Post on 08-Jul-2018
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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
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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 .
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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),
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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)|.
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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).
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[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, .).
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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).
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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.
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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.
,
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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]),
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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.
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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).
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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
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{ 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,
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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
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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
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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
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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)
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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
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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
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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
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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
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trace(∇u) =
10 ∇sus u.
us;
s → ∇sus.
H > /
H > /
, K u H /
H < /
, K u ∇u, ∇.u (1 − H )
1/2
α
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C 1