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 X M
When M satisfies the next, it is called σ additive.
(i) X, Ø ∈M
(ii) A∈M ⇒ X╲A∈M
(iii) An∈M (n=1, 2, …) ⇒∪∞n=1 An∈M
2 <Measurable space>
Set X
Family of σ additive M
Pair ( X, M ) is called measurable space.
3 <Measure space>
Measurable space ( X, M )
Function over M μ
When μ is satisfies the next, it is called measure.
(i) μ (A)∈[0,∞]
(ii) μ (0) = 0
(iii) An∈M , An ∩Am = 0 (n≠m)
μ (∪∞n=1 An) = Σ∞n=1 μ (A)
( X, M, 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 P (A ) probability
Probability space ( Ω, F, P ).
5 <Borel additive>
Measurable space ( X, M ), ( Y, N )
Map f : X→Y
Arbitrary A∈N
f -1 ( A ) = {x∈N ; 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 ( X, M )
Function from X to f
When f satisfies one of the next, it is called M-B(
measurable.
(i) f -1 ( [-∞, a ] )∈M, f -1 ( [-∞, a ) )∈M
(ii) f -1 ( [ a, ∞] )∈M, f -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 ( Ωn, Fn, Pn ).
9 <Random variable>
Probability space ( Ω, F, P )
valued function over Ω X
When X is Ft+-measurable, X is called random variable.
10 <Expectation (Mean)>
Probability space ( Ω, F, P )
| X (ω) | is integrable.
Expectation of random variable EX ∫Ω X(ω)P(dω)
Expectation is also called mean.
11 <Covariance>
Random variable ( X (ω) – EX )2
Variance Expectation of ( X (ω) – EX )2
Random Variable X, Y
Covariant cov ( X, Y ) = E ( X- EX ) (Y-EY ) X (ω) and Y (ω) are integrable.
12 < Probability distribution >
Random variable X
Probability P
Probability distribution function F (x) = P ( X≤x )
13 <Density function>
Probability distribution over R F ( x )
Function ρ(x) satisfies the next, it is called density function for F.
F ( b ) - F ( a ) = ∫ba ρ(x)dx
14 <Gauss distribution>
m∈Rd
d×d positive definite symmetric matrix Σ
Density function over Rd for Σ ( 2π )-d/2 (det Σ )-1/2exp{-1/2 (Σ-1 (x-m), x-m ) }
Gauss distribution N ( m, Σ ) 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}λ∈Λ
When { Fλ}λ∈Λ satisfies the next, it is called independent on probability P.
Arbitrary finite sequence {λ1, …, λn}
Arbitrary Ai∈Fλi ( i = 1, 2, …, n )
P ( A1∩A2∩…∩An ) = P ( A1 ) P ( A2 ) …P ( An )
16 <Brownian motion>
Probability space ( Ω, F, P )
Family of Rd valued random variable {Bt}t≥0
When {Bt}t≥0 satisfies the next, it is called d-dimensional Brownian motion from starting position x.
(i) B0 = x at probability 1 and Bt is continuous on t.
(ii) When 0 = t0≤t1≤…≤tn, {Btk – Btk-1}n k=1 is independent.
(iii) When 0≤s<t, bt – Bs is mean 0, Gauss distribution of covariant matrix ( t-s )I.
17 < Ft >
x∈Rd
d-dimensional Brownian motion that starts from x {Bt}t≥0
Ft is defined by the next.
Ft = σ (Bs ; s≤t )
18 <Markov time>
d-dimensional Brownian motion ( {Bt}t≥0, Px )
Ft = σ (Bs ; s≤t ) , Ft+ = ∩t >0 Ft+ε
[0, ∞] valued random variable τ
When τ satisfies the next, it is called Markov time on Ft+ε.
(i) t≥0
(ii) {ω∈Ω ; τ (ω) ≤t }∈Ω
19 <Martingale>
Martingale is defined by the next.
(i) {Mt} is continuous at probability 1.
(ii) For every t≥0, Mt is Ft+-measurable.
(iii) For every t≥0, Mt is integrable. When t≥s≥0, E ( Mt | Fs+ ) = Ms
20 <Theorem>
Continuous Martingale on Ft+ {Mt}t≥0
T < ∞
{Mt ; o≤t ≤T } is bounded.
For bounded Markov time τ, next is brought.
EMτ = EM0
21 <Confirmation>
Meanings inherent in word : = {Mt}t≥0
All of time inherent in word : = o≤t ≤T<∞
Specific time of word that has meanings : = τ
Specific meaning of specific time : = EMτ
Confirmation of specific meaning : = (EMτ= EM0)
Tokyo June 27, 2008
