Language as the Brownian 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 , P ) Probability space
Element of Ω sample ω
Element of F event A
Probability that event A occurs probability P ( A )
Real number valued Borel measurable function over Ω random variable X = X ( ω )
Random variable is integrable.
Mean (Expectation) of X E[X] = ∫Ω X ( ω ) P ( dω )
2
Measurable space ( S, S )
X : ( Ω, F ) → ( S, 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 ), A∈S distribution
5
Real number space R
Borel set family over R B ( R )
Probability measure over ( R, B ( R ) ) μ
6
Rd-value random variable X
ψX (ξ ) : = E[eiξ・X], ξ∈Rd characteristic function
7
Lebesgue measure dx
Mean m∈R
Variance v >0
Measure over R μ ( dx ) = e -(x-m)2 / 2v dx /
Gauss distribution ( normal distribution)
Gauss distribution ( normal distribution)8
(2p – 1 ) !! : = (2p – 1 ) ・(2p – 3 ) … 3・1
E[X2p] = (2p – 1 ) !! v p moment of X
9
Event A, B∈F
When (A∩B) = P(A) P(B), A and B are independent each other.
10
Integrable and independent random variable X, Y
Product XY integrable
E[XY] = E[X]E[Y]
11
Time t
t ∈[0, ∞)
Family of Rd-value random variable ≥ X = ( Xt ) t ≥ 0 d-dimensional stochastic process
∀ω∈Ω
When Xt (ω) is continuous as function of t., d-dimensional stochastic process is called to be continuous.
12
σ additive family Ft
Ft ⊂F
0 ≤ s ≤ t
F s ⊂Ft
(Ft ) = (Ft ) t ≥ 0 increase information system
13
d-dimensional stochastic process X = ( Xt ) t ≥ 0
∀t ≥ 0
Xt : Ω → Rd is Ft – measurable.
X = ( Xt ) t ≥ 0 is (Ft ) – adapted.
14
Mapping ( t, ω) ∈([0, ∞)×Ω, B([0, ∞)]×F) ↦ Xt ( ω) ∈( Rd, B ( Rd ) )
When the mapping is measurable, X = ( Xt ) t ≥ 0 is called to be measurable.
15
X = ( Xt )
Ft0 = Ft0,X : = σ ( XS ; s≤t )
16
Probability space ( Ω, F , P )
Stochastic process defined over ( Ω, F , P ) (Bt)t ≥ 0 = (Bt(ω)) t ≥ 0
(Bt)t ≥ 0 that satisfies the next, it is called Brownian motion.
(i) P ( B0 = 0 ) = 1
(ii) For ∀ω∈Ω, Bt (ω) is continuous on t.
(iii) For 0 = t0<∀t1<…<tn, ∀n∈N, {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 ( w ) = Wt
(ii) w = ( wt ) t ≥0 ∈W 0
Measure over ( W0, B ( W0 ) ) 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; Bt ∈S }
Hitting place BσS (ω)
Distribution of BσS (ω) uniform stochastic measure
20
d-dimensional Brownian motion B = ( Bt ) t ≥ 0
x∈Rd
Brownian motion from x ( x + Bt ) t ≥ 0
W d = B ( W d )
Space (W d, W d )
Distribution over (W d, W d ) Px
Mean on Px Ex [ ・ ]
Probability space (W d, W d , Px )
Stochastic process over (W d, W d , Px ) Bt ( w ), w∈W d ; Bt ( w ) = wt
Sub σ additive family of W d Ft0 =σ (Bs ; s≤t ) , Ft = Ft0 ⋁ N, t≥0 ; N : = {N∈W d ; Px (N) = 0, ∀x ∈Rd }
Ft* = Ft+ : = ∩s>t Fs
Shift operator over W d θs : W d → W d , s≥0 ; (θs (w) ) t : = wt+s
Bt ∘ θs = Bt+s
21
(Markov property)
∀x∈Rd
∀s≥0
∀Y = Y (w) : W d –measurable bounded function over W d
Ex[Y∘θs ・1A] = Ex[EBs(w)[Y]∘θs ・1A] , ∀A∈F s
By conditional mean
Ex[Y∘θs | Fs] (w) = EBs(w)[Y
Px-a.s.w
22
(Blumenthal’s 0-1 law)
When A∈F0 ( = 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,∞) }
A = {σ(0,∞) = 0 }
A ∈F0*
P (σ(0,∞) = 0 ) = 0 or 1
t↓0
P (σ(0,∞) = 0 ) = 1
From symmetry of Brownian motion Bt = -Bt
[B]
Language that has Brownian motion LB
LB has actual language and imaginary language.
[References]
To be continued
Tokyo August 12, 2008
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