fractional brownian motion - math.univ-brest.fr · contents 1 introduction 2 fbmandsomeproperties 3...

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

Contents

1 Introduction

2 FBM and Some Properties

3 Integral Representation

4 Wiener Integrals

5 Malliavin Calculus

Contents

1 Introduction

4 Wiener Integrals

Stochastic integration

We consider ∫ T

0· dBs .

Stochastic integration

We consider ∫ T

0· dBs .

Here B is a fractional Brownian motion.

Contents

1 Introduction

4 Wiener Integrals

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

Properties of fBm

B is a Brownian motion for H = 1/2.

Properties of fBm

B is a Brownian motion for H = 1/2.B has stationary increments.

Properties of fBm

B is a Brownian motion for H = 1/2.B has stationary increments :

E(|Bt − Bs |2

)= |t − s|2H .

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

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 ].

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.

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

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.

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.

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

≈ h2HH(2H − 1)n2H−2 → 0.

i) If H > 1/2, ρH(n) > 0 and∑∞

n=1 ρ(n) =∞.

≈ 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) <∞.

≈ 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

FBM is not a semimartingale

TheoremB is not a semimartingale for H 6= 1/2.

Proof : (i) Case H > 1/2.

Proof : (i) Case H > 1/2. Let Πt = 0 = t0 < t1 < . . . < tn = t bea partition of [0, t].

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

E( n∑

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 .

FBM is not a semimartingaleTheoremB is not a semimartingale for H 6= 1/2.

E( n∑

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].

Proof : (i) Case H > 1/2. Let Πt = 0 = t0 < t1 < . . . < tn be apartition of [0, t]. Then,

E( n∑

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

Proof : (i) Case H < 1/2.

Proof : (i) Case H < 1/2. We have

In : =n∑

j=1|Bj/n − B(j−1)/n|2

n∑j=1|Bj − Bj−1|2

= n1−2H

n∑j=1|Bj − Bj−1|2

→∞.

In : =n∑

j=1|Bj/n − B(j−1)/n|2

n∑j=1|Bj − Bj−1|2

= n1−2H

n∑j=1|Bj − Bj−1|2

→∞.Due to, the ergodic theorem implies that

n∑j=1|Bj − Bj−1|2 → E ((B1)2) a.s

.Jorge A. León (Cinvestav–IPN) FBM Roscoff 2010 28 / 62

Contents

1 Introduction

4 Wiener Integrals

Mandelbrot-van Ness representation

Bt = CH

[∫ 0

∞(t − s)H−1/2 − (−s)H−1/2dWs +

0(t − s)H−1/2dWs

Mandelbrot-van Ness representation

Bt = CH

[∫ t

∞(t − s)H−1/2 − (−s)H−1/2dWs +

0(t − s)H−1/2dWs

Here W is a Brownian motion.

Representation of fBm on an finite interval

Fix a time interval [0,T ] and consider the fBm B = Bt ; t ∈ [0,T ].

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 ,

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

s(u − s)H− 3

2 uH− 12 du s < t.

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,

s(u − s)H− 3

2 uH− 12 du, s < t.

For H < 1/2,

KH(t, s) = cH

)H− 12

(t − s)H− 12

−(H − 12)s 1

2−H∫ t

suH− 3

2 (u − s)H− 12 du

], s < t.

Contents

1 Introduction

4 Wiener Integrals

Wiener integrals

Let E be the family of the step functions of the form

f =n∑

i=0ai I(tj ,tj+1].

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.

Wiener integrals

L(B) = X ∈ L2(Ω) : X = L2(Ω)− limn→∞

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.

Wiener integrals

L(B) = X ∈ L2(Ω) : X = L2(Ω)− limn→∞

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.

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.

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 .

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 ]).

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

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).

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 .

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

s(u− s)H− 3

2 uH− 12 du, s < t and H > 1/2

KH(t, s) = cH

)H− 12

(t − s)H− 12

−(H − 12)s 1

2−H∫ t

suH− 3

2 (u − s)H− 12 du

], H < 1/2.

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 ].

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 ].

Contents

1 Introduction

4 Wiener Integrals

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.

Derivative operator

F = f (B(φ1), . . . ,B(φn)),

where n ≥ 1, f ∈ C∞b (Rn) and φi ∈ H. The derivative operator isgiven by

DF =n∑

∂f∂xi

(B(φ1), . . . ,B(φn))φi .

Derivative operator

F = f (B(φ1), . . . ,B(φn)),

where n ≥ 1, f ∈ C∞b (Rn) and φi ∈ H. The derivative operator isgiven by

DF =n∑

∂f∂xi

(B(φ1), . . . ,B(φn))φi .

The operator D is closable from L2(Ω) into L2(Ω;H).

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).

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 .

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).

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).

Remark For H = 1/2, H = L2([0,T ]).

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).

Remark For H = 1/2, H = L2([0,T ]). So

E(F δW (u)

(∫ T

s F )usds), F ∈ S, u ∈ L2(Ω× [0,T ]).

Divergence operator

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)

Divergence operator

F δ(u) = δ(Fu) + 〈DF , u〉H.

Divergence operator

F δ(u) = δ(Fu) + 〈DF , u〉H.

Proof : Let G ∈ S, then

Divergence operatorThe divergence operator δ is the adjoint of D. It is defined by theduality relation

F δ(u) = δ(Fu) + 〈DF , u〉H.

E (〈DG ,Fu〉H) = E (〈D(GF ), u〉H − G 〈DF , u〉H)

Divergence operatorThe divergence operator δ is the adjoint of D.

F δ(u) = δ(Fu)− 〈DF , u〉H.

E (〈DG ,Fu〉H) = E (〈D(GF ), u〉H − G 〈DF , u〉H)

= E (G (F δ(u)− 〈DF , u〉H)) .

fractional brownian motion - math.univ-brest.fr · contents 1 introduction 2 fbmandsomeproperties 3...

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