fbm survey

8/19/2019 Fbm Survey

1/24

H (0, 1) H {BH (t); t ∈ [0, 1]}

RH (s, t) = [BH (s)BH (t)] def

= V H

2

s2H + t2H − |t − s|2H


2/24

V H def =

Γ(2 − 2H ) cos(πH )

πH (1 − 2H )

.

BH H

1/H [0, t]

limn→+∞

2n−1 j=0

|BH (( j + 1)2−n ∧ t) − BH ( j2

−n ∧ t)| p =

0 pH > 1,

∞ pH

/ H <

/ . {B(s), s ∈ [0, 1]}

BH (t) =

t0

K H (t, r) dB(r),

K H (t, r) = (t − r)

H − 12

+

Γ(H + 12)F (

1

2 − H, H −

1

2, H +

1

2, 1 −

t

r).

F (α ,β ,γ ,z) C×C×C\{−1, −2, . . .}×{z ∈ C,Arg|1−z| < π}

+∞k=0

(α)k(β )k(γ )kk! z

k

.

(α)k

(α)0 = 1 (α)kdef =

Γ(α + k)

Γ(α) = α(α + 1) . . . (α + k − 1).

K H

t∧r0

K H (t, s)K H (r, s) ds = RH (t, r).

K H

s−|H − /

|(t − s)H − /

.

K H RH .


3/24

0 0.2 0.4 0.6 0.8 10.5

1

1.5

2

2.5

3

3.5

s

0 0.2 0.4 0.6 0.8 1

0.5

1

1.5

2

2.5

3

3.5

s

K H (1, .) K̃ H (1, .) H = 0.2, H = 0.8

H = /

H

BH

BH

BH

RS n(u) def

=2n−1

i=0

ui2−n(BH ((i + 1)2−n) − BH (i2

−n))

u

H > /

,

1/H RS n(u) u 1/β β + H > 1 β ≥ 2.

f g

| f dg| ≤ cf Hol(α)gHol(β ) Hol(α) α + β ≥ 1, RS n(u) u ∈ Hol+(1 − H )

u(s) dBH (s) = lim

n→∞RS n(u),


4/24

u B α,1 α > 1−H, u BH

u(s) dBH (s) =

(−1)α 10

I −α0+ (u0+)(s)I α−11− (B

1−H )(s) ds + u(0

+)BH (1−),

u0+(s) = u(s) − u(0+) u(0+) = lim↓0 u(). B

1−H (s) =

BH (1−) − BH (s) BH (1

−) = lim↓0 BH (1 − ).

u ∈ Hol+(1 − H ), u(s) dBH (s) t →

u(s)1[0,t](s) dBH (s) Hol−(H ).

H (0, 1),

H > /

, H > 1 − H. H > /

u Hol+(1 − H )

u(s) dBH (s)

u(s) dBH (s).

BH ,

Ω [0, 1] R, 0,

f ∞ = supt∈[0,1]

|f (t)|.


5/24

Ω, {BH (ω, t) =ω(t), t ∈ [0, 1]}, H.

H, {R(t, .), t ∈ [0, 1]}

< R(t, .), R(s, .) >= R(t, s) t s.

H, H 0, L2([0, 1]). H

H >

/

H ∈ (0, 1), H f

f (t) =

t0

K (t, s) f̃ (s)ds,

f̃ L2([0, 1]). f H = f̃ L2.

H I H + /

0+ (L2)

(H + /

) 0, (H + /

) L2. H = /

H = /

H L2(Ω, )

H −→ L2(Ω, )

R(t, .) −→ BH (t).

ni=1

αiRH (ti, s) dBH (s) =n

i=1

αiBH (ti).

u ∈ H, (un, n ≥ 1), un R(t, .), u H.

u(s) dBH (s) = L

2(Ω, ) − limn→∞

un(s) dBH (s).


6/24

[0, 1].

B.

H L2([0, 1]) H I 10+

u L2, u f (t) =

t0 u ds = (I

10+u)(t)

H, H.

H { j(t), t ∈ [0, 1]} BH (t)

j(t)H =

W 2t

.

L2([0, 1]) H

i1 : L2([0, 1]) −→ H

h −→ f (t) = t0

K (t, s)h(s)ds.

(L2([0, 1]), i1) H.

H = /

, i1 I 10+,

i1(1[0,t]) = K /

(t, .) H,

R(t, .) i1 K (t, .). H > /

,

H > /

, L2([0, 1])

< f, g >= [0,1]2

f (s)g(t)|t − s|2H −2 dsdt.

i2

i2 : (L2([0, 1]), < , >) −→ H

1[0,t] −→ R(t, .).


7/24

i2, H2 def = (L2([0, 1]), <

, >). (H2, i2) H.

(L2([0, 1]), < , >)

K (t, s). H ≥ /

f ∈ L2,

Kf = I 10+xH − /

I H − /

0+ x /

−H f.

K ∗, K, f g K ∗g

1

0Kf (s)g(s) ds =

1

0f (s)K ∗g(s) ds.

K ∗g(t) =

1t

K (s, t)g(s) ds

K (t, s) = K ∗(t)(s), t t.

K ∗ = x /

−H I H − /

1− xH − /

I 11−.

I 11−(t) = 1[0,t],

K (t, s) = s /

−H

I H − /

1− xH − /

1[0,t](s).

i3 : L2([0, 1]) −→ H

h −→ f (t) = t /

−H

I H − /

1− xH − /

h

(t),

f H = hL2, H, H < / .

i1. H [0, 1],

i3 H, H

i3 H.

: L2([0, 1]) −→ L2(Ω, )

: H2 −→ L2(Ω, )

K (t, .) −→ BH (t) 1[0,t](.) −→ BH (t).


8/24

1[0,t](s) dBH (s) BH (t)

|

1[0,t](s) dBH (s)|

2

= 1[0,t]

2L2 = t =

|BH (t)|

2

= V H t2H .

1[0,t](s) dBH (s), t ≥ 0

min(t, s),

B̃. u L2,

u(s) dBH (s) = u(s) d B̃s,

L2([0, 1]) BH (t) K (t, .) L2([0, 1]) 1[0,t] BH (t).

BH (t) (BH (u), u ≤ r) r < t.

[BH (u)(BH (t) − [BH (t) | BH (u), u ≤ r])] = 0,

L2([0, 1]), H2.

0 ≤ r ≤ t ≤ 1,

[BH (t) | BH (u), u ≤ r] =

K (t, s)1[0,r](s) dBH (s).

H > /

H > /

, u ∈ L2([0, 1]),

K ∗K ∗−1

/

u(s) dBH (s) =

u(s) dBH (s),

K ∗ K, f g L2([0, 1]), 1

0(Kf )(s)g(s) ds =

10

f (s)(K ∗g)(s) ds.


9/24

K K −1 /

◦ K, K /

I 10+, K = I −10+ K

Kf (t) = (Kf )(t).

H > /

,

Ku(t) = tH − /

I H − /

0+ (x

/

−H u)(t),

K∗u(t) = t /

−H I H − /

1− (xH − /

u)(t),

K∗u(t) = t

/

−H

Γ(H − /

)

1t

(r − t)H − /

rH − /

u(r) dr.

H =

/

, K

H <

/

, K

L2 I /

−H 0+ (L

2)

K K /

K ∗(t)(s) = K (t, s) K ∗

/

(t)(s) = I 11−(t)(s) = 1 [0,t](s),

t t. K∗1[0,t] = K (t, .)

ni=1 ui1[0,ti]. K

∗

L2([0, 1])

BH (t)

1[0,t]

H = /

, K /

(t, .) = 1 [0,t]

u(s) dBH (s) =

u(s) dBH (s) =

limn→+∞

2n−1i=0

u(i2−n)(BH ((i + 1)2−n) − BH (i2

−n)),

u.

,


10/24

B u

u(s) ◦ dB(s)

u(s) ◦ dB(s)

u

u(s) ◦ dB(s) u. Dom δ f (B(t1), . . . , B(tn)) ti [0, 1] i ∈ {1, . . . , n} f

Rn

R

.

,

u K∗u Dom δ,

u(s) ◦ dBH (s)

def =

K∗u(s) ◦ dB(s),

K∗u. K K (t, s) = 0 s > t

K∗t−u ≡ K∗1−(u1[0,t]),


11/24

K∗t− I αb− K

L2([0, t]), t0

Kf (s)g(s) ds = t0

f (s)K∗t−g(s) ds

f g.

t0

u(s) ◦ dBH (s) =

K∗t−u(s) ◦ dB(s).

H > /

BH

. u(s) ḂH

(s) ds ḂH

BH K −1

/

,

u(s) ḂH (s) ds =

u(s) K −1

/

◦ K ( Ḃ)(s) ds.

K −1 /

◦ K, K,

u(s) ḂH (s) ds =

K∗u(s) Ḃ(s) ds,

H > /

,

Kf = I 10+xH − /

I H − /

0+ x

/

−H f,

K∗f (t) f t 1

K∗f (t) = t

/

−H

Γ(H − /

)

1t

xH − /

(x − t)H − /

f (x) dx.

H < /

, u

t0 u(s) ◦ dBH (s) F t K

∗t−u

K,

10 wvdx v(x) = x

0 v(y) dy.

10

wv dx =

v(x)

1x

w(y) dy dx,

u.


12/24

H, u K u H > /

u I /

−H 0+ (L2) H <

/

u

u(s) ◦ dBH (s) = lim

|πn|→0

ti∈πn

u(ti)(BH (ti+1) − BH (ti)).

u

u(s) ◦ dBH (s) =

lim|πn|→0

ti∈πnu(ti)(BH (ti+1) − BH (ti)) −

10

Dsu(s) ds.

D

u

u(s) ◦ dBH (s) =

u(s) ◦ dBH (s) +

10

Dsu(s) ds

t

0u(s) ◦ dBH (s) =

t

0u(s) ◦ dBH (s) +

t

0Dsu(s) ds.

LI n(u) =n−1i=0

n

(i+1)/ni/n

u(s) ds

BH (i + 1

n ) − BH (

i

n)

.

dBH (s).

H > /

, u ∈ L2(Ω × [0, 1], ⊗dt) ∩Dom δ ∇u L p(Ω; L p([0, 1]2)) p(H − /

) > 1; u {LI n(u), n ≥ 1} L

2(Ω, )

u(s) ◦ dBH (s).

H < /

, u ∈ D2,1(B /

−H,2), u

{LI n(u), n ≥ 1} L2(Ω, )

u(s) ◦

dBH (s).


13/24

BH .

BH

BH

.

C ∞0 ([0, 1]2) = {k ∈ C ∞, k(1, r) = k(0, r) = 0, r ∈ [0, 1]},

W C ∞0 ([0, 1]2)

k2W = k2L2([0,1]2) +

1

0

∂ s 1

0

k2(s, r) dr ds.

W (kn, n ∈ N) C

∞0 ([0, 1]

2),

k W, 10 u(s)∂ skn(s, .) ds ∈ Dom δ n

W kn (u) =

10

10

u(s)∂ skn(s, .) ds

◦ dB(s)

L2(Ω, ) (kn, n ≥ 0).

10 u(s)∂ skn(s, .) ds K

∗u K Ku W

W kn (u).

H = /

,

u

[Λu

1

] = 1 Λu

t

= exp( t

0

u(s) dBH (s) − 1

2

t

0

u(s)2 ds).

u

u

d

F H t

= Λut .

{BH (t) − t0 K (t, s)u(s) ds, t ∈ [0, 1]} u

BH . v


14/24

{ t0 K (t, s)v(s) dB(s) −

t0 K (t, s)u(s)v(s) ds, t ∈ [0, 1]} u

{

t0 K (t, s)v(s)u(s) dB(s), t ∈ [0, 1]} .

.

BH B (Ω; F , ) B B.

F,

F − [F ] =

10

u(s) dBH (s)

F D2,1 u

u(s) = [∇sF | F s] .

BH

H

1/2.

F (BH (t)) = F (BH (0)) +

F (BH (s))1[0,t](s) dBH (s),

F C 1.

.

X t = x0 +

t

0K (t, s)ξ (s) ds +

t

0K (t, s)σ(s) ◦ dB(s),

σ BH .

t0 α(s) ds,


15/24

ρ(s) = s1−2H

L2ρ([0, 1]) = {h : [0, 1] → R,

10

h2(s)ρ(s) ds /

L2ρ([0, 1]) ⊂ L2([0, 1]) ρ(s) ≥ 1

s ∈ [0, 1] D2,k,ρ ψ D2,k n ≤ k, n ψ L2ρ([0, 1])

⊗n ψ

ψ22,k,H,ρ = ψ2L2(

) +

ki=1

∇(i)ψ2L2ρ([0,1])⊗i

.

H > 1/2 F : R −→ R X ξ D2,1,ρ(L

2ρ([0, 1])) σ D2,2,ρ(L

2ρ([0, 1]))

F (X t) − F (x0) =

t0

F (X s)(Kξ )(s) ds

+

t0

σ(s)K∗t−(F ◦ X K H (∇sξ ))(s) ds

+

t0

σ(s)K∗t−(F ◦ X )(s) ◦ dB(s)

+

t

0K∗t−(F

◦ XK (., s))(s) σ(s)2 ds

+

t0

K∗t−

F ◦ X

10

∇sσ(r)K H (., r) ◦ dB(r)

(s)σ(s) ds.

t ∈ [0, 1]

Z t = z0 +

t0

ξ (s) ds +

σ(s) ◦ dBH (s),

ξ D2,1,ρ(L2([0, 1])) σ ∈ D2,2,ρ(L

2ρ([0, 1])). F

t


16/24

F (Z t) = F (z0) + t0

F (Z s)ξ (s) ds + t0

F (Z s)σ(s) ◦ dBH (s)

+

t0

F (Z s)DsZ sσ(s) ds.

F (Z t) = F (z0) +

t0

F (Z s)ξ (s) ds +

t0

F (Z s)σ(s) ◦ dBH (s).

X (t) = BH (t),

F (BH (t)) = F (0) +

t0 F

(BH (s)) ◦ dBH (s)

+ HV H

t0

F (BH (s))s2H −1 ds.

/

≤ H < /

, BH

X

X t = θt + BH (t)

X θ X [0, t] φ (Kφ)(t) = t

d X d

F t

= exp(θ

t0

φ(s) dBH (s) − θ2

2

t0

φ(s)2 ds)

d

d X

F t

= exp(−θ

t0

φ(s) dBH (s) + θ2

2

t0

φ(s)2 ds),

[F ] = F (X (w)) exp(−θ t

0

φ(s) dBH (s) + θ2

2

t

0

φ(s)2 ds) . θ̂t θ

θ̂t = 1 t0 φ(s)

2 ds

t0

φ(s) dBH (s) = 1 t0 φ(s)

2 ds

t0

K∗−1φ(s) dBH (s).

X


17/24

φ K∗−1φ

φ(s) = Γ(3/2 − H )Γ(2 − 2H )

s /

−H

K∗−1φ(s) = Γ(3/2 − H )

Γ(2 − 2H )Γ(H + /

)

s(1 − s)

/

−H .

Y X

X,

X Y.

X t = BH 1(t)

Y t =

t0

K H 2(t, s)h(X s) ds + BH 2(t),

BH 1 BH 2

H 1

H 2.

d

d

F t

def = Λht = exp

t0

h(X s)dBH 2(s) − 1

2

t0

h(X s)2 ds

,

B̃H 1(t) = BH 1(t) B̃H 2(t) = BH 2(t) −

t0

K H 2(t, s)h(X s)ds,

H 1 H 2.

Bi

Bi(t) =

1[0,t](s) dBH (s).

(t1, . . . , tn) ∈ [0, 1]n, 2n

Z t1,...,tnr = (

r0

K H 1(t j, s) dB1(s),

r0

K H 2(t j, s) dB2(s)) j=1,...,n


18/24

A

t1,...,tn

r = (0,

r

0 K H 2(t j , s) h(X s) ds) j=1,...,n.

f,

f (Z t1,...,tnr − A

t1,...,tnr )

=

f (Z t1,...,tnr ))

.

r = supi ti

f ( B̃H 1(t j ), i = 1, 2, 1 ≤ j ≤ n)

= [f (BH i(t j), i = 1, 2, 1 ≤ j ≤ n)] .

( B̃H 1, B̃H 2)

(BH 1 , BH 2) .

, (X, Y )

( B̃H 1 , B̃H 2). H 1 > 1/2 f (X t)Λ

ht

σt(f ) def

=

f (X t)Λ

ht | Y s, s ≤ t

.

σt(f )

dσt(f ) = σt(f ψ(X t])) dt + σt(f.h) dBH 2(t), ,

ψ(X t]) X t. BH 1 . X , X W = C 0([0, 1],R)

X tdef =

t0

K H 1(., s) dB1(s),

BH 1 X t(t) = BH 1(t).

F : W → R, σ̃t(F ) def

=

F (X t)Λ

ht | Y t

σ̃t(F ) = F (x0) + t0 σ̃

s(F.h ◦ ps) dB2

(s)

+ 1

2

t0

σ̃s

(K H 1(., s))

∗ D2αF (.)(K H 1(., s))

ds,

ps(x) = x(s) x ∈ W D2αF

F B α,2. h = Id, F (x) = exp(iβx(1)) x(1)


19/24

1 x W, X(t, β ) = σ̃t(F ),

dX(t, β ) = −β 2K H (1, t)2X(t, β ) dt − i ∂ X∂β

(t, β ) dBH 2(t),

X(0, β ) = x0.

X Y BH 2


20/24

f ∈ L1([0, 1]) f

(I α0+f )(x) def

= 1

Γ(α)

x0

f (t)(x − t)α−1dt , x ≥ 0,

(I αb−f )(x) def

= 1

Γ(α)

bx

f (t)(t − x)α−1dt , x ≤ b,

α > 0 I 0 = Id . α ≥ 0 f ∈ L p([0, 1]) g ∈ Lq([0, 1]) p−1 + q −1 ≤ α

10

f (s)(I α0+g)(s) ds =

10

(I α1−f )(s)g(s) ds.

I α0+(L p) def

= B α,p

f Bα,p = I −α0+ f Lp .

0 < α 0. Hol(ν ) 0,

f Hol(ν ) = supt=s

|f (t) − f (s)|

|t − s|ν .

Hol−(ν ) Hol(η) η < ν Hol+(ν ) Hol(η) η > ν.

0 < α < β


21/24

n f

10 tn0 . . .

t20 f (t1, . . . , tn) dBt1dBt2 . . . dBtn.

L2s([0, 1]n) L2([0, 1]n).

f ∈ L2s([0, 1]n),

I n(f ) = n!

10

tn0

. . .

t20

f (t1, . . . , tn) dBt1dBt2 . . . dBtn.

L2([0, 1]n), I n(f ) I n(f s)

f s f. F (Ω, ) L2(Ω, )

F =+∞

n=0

I n(f n),

n, f n ∈ L2s([0, 1]

n).

F F D2,1

∞n=1

nn!f n2L2([0,1]n) < +∞,

∇tF =

∞

n=1

nI n−1(f n(., t)),

10 ∇tF

2 dt

.

k > 1, k F, ∇(k)F, k. D2,k

∇(k)F

D. Kf = (Kf ).

u ∈ L2(Ω × [0, 1]) K∇u

|

10

(K ◦ ∇)su(s) ds|

< ∞.

∇ L2(Ω, ) L2(Ω×[0, 1]). δ, L2(Ω×[0, 1]) L2(Ω, ). δ


22/24

δ u ∈ L2(Ω × [0, 1])

| 1

0∇tF ut dt

| ≤ cF L2(Ω),

F ∈ D2,1, c u. u Dom δ, δ (u) L2(Ω, )

[F δ (u)] =

10

∇tF ut dt

.

u ∈ L2(Ω × [0, 1]), f m(t1, . . . , tm, t), m, f m m

ut =

∞n=0

I n(f n(., t)),

L2(Ω × [0, 1])

10

u2t dt

=

∞n=0

n!f nL2([0,1]n+1).

δ

u ∈ L2(Ω × [0, 1]) u Dom δ

δ (u) =∞

n=0

I n+1( f̃ n)

L2(Ω, ), f̃ n f n,

f̃ n(t1, . . . , tn, t) = 1

n + 1

f n(t1, . . . , tn, t)

+n

i=1

f n(t1, . . . , ti−1, t , ti+1, . . . , tn)

.

δ (u) u(s) ◦ dB(s).

D

DF (s) = K(∇.F )(s),

K


23/24

trace(∇u) =

10 ∇sus u.

us;

s → ∇sus.

H > /

H > /

, K u H /

H < /

, K u ∇u, ∇.u (1 − H )

1/2

α


24/24

C 1

fbm survey

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