Some terminology on information
This note aims at explaining a few of the concepts relating to information in stochastic processes. The goal is not to teach terminology -- you won't get "progressively measurable" for the exam -- but to eliminate the need for clarification; for your purposes, we can safely use the "predictable" definition for "not looking into the future" (at first glance, a more natural interpretation would be the "adapted" concept, but that interpretation is not useful in e.g. finance and insurance, if there is a possibility that information will be revealed suddenly).
The word "measurable" originally had to do with e.g. what subsets of the real line you can measure a length to (there are subsets so bizarre that any reasonable "length function" is ill-defined on them). However, it turned out that the same concept can be used to model information. The standard mathematical term for an information set which increases in time (so that more information may be added, but nothing ever forgotten), is "filtration" (or sometimes "current"). We take the information as given, so that we do not need to specify "with respect to which filtration"; a primer on the impact of changing the filtration is given below.
Consider an example where you observe vulnerable to a sudden incident, e.g. lightening. At some random time T, lightening strikes, completely destroying the object; immediately after, at T+, the owner acts by announcing the replacement object, which you at time T do not know. Imagine that the value of the object can then be modeled by a stochastic process X which is positive and continuous, except at time T, where it is = 0. This process would be right-continuous if not for the owner's action; the controlled process is neither left-continuous nor right-continuous.
The information inflow is then also "continuous" except at the random time T (informally stated, but can be made precise).
This example may be used to explain the following terminology which is frequently encountered in the literature:
- "Predictable": If, for any t, known already at the "previous moment". The process Y defined by Y(t) = X(t -) is predictable. An insurance policy should be predictable, because it cannot be bought when you observe the lightening. Same goes for a trading portfolio; if you just observed a high price, which for sudden drops "now", then you are already too late.
- "Progressively measurable": Slightly stronger than "adapted" (see below), but for your purposes the differences are of technical nature.
- "Adapted": Known now. The process X is adapted. (Generally, any process is adapted to its own filtration.)
Note: {Adapted & left-continuous} => predictable. - "Non-anticipating": sligtly less formal synonym for "adapted" or for "progressively measurable"
- "Optional": If known "the next moment". The process Z defined by Z(t) = X(t+) is optional. (Like: assume that the owner decides it at time T based on his/her financial situation; then the owner knows at T what X(T+) will be, so (s)he has the optional information; we only have the adapted information.)
Note. {Optional & right-continuous} => adapted
Finally, just to give a brief glimpse on the impact of changing the filtration:
Assume first that we don't observe the lightening, only X. Then at time T where X(T)=0, we know that lightening strikes, because X is assumed to be > 0 otherwise in this example. So observing the lightening does not give us any more information in the above example.
However, modify the example: weaken the assumption that X>0 except at T, and assume X merely nonnegative. Assume still X continuous except at T. Then observing the lightening does reveal information; if X(t) = 0 and no lightening, we know that X(t+)=0 by continuity. Now impose the additional condition that it is known that when lightening strikes, the owner will choose X(T+)=1. Then under the information of observing both X and lightening, Z(t) is not only optional but also adapted, since at time t we know X(t+) (equal to 1 if lightening strikes, 0 otherwise); if we don't observe whether the lightening strikes or not, we cannot tell, and Z is not adapted.