In filtering theory, we have a filtered probability space and a signal process . The sigma-algebra represents the collection of events which are observable up to and including time t. The process X is not assumed to be adapted, so need not be directly observable. For example, we may only be able to measure an observation process , which incorporates some noise , and generates the filtration , so is adapted. The problem, then, is to compute an estimate for based on the observable data at time t. Looking at the expected value of X conditional on the observable data, we obtain the following estimate for X at each time ,
The process Y is adapted. However, as (1) only defines Y up to a zero probability set, it does not give us the paths of Y, which requires specifying its values simultaneously at the uncountable set of times in . Consequently, (1) does not tell us the distribution of Y at random times. So, it is necessary to specify a good version for Y.
Optional projection gives a uniquely defined process which satisfies (1), not just at every time t in , but also at all stopping times. The full theory of optional projection for jointly measurable processes requires the optional section theorem. As I will demonstrate, in the case where X is right-continuous, optional projection can be done by more elementary methods.
Throughout this post, it will be assumed that the underlying filtered probability space satisfies the usual conditions, meaning that it is complete and right-continuous, . Stochastic processes are considered to be defined up to evanescence. That is, two processes are considered to be the same if they are equal up to evanescence. In order to apply (1), some integrability requirements need to imposed on X. Often, to avoid such issues, optional projection is defined for uniformly bounded processes. For a bit more generality, I will relax this requirement a bit and use prelocal integrability. Recall that, in these notes, a process X is prelocally integrable if there exists a sequence of stopping times increasing to infinity and such that
is integrable. This is a strong enough condition for the conditional expectation (1) to exist, not just at each fixed time, but also whenever t is a stopping time. The main result of this post can now be stated.
Theorem 1 (Optional Projection) Let X be a right-continuous and prelocally integrable process. Then, there exists a unique right-continuous process Y satisfying (1).
Uniqueness is immediate, as (1) determines Y, almost-surely, at each fixed time, and this is enough to uniquely determine right-continuous processes up to evanescence. Existence of Y is the important part of the statement, and the proof will be left until further down in this post.
The process defined by Theorem 1 is called the optional projection of X, and is denoted by . That is, is the unique right-continuous process satisfying
for all times t. In practise, the process X will usually not just be right-continuous, but will also have left limits everywhere. That is, it is cadlag.
Theorem 2 Let X be a cadlag and prelocally integrable process. Then, its optional projection is cadlag.
A simple example of optional projection is where is constant in t and equal to an integrable random variable U. Then, is the cadlag version of the martingale . (more…)