fractional brownian motion - math.univ-brest.fr · contents 1 introduction 2 fbmandsomeproperties 3...
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Fractional Brownian motion
Jorge A. León
Departamento de Control AutomáticoCinvestav del IPN
Spring School “Stochastic Control in Finance”, Roscoff 2010
Jorge A. León (Cinvestav–IPN) FBM Roscoff 2010 1 / 62
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Contents
1 Introduction
2 FBM and Some Properties
3 Integral Representation
4 Wiener Integrals
5 Malliavin Calculus
Jorge A. León (Cinvestav–IPN) FBM Roscoff 2010 2 / 62
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Contents
1 Introduction
2 FBM and Some Properties
3 Integral Representation
4 Wiener Integrals
5 Malliavin Calculus
Jorge A. León (Cinvestav–IPN) FBM Roscoff 2010 3 / 62
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Stochastic integration
We consider ∫ T
0· dBs .
Jorge A. León (Cinvestav–IPN) FBM Roscoff 2010 4 / 62
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Stochastic integration
We consider ∫ T
0· dBs .
Here B is a fractional Brownian motion.
Jorge A. León (Cinvestav–IPN) FBM Roscoff 2010 5 / 62
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Contents
1 Introduction
2 FBM and Some Properties
3 Integral Representation
4 Wiener Integrals
5 Malliavin Calculus
Jorge A. León (Cinvestav–IPN) FBM Roscoff 2010 6 / 62
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Fractional Brownian motion
DefinitionA Gaussian stochastic process B = Bt ; t ≥ 0 is called a fractionalBrownian motion (fBm) of Hurst parameter H ∈ (0, 1) if it has zeromean and covariance fuction
RH(t, s) = E (BtBs) =12(t2H + s2H − |t − s|2H
).
Jorge A. León (Cinvestav–IPN) FBM Roscoff 2010 7 / 62
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Properties of fBm
DefinitionA Gaussian stochastic process B = Bt ; t ≥ 0 is called a fractionalBrownian motion (fBm) of Hurst parameter H ∈ (0, 1) if it has zeromean and covariance fuction
RH(t, s) = E (BtBs) =12(t2H + s2H − |t − s|2H
).
B is a Brownian motion for H = 1/2.
Jorge A. León (Cinvestav–IPN) FBM Roscoff 2010 8 / 62
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Properties of fBm
DefinitionA Gaussian stochastic process B = Bt ; t ≥ 0 is called a fractionalBrownian motion (fBm) of Hurst parameter H ∈ (0, 1) if it has zeromean and covariance fuction
RH(t, s) = E (BtBs) =12(t2H + s2H − |t − s|2H
).
B is a Brownian motion for H = 1/2.B has stationary increments.
Jorge A. León (Cinvestav–IPN) FBM Roscoff 2010 9 / 62
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Properties of fBm
DefinitionA Gaussian stochastic process B = Bt ; t ≥ 0 is called a fractionalBrownian motion (fBm) of Hurst parameter H ∈ (0, 1) if it has zeromean and covariance fuction
RH(t, s) = E (BtBs) =12(t2H + s2H − |t − s|2H
).
B is a Brownian motion for H = 1/2.B has stationary increments :
E(|Bt − Bs |2
)= |t − s|2H .
Jorge A. León (Cinvestav–IPN) FBM Roscoff 2010 10 / 62
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Properties of fBmDefinitionA Gaussian stochastic process B = Bt ; t ≥ 0 is called a fractionalBrownian motion (fBm) of Hurst parameter H ∈ (0, 1) if it has zeromean and covariance fuction
RH(t, s) = E (BtBs) =12(t2H + s2H − |t − s|2H
).
B is a Brownian motion for H = 1/2.B has stationary increments :
E(|Bt − Bs |2
)= |t − s|2H .
For any ε ∈ (0,H) and T > 0, there exists Gε,T such that
|Bt − Bs | ≤ Gε,T |t − s|H−ε, t, s ∈ [0,T ].
Jorge A. León (Cinvestav–IPN) FBM Roscoff 2010 11 / 62
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Properties of fBm
DefinitionA Gaussian stochastic process B = Bt ; t ≥ 0 is called a fractionalBrownian motion (fBm) of Hurst parameter H ∈ (0, 1) if it has zeromean and covariance fuction
RH(t, s) = E (BtBs) =12(t2H + s2H − |t − s|2H
).
B is a Brownian motion for H = 1/2.B has stationary increments.B is Hölder continuous for any exponent less than H .B is self-similar (with index H). That is, for any a > 0,a−HBat ; t ≥ 0 and Bt ; t ≥ 0 have the same distribution.
Jorge A. León (Cinvestav–IPN) FBM Roscoff 2010 12 / 62
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Properties of fBmDefinitionA Gaussian stochastic process B = Bt ; t ≥ 0 is called a fractionalBrownian motion (fBm) of Hurst parameter H ∈ (0, 1) if it has zeromean and covariance fuction
RH(t, s) = E (BtBs) =12(t2H + s2H − |t − s|2H
).
B has stationary increments.B is Hölder continuous for any exponent less than H .B is self-similar (with index H). That is, for any a > 0,a−HBat ; t ≥ 0 and Bt ; t ≥ 0 have the same distribution.The covariance of its increments on intervals decaysasymptotically as a negative power of the distance between theintervals.
Jorge A. León (Cinvestav–IPN) FBM Roscoff 2010 13 / 62
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Properties of fBm
B is a Brownian motion for H = 1/2.B has stationary increments.B is Hölder continuous for any exponent less than H .B is self-similar (with index H). That is, for any a > 0,a−HBat ; t ≥ 0 and Bt ; t ≥ 0 have the same distribution.The covariance of its increments on intervals decaysasymptotically as a negative power of the distance between theintervals : Let t − s = nh and
ρH(n) = E [(Bt+h − Bt)(Bs+h − Bs)]
≈ h2HH(2H − 1)n2H−2 → 0.
Jorge A. León (Cinvestav–IPN) FBM Roscoff 2010 14 / 62
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Properties of fBmB is a Brownian motion for H = 1/2.B has stationary increments.B is Hölder continuous for any exponent less than H .B is self-similar (with index H). That is, for any a > 0,a−HBat ; t ≥ 0 and Bt ; t ≥ 0 have the same distribution.The covariance of its increments on intervals decaysasymptotically as a negative power of the distance between theintervals : Let t − s = nh and
ρH(n) = E [(Bt+h − Bt)(Bs+h − Bs)]
≈ h2HH(2H − 1)n2H−2 → 0.
i) If H > 1/2, ρH(n) > 0 and∑∞
n=1 ρ(n) =∞.
Jorge A. León (Cinvestav–IPN) FBM Roscoff 2010 15 / 62
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Properties of fBmB is a Brownian motion for H = 1/2.B has stationary increments.B is Hölder continuous for any exponent less than H .B is self-similar (with index H). That is, for any a > 0,a−HBat ; t ≥ 0 and Bt ; t ≥ 0 have the same distribution.The covariance of its increments on intervals decaysasymptotically as a negative power of the distance between theintervals : Let t − s = nh and
ρH(n) = E [(Bt+h − Bt)(Bs+h − Bs)]
≈ h2HH(2H − 1)n2H−2 → 0.
i) If H > 1/2, ρH(n) > 0 and∑∞
n=1 ρ(n) =∞.ii) If H < 1/2, ρH(n) < 0 and
∑∞n=1 ρ(n) <∞.
Jorge A. León (Cinvestav–IPN) FBM Roscoff 2010 16 / 62
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Properties of fBmB is a Brownian motion for H = 1/2.B has stationary increments.B is Hölder continuous for any exponent less than H .B is self-similar (with index H). That is, for any a > 0,a−HBat ; t ≥ 0 and Bt ; t ≥ 0 have the same distribution.The covariance of its increments on intervals decaysasymptotically as a negative power of the distance between theintervals : Let t − s = nh and
ρH(n) = E [(Bt+h − Bt)(Bs+h − Bs)]
≈ h2HH(2H − 1)n2H−2 → 0.
i) If H > 1/2, ρH(n) > 0 and∑∞
n=1 ρ(n) =∞.ii) If H < 1/2, ρH(n) < 0 and
∑∞n=1 ρ(n) <∞.
B has no bounded variation paths.Jorge A. León (Cinvestav–IPN) FBM Roscoff 2010 17 / 62
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FBM is not a semimartingale
TheoremB is not a semimartingale for H 6= 1/2.
Jorge A. León (Cinvestav–IPN) FBM Roscoff 2010 18 / 62
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FBM is not a semimartingale
TheoremB is not a semimartingale for H 6= 1/2.
Proof : (i) Case H > 1/2.
Jorge A. León (Cinvestav–IPN) FBM Roscoff 2010 19 / 62
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FBM is not a semimartingale
TheoremB is not a semimartingale for H 6= 1/2.
Proof : (i) Case H > 1/2. Let Πt = 0 = t0 < t1 < . . . < tn = t bea partition of [0, t].
Jorge A. León (Cinvestav–IPN) FBM Roscoff 2010 20 / 62
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FBM is not a semimartingale
TheoremB is not a semimartingale for H 6= 1/2.
Proof : (i) Case H > 1/2. Let Πt = 0 = t0 < t1 < . . . < tn = t bea partition of [0, t]. Then,
E( n∑
i=1|Bti − Bti−1|2
)=
n∑i=1|ti − ti−1|2H
≤ |Π|2H−1n∑
i=1|ti − ti−1|
= t|Π|2H−1 → 0
Jorge A. León (Cinvestav–IPN) FBM Roscoff 2010 21 / 62
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FBM is not a semimartingale
TheoremB is not a semimartingale for H 6= 1/2.
Proof : (i) Case H > 1/2. Let Πt = 0 = t0 < t1 < . . . < tn = t bea partition of [0, t]. Then,
E( n∑
i=1|Bti − Bti−1|2
)=
n∑i=1|ti − ti−1|2H
≤ |Π|2H−1n∑
i=1|ti − ti−1|
= t|Π|2H−1 → 0
If B were a semimartingale. Then, B = M + V .
Jorge A. León (Cinvestav–IPN) FBM Roscoff 2010 22 / 62
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FBM is not a semimartingaleTheoremB is not a semimartingale for H 6= 1/2.
Proof : (i) Case H > 1/2. Let Πt = 0 = t0 < t1 < . . . < tn = t bea partition of [0, t]. Then,
E( n∑
i=1|Bti − Bti−1|2
)=
n∑i=1|ti − ti−1|2H
≤ |Π|2H−1n∑
i=1|ti − ti−1|
= t|Π|2H−1 → 0
If B were a semimartingale. Then, B = M + V . Thus
0 = [B] = [M].
Jorge A. León (Cinvestav–IPN) FBM Roscoff 2010 23 / 62
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FBM is not a semimartingaleTheoremB is not a semimartingale for H 6= 1/2.
Proof : (i) Case H > 1/2. Let Πt = 0 = t0 < t1 < . . . < tn be apartition of [0, t]. Then,
E( n∑
i=1|Bti − Bti−1|2
)=
n∑i=1|ti − ti−1|2H
≤ |Π|2H−1n∑
i=1|ti − ti−1|
= t|Π|2H−1 → 0
If B were a semimartingale. Then, B = M + V . Thus
0 = [B] = [M].
Consequently B = V .Jorge A. León (Cinvestav–IPN) FBM Roscoff 2010 24 / 62
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FBM is not a semimartingale
TheoremB is not a semimartingale for H 6= 1/2.
Proof : (i) Case H < 1/2.
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FBM is not a semimartingale
TheoremB is not a semimartingale for H 6= 1/2.
Proof : (i) Case H < 1/2. We have
Jorge A. León (Cinvestav–IPN) FBM Roscoff 2010 26 / 62
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FBM is not a semimartingale
TheoremB is not a semimartingale for H 6= 1/2.
Proof : (i) Case H < 1/2. We have
In : =n∑
j=1|Bj/n − B(j−1)/n|2
(d)=
1n2H
n∑j=1|Bj − Bj−1|2
= n1−2H
1n
n∑j=1|Bj − Bj−1|2
→∞.
Jorge A. León (Cinvestav–IPN) FBM Roscoff 2010 27 / 62
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FBM is not a semimartingaleTheoremB is not a semimartingale for H 6= 1/2.
Proof : (i) Case H < 1/2. We have
In : =n∑
j=1|Bj/n − B(j−1)/n|2
(d)=
1n2H
n∑j=1|Bj − Bj−1|2
= n1−2H
1n
n∑j=1|Bj − Bj−1|2
→∞.Due to, the ergodic theorem implies that
1n
n∑j=1|Bj − Bj−1|2 → E ((B1)2) a.s
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Contents
1 Introduction
2 FBM and Some Properties
3 Integral Representation
4 Wiener Integrals
5 Malliavin Calculus
Jorge A. León (Cinvestav–IPN) FBM Roscoff 2010 29 / 62
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Mandelbrot-van Ness representation
Bt = CH
[∫ 0
∞(t − s)H−1/2 − (−s)H−1/2dWs +
∫ t
0(t − s)H−1/2dWs
].
Jorge A. León (Cinvestav–IPN) FBM Roscoff 2010 30 / 62
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Mandelbrot-van Ness representation
Bt = CH
[∫ t
∞(t − s)H−1/2 − (−s)H−1/2dWs +
∫ t
0(t − s)H−1/2dWs
].
Here W is a Brownian motion.
Jorge A. León (Cinvestav–IPN) FBM Roscoff 2010 31 / 62
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Representation of fBm on an finite interval
Fix a time interval [0,T ] and consider the fBm B = Bt ; t ∈ [0,T ].
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Representation of fBm on an finite interval
Fix a time interval [0,T ] and consider the fBm B = Bt ; t ∈ [0,T ].Then there exists a Bm Wt ; t ∈ [0,T ] such that
Bt =∫ t
0KH(t, s)dWs ,
Jorge A. León (Cinvestav–IPN) FBM Roscoff 2010 33 / 62
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Representation of fBm on an finite interval
Fix a time interval [0,T ] and consider the fBm B = Bt ; t ∈ [0,T ].Then there exists a Bm Wt ; t ∈ [0,T ] such that
Bt =∫ t
0KH(t, s)dWs ,
whereFor H > 1/2,
KH(t, s) = cHs 12−H
∫ t
s(u − s)H− 3
2 uH− 12 du s < t.
Jorge A. León (Cinvestav–IPN) FBM Roscoff 2010 34 / 62
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Representation of fBm on an finite intervalFix a time interval [0,T ] and consider the fBm B = Bt ; t ∈ [0,T ].Then there exists a Bm Wt ; t ∈ [0,T ] such that
Bt =∫ t
0KH(t, s)dWs ,
whereFor H > 1/2,
KH(t, s) = cHs 12−H
∫ t
s(u − s)H− 3
2 uH− 12 du, s < t.
For H < 1/2,
KH(t, s) = cH
[( ts
)H− 12
(t − s)H− 12
−(H − 12)s 1
2−H∫ t
suH− 3
2 (u − s)H− 12 du
], s < t.
Jorge A. León (Cinvestav–IPN) FBM Roscoff 2010 35 / 62
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Contents
1 Introduction
2 FBM and Some Properties
3 Integral Representation
4 Wiener Integrals
5 Malliavin Calculus
Jorge A. León (Cinvestav–IPN) FBM Roscoff 2010 36 / 62
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Wiener integrals
Let E be the family of the step functions of the form
f =n∑
i=0ai I(tj ,tj+1].
Jorge A. León (Cinvestav–IPN) FBM Roscoff 2010 37 / 62
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Wiener integrals
Let E be the family of the step functions of the form
f =n∑
i=0ai I(tj ,tj+1].
The Wiener integral with respect to B
I(f ) =n∑
i=0ai(Bti+1 − Bti )
and the space
L(B) = X ∈ L2(Ω) : X = L2(Ω)− limn→∞
I(fn), for some fn ⊂ E.
Jorge A. León (Cinvestav–IPN) FBM Roscoff 2010 38 / 62
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Wiener integrals
L(B) = X ∈ L2(Ω) : X = L2(Ω)− limn→∞
I(fn), for some fn ⊂ E.
Proposition (Pipiras and Taqqu)Suppose that H is a inner product space with inner product (·, ·)such that :i) E ⊂ H and (f , g) = E (I(f )I(g)), for f , g ∈ E .ii) E is dense in H.
Then H is isometric to L(B) if and only if H is complete.
Jorge A. León (Cinvestav–IPN) FBM Roscoff 2010 39 / 62
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Wiener integrals
L(B) = X ∈ L2(Ω) : X = L2(Ω)− limn→∞
I(fn), for some fn ⊂ E.
Proposition (Pipiras and Taqqu)Suppose that H is a inner product space with inner product (·, ·)such that :i) E ⊂ H and (f , g) = E (I(f )I(g)), for f , g ∈ E .ii) E is dense in H.
Then H is isometric to L(B) if and only if H is complete. Moreover,the isometry is an extension of I.
Jorge A. León (Cinvestav–IPN) FBM Roscoff 2010 40 / 62
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Wiener integralsProposition (Pipiras and Taqqu)Suppose that H is a inner product space with inner product (·, ·)such that :i) E ⊂ H and (f , g) = E (I(f )I(g)), for f , g ∈ E .ii) E is dense in H.
Then H is isometric to L(B) if and only if H is complete. Moreover,the isometry is an extension of I.
Remarksa) For H < 1/2,
H = f ∈ L2([0,T ]) : f (s) = cHs 12−H(I
12−HT− uH− 1
2φf (u))(s)
for some φf ∈ L2with (IαT−g)(s) = 1
Γ(α)
∫ Ts (x − s)α−1g(x)dx .
Jorge A. León (Cinvestav–IPN) FBM Roscoff 2010 41 / 62
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Wiener integralsProposition (Pipiras and Taqqu)Suppose that H is a inner product space with inner product (·, ·)such that :i) E ⊂ H and (f , g) = E (I(f )I(g)), for f , g ∈ E .ii) E is dense in H.
Then H is isometric to L(B) if and only if H is complete. Moreover,the isometry is an extension of I.
Remarksa) For H < 1/2,
H = f ∈ L2([0,T ]) : f (s) = cHs 12−H(I
12−HT− uH− 1
2φf (u))(s)
for some φf ∈ L2with the inner product (f , g) = (φf , φg )L2([0,T ]).
Jorge A. León (Cinvestav–IPN) FBM Roscoff 2010 42 / 62
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Wiener integralsProposition (Pipiras and Taqqu)Suppose that H is a inner product space with inner product (·, ·)such that :i) E ⊂ H and (f , g) = E (I(f )I(g)), for f , g ∈ E .ii) E is dense in H.
Then H is isometric to L(B) if and only if H is complete. Moreover,the isometry is an extension of I.
Remarksb) For H > 1/2,
H = f ∈ D′ : ∃f ∗ ∈ W 1/2−H,2(R) with supp(f ) ⊂ [0,T ]
such that f = f ∗|[0,T ]
with the inner product (f , g) = cH∫
R F f ∗(x)Fg∗(x)|x |1−2Hdx .Jorge A. León (Cinvestav–IPN) FBM Roscoff 2010 43 / 62
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Wiener integralsa) For H < 1/2,
H = f ∈ L2([0,T ]) : f (s) = cHs 12−H(I
12−HT− uH− 1
2φf (u))(s)
for some φf ∈ L2
with the inner product (f , g) = (φf , φg )L2([0,T ]).
b) For H > 1/2,
H = f ∈ D′ : ∃f ∗ ∈ W 1/2−H,2(R) with supp(f ) ⊂ [0,T ]
such that f = f ∗|[0,T ]
with the inner product (f , g) = cH∫
R F f ∗(x)Fg∗(x)|x |1−2Hdx .c) W s,2(R) = f ∈ S : (1 + |x |2)s/2F f (x) ∈ L2(R).
Jorge A. León (Cinvestav–IPN) FBM Roscoff 2010 44 / 62
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Representation of Wiener integrals
Moreover, there exists an isometry K ∗H : H → L2([0,T ]) and aBrownian motion W such that :
1 I(f ) =∫ T0 (K ∗H f )(s)dWs .
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Representation of Wiener integralsMoreover, there exists an isometry K ∗H : H → L2([0,T ]) and aBrownian motion W such that :
1 I(f ) =∫ T0 (K ∗H f )(s)dWs .
2 K ∗H I[0,t] = KH(t, ·) with
KH(t, s) = cHs 12−H
∫ t
s(u− s)H− 3
2 uH− 12 du, s < t and H > 1/2
and
KH(t, s) = cH
[( ts
)H− 12
(t − s)H− 12
−(H − 12)s 1
2−H∫ t
suH− 3
2 (u − s)H− 12 du
], H < 1/2.
Jorge A. León (Cinvestav–IPN) FBM Roscoff 2010 46 / 62
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Representation of Wiener integrals
Moreover, there exists an isometry K ∗H : H → L2([0,T ]) and aBrownian motion W such that :
1 I(φ) =∫ T0 (K ∗Hφ)(s)dWs .
2 K ∗H I[0,t] = KH(t, ·) .3 For H < 1/2,
K ∗H f = φf ,
with f (s) = cHs 12−H(I
12−HT− uH− 1
2φf (u))(s), s ∈ [0,T ].
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Representation of Wiener integrals
Moreover, there exists an isometry K ∗H : H → L2([0,T ]) and aBrownian motion W such that :
1 I(φ) =∫ T0 (K ∗Hφ)(s)dWs .
2 K ∗H I[0,t] = KH(t, ·) .3 For H < 1/2,
K ∗H f = φf ,
with f (s) = cHs 12−H(I
12−HT− uH− 1
2φf (u))(s), s ∈ [0,T ].
4 For H > 1/2 and φ ∈ E ,
(K ∗Hφ)(s) = cHs1/2−H(IH− 12
T− uh− 12φ(u))(s), s ∈ [0,T ].
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Contents
1 Introduction
2 FBM and Some Properties
3 Integral Representation
4 Wiener Integrals
5 Malliavin Calculus
Jorge A. León (Cinvestav–IPN) FBM Roscoff 2010 49 / 62
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Derivative operator
Let S be the set of smooth functional of the form
F = f (B(φ1), . . . ,B(φn)),
where n ≥ 1, f ∈ C∞b (Rn) and φi ∈ H.
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Derivative operator
Let S be the set of smooth functional of the form
F = f (B(φ1), . . . ,B(φn)),
where n ≥ 1, f ∈ C∞b (Rn) and φi ∈ H. The derivative operator isgiven by
DF =n∑
i=1
∂f∂xi
(B(φ1), . . . ,B(φn))φi .
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Derivative operator
Let S be the set of smooth functional of the form
F = f (B(φ1), . . . ,B(φn)),
where n ≥ 1, f ∈ C∞b (Rn) and φi ∈ H. The derivative operator isgiven by
DF =n∑
i=1
∂f∂xi
(B(φ1), . . . ,B(φn))φi .
The operator D is closable from L2(Ω) into L2(Ω;H).
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Divergence operator
The divergence operator δ is the adjoint of D. It is defined by theduality relation
E (F δ(u)) = E (〈DF , u〉H) , F ∈ S, u ∈ L2(Ω,H).
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Transfer principle
Let W be the Brownian motion such that
Bt =∫ t
0KH(t, s)dWs t ∈ [0,T ].
Then,1 DomD=DomDW and
K ∗HDF = DW F .
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Transfer principle
Let W be the Brownian motion such that
Bt =∫ t
0KH(t, s)dWs t ∈ [0,T ].
Then,1 DomD=DomDW and
K ∗HDF = DW F .
2 φ ∈Domδ if and only if K ∗Hφ ∈DomδW and
δ(φ) = δW (K ∗Hφ).
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Transfer principle
1 DomD=DomDW and
K ∗HDF = DW F .
2 φ ∈Domδ if and only if K ∗Hφ ∈DomδW and
δ(φ) = δW (K ∗Hφ).
The divergence operator δ is the adjoint of D. It is defined by theduality relation
E (F δ(u)) = E (〈DF , u〉H) , F ∈ S, u ∈ L2(Ω,H).
Remark For H = 1/2, H = L2([0,T ]).
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Transfer principle1 DomD=DomDW and K ∗HDF = DW F .2 φ ∈Domδ if and only if K ∗Hφ ∈DomδW and
δ(φ) = δW (K ∗Hφ).
The divergence operator δ is the adjoint of D. It is defined by theduality relation
E (F δ(u)) = E (〈DF , u〉H) , F ∈ S, u ∈ L2(Ω,H).
Remark For H = 1/2, H = L2([0,T ]). So
E(F δW (u)
)= E
(∫ T
0(DW
s F )usds), F ∈ S, u ∈ L2(Ω× [0,T ]).
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Divergence operator
The divergence operator δ is the adjoint of D. It is defined by theduality relation
E (F δ(u)) = E (〈DF , u〉H) , F ∈ S, u ∈ L2(Ω,H).
PropositionLet u ∈Dom δ and F ∈ Dom D such that (Fu) ∈ L2(Ω;H) and(F δ(u)− 〈DF , u〉H) ∈ L2(Ω). Then
F δ(u) = δ(Fu)
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Divergence operator
The divergence operator δ is the adjoint of D. It is defined by theduality relation
E (F δ(u)) = E (〈DF , u〉H) , F ∈ S, u ∈ L2(Ω,H).
PropositionLet u ∈Dom δ and F ∈ Dom D such that (Fu) ∈ L2(Ω;H) and(F δ(u)− 〈DF , u〉H) ∈ L2(Ω). Then
F δ(u) = δ(Fu) + 〈DF , u〉H.
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Divergence operator
The divergence operator δ is the adjoint of D. It is defined by theduality relation
E (F δ(u)) = E (〈DF , u〉H) , F ∈ S, u ∈ L2(Ω,H).
PropositionLet u ∈Dom δ and F ∈ Dom D such that (Fu) ∈ L2(Ω;H) and(F δ(u)− 〈DF , u〉H) ∈ L2(Ω). Then
F δ(u) = δ(Fu) + 〈DF , u〉H.
Proof : Let G ∈ S, then
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Divergence operatorThe divergence operator δ is the adjoint of D. It is defined by theduality relation
E (F δ(u)) = E (〈DF , u〉H) , F ∈ S, u ∈ L2(Ω,H).
PropositionLet u ∈Dom δ and F ∈ Dom D such that (Fu) ∈ L2(Ω;H) and(F δ(u)− 〈DF , u〉H) ∈ L2(Ω). Then
F δ(u) = δ(Fu) + 〈DF , u〉H.
Proof : Let G ∈ S, then
E (〈DG ,Fu〉H) = E (〈D(GF ), u〉H − G 〈DF , u〉H)
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Divergence operatorThe divergence operator δ is the adjoint of D.
E (F δ(u)) = E (〈DF , u〉H) , F ∈ S, u ∈ L2(Ω,H).
PropositionLet u ∈Dom δ and F ∈ Dom D such that (Fu) ∈ L2(Ω;H) and(F δ(u)− 〈DF , u〉H) ∈ L2(Ω). Then
F δ(u) = δ(Fu)− 〈DF , u〉H.
Proof : Let G ∈ S, then
E (〈DG ,Fu〉H) = E (〈D(GF ), u〉H − G 〈DF , u〉H)
= E (G (F δ(u)− 〈DF , u〉H)) .
Jorge A. León (Cinvestav–IPN) FBM Roscoff 2010 62 / 62