Martingales in finite probability space

Lecturer: Dr. Hong-Gwa YehDepartment of MathematicsNational Central Universityhgyeh@math.ncu.edu.tw

2

a

we only considernote : .Ω<∞

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4

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( )Dα

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7

havingdistinct

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9

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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2

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

21(Gn,p)

22(Gn,p)

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References