivalent, ways of defining the optional sigma-algebra. Throughout this post we work with respect to a complete filtered probability space , and all stochastic processes will be assumed to be either real-valued or to take values in the extended reals .
Theorem 1 The following collections of sets and processes each generate the same sigma-algebra on .
The optional-sigma algebra was previously defined to be generated by the right-continuous adapted processes. However, any of the four collections of sets and processes stated in Theorem 1 can equivalently be used, and the definitions given in the literature do vary. So, I will restate the definition making use of this equivalence.
Definition 2 The optional sigma-algebra, , is the sigma-algebra on generated by any of the collections of sets/processes in Theorem 1.
A stochastic process is optional iff it is -measurable.