Saturday 28 April 2018

Stochastic Meaning Theory 2 Period of Meaning 13th for KARCEVSKIJ Sergej On what there exists confirmation of meaning in word


Period of Meaning
13th for KARCEVSKIJ Sergej
On what there exists confirmation of meaning in word


1 <σ additive>
Set     X
A family of subset of     M
When M satisfies the next, it is called σ additive.
(i) XØ M
(ii) AM  XAM
(iii) An(n=1, 2, …) n=1 AnM

2 <Measurable space>
Set     X
Family of σ additive     M
Pair ( XM ) is called measurable space.

3 <Measure space>
Measurable space      ( XM )
Function over M     μ
When μ is satisfies the next, it is called measure.
(i) μ (A)[0,]
(ii) μ (0) = 0
(iii) AnAAm = 0  (nm)
μ (n=1 An) = Σn=1 μ (A)
XM, μ ) is called measure space.
Measure that is 1 by all the measures is called probability measure.

4 <Probability space>
Measure space in which all the measures are 1 is called probability space.
Set     Ω
Element of Ω     ω
σ-field      F
Element of F     A
Function over F   P 
Measure ()     probability
Probability space     ( ΩFP ).

5 <Borel additive>
Measurable space     ( XM ), ( YN )
Map     XY
Arbitrary AN
f -1 ( A ) = {xN ; f (x)A }M
Map f is called M-N measurable.
A family of subsets of X     U
σ ( U ) = ( M ; M is σ additive that contains U )
σ ( U ) is also notated B ( X ) that is called Borel σ additive.
= [-, +]
Borel σ additive of  is notated B(.
Element of Borel σ additive is called Borel set.

6 < M-B(measurable>
Measurable space     ( XM )
Function from X to      f
When f satisfies one of the next, it is called M-B(measurable.
(i) -1 ( [-] )M,  -1 ( [-a ) )M
(ii) -1 ( [ a] )M,  -1 ( (a] )M

7 <F-measurable>
When function f : X is M-B(measurable, it is called M-measurable function, that is generally notated F-measurable.

8 < Ft+-measurable>
Countable sequence of probability space      ( ΩnFnPn ).

9 <Random variable>
Probability space     ( ΩFP )
valued function over Ω     X
When X is Ft+-measurable, X is called random variable.

10 <Expectation (Mean)>
Probability space     ( ΩFP )
| X (ω) | is integrable.
Expectation of random variable EX     Ω X(ω)P()
Expectation is also called mean.

11 <Covariance>
Random variable     ( X (ω) – EX )2
Variance     Expectation of ( X (ω) – EX )2    
Random Variable     XY
Covariant    cov ( XY ) = E ( XEX ) (Y-EY )    X (ω) and Y (ω) are integrable.

12 < Probability distribution >
Random variable     X
Probability     P
Probability distribution function     F (x) = P ( X)

13 <Density function>
Probability distribution over R     F x )
Function ρ(x) satisfies the next, it is called density function for F.
b ) - F a ) = bρ(x)dx

14 <Gauss distribution>
mRd
d×d positive definite symmetric matrix    Σ
Density function over Rfor Σ     ( 2π )-d/2 (det Σ )-1/2exp{-1/2 (Σ-1 (x-m), x-m ) }
Gauss distribution N ( mΣ )      distribution that has ( 2π )-d/2 (det Σ )-1/2exp{-1/2 (Σ-1 (x-m), x-m ) }

15 <Independent>
Set    Λ
Element of Λ     λ
Sub-family of σ additive F     Fλ
Sequence of Fλ     { Fλ}λΛ
When { Fλ}λΛ satisfies the next, it is called independent on probability P.
Arbitrary finite sequence {λ1, …, λn}
Arbitrary AiFλi  = 1, 2, …, )
P ( A1A2∩…∩An ) = AA2 ) P ( A  

16 <Brownian motion>
Probability space     ( ΩFP )
Family of Rd valued random variable     {Bt}t0
When {Bt}tsatisfies the next, it is called d-dimensional Brownian motion from starting position x.
(i) B0 = x at probability 1 and Bis continuous on t.
(ii) When 0 = t0t1tn, {Btk – Btk-1}n k=1 is independent.
(iii) When 0s<tbt – Bs is mean 0, Gauss distribution of covariant matrix ( t-s )I.

17 < F>
xRd
d-dimensional Brownian motion that starts from x     {Bt}t0
Ft is defined by the next.
Fσ (Bst )

18 <Markov time>
d-dimensional Brownian motion    ( {Bt}t0Px )
Fσ (Bst ) , Ft+  = ∩>0 Ft+ε
[0, ] valued random variable     τ
When τ satisfies the next, it is called Markov time on Ft+ε.
(i) t0
(ii) {ωΩ τ (ω) }Ω

19 <Martingale>
Martingale is defined by the next.
(i)  {Mt} is continuous at probability 1.
(ii) For every t0, Mt is Ft+-measurable.
(iii) For every t0, Mt is integrable. When ts0, E ( Mt Fs) = Ms

20 <Theorem>
Continuous Martingale on Ft+     {Mt}t0
{M; o} is bounded.
For bounded Markov time τ, next is brought.
EMτ = EM0

21 <Confirmation>
Meanings inherent in word : =  {Mt}t0
All of time inherent in word : = oT<∞
Specific time of word that has meanings : = τ  
Specific meaning of specific time : = EMτ
Confirmation of specific meaning : = (EMτEM0)

Tokyo June 27, 2008

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