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Scott Mcclintock science forum beginner
Joined: 25 Oct 2005
Posts: 3

Posted: Fri May 05, 2006 8:55 pm Post subject:
weak * topology



Im working through 'NoArbitrage and Equivalent Martingale Measures: An
Elementary proof of The HarrisonPliska Theorem', by Kabanov and Kramkov and
having a difficult time with one of their results/statements. It involves
weak * topologies which I readily admit are well beyond the frontiers of my
knowledge. Ive taken a fair bit of graduate analysis/probability but havent
formally taken any functional analysis or topology.
Unfortunately there's a bit of notation.
Let E consist of all e, elements of R^n, s.t e=(+/1,....,+/1). There are
2^n such elements. For a fixed e in E consider A_e as the set of all points
in R^n
(x1,....,xn) s.t. xi>=0 if ei=1 and xi < 0 if ei = 1. These A_e, then,
form a disjoint partition of R^n.
Consider a sequence of measurable functions(random variables) that map from
some space, omega, to R^n, that are measurable wrt some sigmaalgebra G.
Call these functions a_n(Note Im suppressing the additional subscript of e,
but they depend on e as well). Each particular a_n takes values in (A_e
intersect the unit sphere) union zero. Basically they take values on the
unit sphere restricted to A_e with the possibility of being 0.
The paper then reads, "The sequences a_n are bounded and hence are compact
in the topology sigma(Linfinity, L_1). Taking a subsequence we can consider
that every a_n converges in sigma(Linfinity,L_1) to a G measurable random
variable a which takes values in A_e union {0} (by the property of weak
convergence to preserve positivity)."
Ive become familar enough with the weak * topology and the Banach Alagolu
Thm(Mispelled, I imagine) to be happy with the convergence of the a_n to a
in the weak * topology. What Im not sure, however, is how you can say or
see what values the function a takes. The paper doesnt actually need the
preservation of positivity. So it would be fine if the function took values
in Closure(A_e) union {0}. But Im not sure how to see that it takes values
(exclusively) there either. It certainly makes intuitive sense but Im not
sure how to 'prove' it. Any ideas, suggestions, results would be
appreciated. (Watered down as much as possible and perhaps with some kind
of citation so I could look it up).
Thanks so much for your time,
Scott 

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Janusz Kawczak science forum beginner
Joined: 18 Feb 2006
Posts: 10

Posted: Thu May 11, 2006 5:11 am Post subject:
Re: weak * topology



It would be much easier to understand what you are trying to say by
citing/giving link to the original paper. Why a_n are bounded? You mean,
"in probability/a.s." or what sense? And, many other questions.
In summary, I believe this is a trivial fact but need to see the setup
for the statement.
Janusz.
Scott Mcclintock wrote:
Quote:  Im working through 'NoArbitrage and Equivalent Martingale Measures: An
Elementary proof of The HarrisonPliska Theorem', by Kabanov and Kramkov and
having a difficult time with one of their results/statements. It involves
weak * topologies which I readily admit are well beyond the frontiers of my
knowledge. Ive taken a fair bit of graduate analysis/probability but havent
formally taken any functional analysis or topology.
Unfortunately there's a bit of notation.
Let E consist of all e, elements of R^n, s.t e=(+/1,....,+/1). There are
2^n such elements. For a fixed e in E consider A_e as the set of all points
in R^n
(x1,....,xn) s.t. xi>=0 if ei=1 and xi < 0 if ei = 1. These A_e, then,
form a disjoint partition of R^n.
Consider a sequence of measurable functions(random variables) that map from
some space, omega, to R^n, that are measurable wrt some sigmaalgebra G.
Call these functions a_n(Note Im suppressing the additional subscript of e,
but they depend on e as well). Each particular a_n takes values in (A_e
intersect the unit sphere) union zero. Basically they take values on the
unit sphere restricted to A_e with the possibility of being 0.
The paper then reads, "The sequences a_n are bounded and hence are compact
in the topology sigma(Linfinity, L_1). Taking a subsequence we can consider
that every a_n converges in sigma(Linfinity,L_1) to a G measurable random
variable a which takes values in A_e union {0} (by the property of weak
convergence to preserve positivity)."
Ive become familar enough with the weak * topology and the Banach Alagolu
Thm(Mispelled, I imagine) to be happy with the convergence of the a_n to a
in the weak * topology. What Im not sure, however, is how you can say or
see what values the function a takes. The paper doesnt actually need the
preservation of positivity. So it would be fine if the function took values
in Closure(A_e) union {0}. But Im not sure how to see that it takes values
(exclusively) there either. It certainly makes intuitive sense but Im not
sure how to 'prove' it. Any ideas, suggestions, results would be
appreciated. (Watered down as much as possible and perhaps with some kind
of citation so I could look it up).
Thanks so much for your time,
Scott



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