Aaron
Senior Member
Posts: 6371
Joined: Jul 2001
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Tue Apr 22, 03 10:44 PM
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"Measurable" and "adapted" refer to the process, not a realization. There are many different measures and process can be measurable (or not) with respect to, and many filtrations that a process can be adapted (or not) with respect to.
The simplest answer is these are technical regularity conditions that you don't have to worry about unless you explore untested realms of exotic models.
The next simplest answer is that these conditions allow you to do common sense things. If a process is measurable, then the probability distribution at any point in time can be integrated. It makes sense to talk about the probability that X(3)>12. If a process is adapted then you can answer questions about the path. For example, given the values of X(t) for 0>=t>=3, you can tell if x(t) was ever >12 in that interval.
An example of a non-adapted, non-measurable process is a random walk in which the variance at time t is equal to the lowest integer b such that t = a/b.
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Aaron Brown
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