Friday 20 September 2019

Stochastic Meaning Theory 5 Language as Brown Motion For ZHANG Taiyan 2



Language as Brown Motion


[A]
1
Abstractive space     Ω
σ additive family that consists of subset of Ω     F
Measure that is defined over F     P
P satisfies P (Ω ) = 1. 
P      probability measure over ( Ω, F )
Ω      sample space     
Ω, F , )     Probability space
Element of Ω     sample ω
Element of F     event A
Probability that event A occurs     probability P ( )
Real number valued Borel measurable function over Ω     random variable X = X ω )
Random variable is integrable.
Mean (Expectation) of X     E[X] = Ω ω ) P  )
2
Measurable space     ( S)
X : Ω, F )  S)
X is measurable.
X      S-value random variable.
Random variable     X1, …, Xd
X : = (X1, …, Xd )     Rd-value random variable
3
Rd-value random variable     X
E[X i 2] < 
E[(X - E[X])2]     variance
4
S-value random variable.     X
PX : = P ( X A ), AS     distribution
5
Real number space     R
Borel set family over R    R )
Probability measure over ( RR ) )     μ
6
Rd-value random variable     X
ψX (ξ ) : = E[eiξ・X], ξ∈Rd     characteristic function
7
Lebesgue measure     dx
Mean     mR
Variance     v >0
Measure over R     μ dx ) = -(x-m)2 / 2v dx /     Gauss distribution ( normal distribution)
8
(2p – 1 ) !! : = (2p – 1 ) (2p – 3 ) … 31
E[X2p] = (2p – 1 ) !! p     moment of X
9
Event     ABF
When (AB) = P(A) P(B), A and B are independent each other.
10
Integrable and independent random variable     XY
Product XY     integrable
E[XY] = E[X]E[Y]
11
Time     t
[0, ∞)
Family of Rd-value random variable     X = ( Xt ) t  0     d-dimensional stochastic process
ωΩ
When X(ω) is continuous as function of t.d-dimensional stochastic process is called to be continuous.
12
σ additive family     Ft
F F
 s  t
F s  Ft
(Ft  ) = (Ft  ) t  0   increase information system
13
d-dimensional stochastic process     X = ( Xt ) t  0    
t ≥ 0
XΩ  Rd  is F– measurable.
X = ( Xt ) t  0 is (F) – adapted.
14
Mapping ( tω([0, ∞)×ΩB([0, ∞)]×F Xt ( ωRdRd ) )
When the mapping is measurable, X = ( Xt ) t  0  is called to be measurable.
15
X = ( Xt )
FtFt0,X : = σ XS st )
16
Probability space      ( Ω, F , )    
Stochastic process defined over  ( Ω, F , )      (Bt)≥ = (Bt(ω)) ≥ 0
(Bt)≥ that satisfies the next, it is called Brownian motion.
(i) P ( B0 = 0  ) = 1
(ii) For ωΩBt (ω) is continuous on t.
(iii) For 0 = t0<t1<…<tnnN, {Bti-Bti-1} satisfies the next.
a) {Bti-Bti-1} are independent each other.
b) {Bti-Bti-1} are followed by mean 0 and variance ti-ti-1 of Gauss distribution.
17
(Existence theorem)
Over adequate probability space, there exists Brownian motion.
18
Ω = W0
F = B ( W0 )
Brownian motion has the next.
(i)Bt ) = Wt
(ii) = ( wt ) 0 0
Measure over ( W0, B ( W) )      P
P is called Wiener measure.
19
d-dimensional Brownian motion     B = ( Bt ) t  0
d×d orthogonal matrix     A
ABt     d-dimensional Brownian motion
Sphere     S : = δ B (0, r),  B (0, r) = {|x r }
Hitting time     σS (ω) : = inf{t >0; BS }
Hitting place    BσS (ω)
Distribution of BσS (ω)      uniform stochastic measure
20
d-dimensional Brownian motion     B = ( Bt ) t  0
xRd
Brownian motion from x     ( x + Bt ) t  0
 d = B ( W d )
Space     (dd )
Distribution over  (dd )     Px
Mean on Px     Ex [  ]
Probability space     (dd , P)
Stochastic process over (dd , P)    Bt ( w ), wBt ( w ) = wt
Sub σ additive family of d     Ft=σ (Bs ; s) , Ft Ft Nt≥0 ; N : = {Nd ; Px (N) = 0, Rd }
Ft* = Ft+ : = s>Fs
Shift operator over d     θs : → s0 ; (θs (w) ) t : = wt+s
Bt   θ = Bt+s
21
(Markov property)
xRd
s0
Y (w) : d –measurable bounded function over d
Ex[Yθ1A] = Ex[EBs(w)[Y]θ1A] , AF s
By conditional mean
Ex[YθFs] (w) = EBs(w)[Y
Px-a.s.w
22
(Blumenthal’s 0-1 law)
When AF0 ( = F0* ), Px (A) = 0 or 1
23
Random variant of 1-dimensional Brownian motion starting from the origin     B
σ (0,: = inf {t >0; Bt(0,) }
= {σ(0,= 0 }
F0*
(σ(0,= 0 ) = 0 or 1
t0
P (σ(0,= 0 ) = 1
From symmetry of Brownian motion Bt = -Bt

[B]
Language that has Brownian motion     LB
Lhas actual language and imaginary language.

[References]


To be continued
Tokyo August 12, 2008

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