By Darwin C. G.

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How to extend the definition of optionality to continuous time? 2) won't do. But I have mentioned earlier that it can be modified by changing "T = t" into "T < i" to an equivalent condition when time is discrete. The latter will serve in continuous time. For example, the first, second,.. jump times of the Poisson process are all optional in this sense. But we must stop here before getting into complications in continuous time. If you are eager to find out, you should try to write down a precise definition of those "jump times" just casually spoken of, and verify that they are indeed optional according to the definition: {w : T(w) < t} G Tt, for each t G [0, oo).

Introduction to random time And pronto, the solution is Pn{t) = nn(\t), n € N°. 17). Where is the process? " is one answer. For us, however. " The Poisson Process is a Homogeneous Markov Process as discussed in §4, though it runs in continuous time. 1, to be translated appropriately, where the rightcontinuity of paths is essential. 7) into its more useful form: Pm{Rt = m + n}=Pn(t) = nn(Xt). ) sure that after another lapse of time t, the process will be found in state m + n with the probability given by the Poisson distribution.

This is the constructive way of defining TT- It is of course equivalent to the "characterizing" definition: A G TT iff for each t e N we have An{T = t} &Tt. 2) Since Tt increases with t, the above is seen to be equivalent to the modified form in which {T — t} is changed into {T < t}. ) Why don't we define TT to be the tribe generated by {Xs,s < T}? "Good question", but how do you define the latter bunch of random variables? Here is a better try: define it as the tribe generated by {X(TAt), t £ N}.