Hey,
I'm a little confused about the correct notion of interior probabilities in a scenario tree. Assume, that the scenario tree (see link) is given and the space of the final outcome at the end of period 3 is described by the probability space (Omega, F, P), i.e.,
Omega = {(1,1,1),(1,1,2),...,(2,2,2)}
F = {(1,1,1), ..., (2,2,2),{(1,1,1),(1,1,2)},...,{(2,2,1),(2,2,2)},{(1,1,1),...(1,2,2)},{(2,1,1),...,(2,2,2)}, Omega, empty set}
P((1,1,1))= ... = P((2,2,2)) = 1/8 (each scenario is equiprobable).
The scenario tree induces a filtration (F_0, F_1, F_2, F_3) with
F_0 = {Omega, empty set}
F_1 = {{(1,1,1),...(1,2,2)},{(2,1,1),...,(2,2,2)}, Omega, empty set}
...
F_3 = F.
The question is if it is mathematical correct to write P((1,1)), P({1,1}) instead of P({(1,1,1),(1,1,2)}). The probability measure is a mapping from F, so I think this is problematic, but I found this notion very often in the literature.
Thanks for your expert opinion.
http://dtrees.com/fileadmin/images/Bilder/StochasticOptimizationMetho2_01.png
Hi Jelto,
your proposal P({(1,1,1),(1,1,2)}) is perfect, however without loss it can be simplified to P{(1,1,1),(1,1,2)} (which is popular in some communities). BUT, this has nothing to do with mathematical correctnes! This is a subject of the assumed notation. As long as the applied letters do not lead to confusion - every notation is acceptable. Only the content of claims counts! Obviously, if the picture you have attached were missed, then the notation you have cited could be unreadable.
Another suggestions: 11\cdot , 1 \cdot 2 etc. is readable AND simplefor coding evetnts like {(1,1,1),(1,1,2)} or {(1,1,2),(1,2,2)}, respectively.As it shows, within this convention events from other sub-fields can be denoted, too.
Best regards
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