Martingales in finite probability space - NCU · Martingales in finite probability space Lecturer:...
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Martingales in finite probability space Lecturer: Dr. Hong-Gwa Yeh Department of Mathematics National Central University [email protected]
Transcript of Martingales in finite probability space - NCU · Martingales in finite probability space Lecturer:...
Martingales in finite probability space
Lecturer: Dr. Hong-Gwa YehDepartment of MathematicsNational Central [email protected]
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a
we only considernote : .Ω<∞
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( )Dα
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havingdistinct
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distinct
(1)
(2)
(3)
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ℑ ℑ
ℑ
ℑ
ℑ
a algebra of a finite spaceσ −
Warning: we only use this notation for finite probability space.
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P(A|X1,X2,…,XM)
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aE(X|Y1,Y2,…,YM)
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:ω ω∈Ω
:ω ω= ∈Ω Ω =
s
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Martingale (I) general setting
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Martingale (II) special case
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Recall
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1( | )
Suppose ( , , , ).
Then impl
( | )
ies
i ki
tk k k k
k k
k k ki
k
t
DX
E X
E X
D D D
D I
X
α
+
+=
ℑ
=
=
ℑ
=
∑
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0 1 2Note .ℑ ⊆ ℑ ⊆ ℑ
An illustration of a martingale
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Doob Martingales
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References
