Fractional Brownian motion Jorge A. Len Departamento de Control - - PowerPoint PPT Presentation

Fractional Brownian motion

Jorge A. León

Departamento de Control Automático Cinvestav 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

Jorge A. León (Cinvestav–IPN) FBM Roscoff 2010 2 / 62

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

Stochastic integration

We consider

T

0 · dBs.

Jorge A. León (Cinvestav–IPN) FBM Roscoff 2010 4 / 62

Stochastic integration

We consider

T

0 · dBs.

Here B is a fractional Brownian motion.

Jorge A. León (Cinvestav–IPN) FBM Roscoff 2010 5 / 62

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

Fractional Brownian motion

Definition

A Gaussian stochastic process B = {Bt; t ≥ 0} is called a fractional Brownian motion (fBm) of Hurst parameter H ∈ (0, 1) if it has zero mean and covariance fuction RH(t, s) = E (BtBs) = 1 2

• t2H + s2H − |t − s|2H

.

Jorge A. León (Cinvestav–IPN) FBM Roscoff 2010 7 / 62

Properties of fBm

Definition

A Gaussian stochastic process B = {Bt; t ≥ 0} is called a fractional Brownian motion (fBm) of Hurst parameter H ∈ (0, 1) if it has zero mean and covariance fuction RH(t, s) = E (BtBs) = 1 2

• t2H + s2H − |t − s|2H

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

Jorge A. León (Cinvestav–IPN) FBM Roscoff 2010 8 / 62

Properties of fBm

Definition

A Gaussian stochastic process B = {Bt; t ≥ 0} is called a fractional Brownian motion (fBm) of Hurst parameter H ∈ (0, 1) if it has zero mean and covariance fuction RH(t, s) = E (BtBs) = 1 2

• 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

Properties of fBm

Definition

A Gaussian stochastic process B = {Bt; t ≥ 0} is called a fractional Brownian motion (fBm) of Hurst parameter H ∈ (0, 1) if it has zero mean and covariance fuction RH(t, s) = E (BtBs) = 1 2

• 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

Properties of fBm

Definition

A Gaussian stochastic process B = {Bt; t ≥ 0} is called a fractional Brownian motion (fBm) of Hurst parameter H ∈ (0, 1) if it has zero mean and covariance fuction RH(t, s) = E (BtBs) = 1 2

• 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

Properties of fBm

Definition

A Gaussian stochastic process B = {Bt; t ≥ 0} is called a fractional Brownian motion (fBm) of Hurst parameter H ∈ (0, 1) if it has zero mean and covariance fuction RH(t, s) = E (BtBs) = 1 2

• 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

Properties of fBm

Definition

A Gaussian stochastic process B = {Bt; t ≥ 0} is called a fractional Brownian motion (fBm) of Hurst parameter H ∈ (0, 1) if it has zero mean and covariance fuction RH(t, s) = E (BtBs) = 1 2

• 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 decays asymptotically as a negative power of the distance between the intervals.

Jorge A. León (Cinvestav–IPN) FBM Roscoff 2010 13 / 62

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 decays asymptotically as a negative power of the distance between the intervals : 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

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 decays asymptotically as a negative power of the distance between the intervals : 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

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 decays asymptotically as a negative power of the distance between the intervals : 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

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 decays asymptotically as a negative power of the distance between the intervals : 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

FBM is not a semimartingale

Theorem

B is not a semimartingale for H = 1/2.

Jorge A. León (Cinvestav–IPN) FBM Roscoff 2010 18 / 62

FBM is not a semimartingale

Theorem

B is not a semimartingale for H = 1/2. Proof : (i) Case H > 1/2.

Jorge A. León (Cinvestav–IPN) FBM Roscoff 2010 19 / 62

FBM is not a semimartingale

Theorem

B is not a semimartingale for H = 1/2. Proof : (i) Case H > 1/2. Let Πt = {0 = t0 < t1 < . . . < tn = t} be a partition of [0, t].

Jorge A. León (Cinvestav–IPN) FBM Roscoff 2010 20 / 62

FBM is not a semimartingale

Theorem

B is not a semimartingale for H = 1/2. Proof : (i) Case H > 1/2. Let Πt = {0 = t0 < t1 < . . . < tn = t} be a partition of [0, t]. Then, E

n

• i=1

|Bti − Bti−1|2

• =

n

• i=1

|ti − ti−1|2H ≤ |Π|2H−1

n

• i=1

|ti − ti−1| = t|Π|2H−1 → 0

Jorge A. León (Cinvestav–IPN) FBM Roscoff 2010 21 / 62

FBM is not a semimartingale

Theorem

B is not a semimartingale for H = 1/2. Proof : (i) Case H > 1/2. Let Πt = {0 = t0 < t1 < . . . < tn = t} be a partition of [0, t]. Then, E

n

• i=1

|Bti − Bti−1|2

• =

n

• i=1

|ti − ti−1|2H ≤ |Π|2H−1

n

• 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

FBM is not a semimartingale

Theorem

B is not a semimartingale for H = 1/2. Proof : (i) Case H > 1/2. Let Πt = {0 = t0 < t1 < . . . < tn = t} be a partition of [0, t]. Then, E

n

• i=1

|Bti − Bti−1|2

• =

n

• i=1

|ti − ti−1|2H ≤ |Π|2H−1

n

• 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

FBM is not a semimartingale

Theorem

B is not a semimartingale for H = 1/2. Proof : (i) Case H > 1/2. Let Πt = {0 = t0 < t1 < . . . < tn} be a partition of [0, t]. Then, E

n

• i=1

|Bti − Bti−1|2

• =

n

• i=1

|ti − ti−1|2H ≤ |Π|2H−1

n

• 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

FBM is not a semimartingale

Theorem

B is not a semimartingale for H = 1/2. Proof : (i) Case H < 1/2.

Jorge A. León (Cinvestav–IPN) FBM Roscoff 2010 25 / 62

FBM is not a semimartingale

Theorem

B is not a semimartingale for H = 1/2. Proof : (i) Case H < 1/2. We have

Jorge A. León (Cinvestav–IPN) FBM Roscoff 2010 26 / 62

FBM is not a semimartingale

Theorem

B is not a semimartingale for H = 1/2. Proof : (i) Case H < 1/2. We have In : =

n

• j=1

|Bj/n − B(j−1)/n|2 (d) = 1 n2H

n

• j=1

|Bj − Bj−1|2 = n1−2H

 1

n

n

• j=1

|Bj − Bj−1|2

  → ∞.

Jorge A. León (Cinvestav–IPN) FBM Roscoff 2010 27 / 62

FBM is not a semimartingale

Theorem

B is not a semimartingale for H = 1/2. Proof : (i) Case H < 1/2. We have In : =

n

• j=1

|Bj/n − B(j−1)/n|2 (d) = 1 n2H

n

• j=1

|Bj − Bj−1|2 = n1−2H

 1

n

n

• j=1

|Bj − Bj−1|2

  → ∞.

Due to, the ergodic theorem implies that 1 n

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

2

FBM and Some Properties

3

Integral Representation

4

Wiener Integrals

5

Malliavin Calculus

Jorge A. León (Cinvestav–IPN) FBM Roscoff 2010 29 / 62

Mandelbrot-van Ness representation

Bt = CH

∞{(t − s)H−1/2 − (−s)H−1/2}dWs +

t

0 (t − s)H−1/2dWs

• .

Jorge A. León (Cinvestav–IPN) FBM Roscoff 2010 30 / 62

Mandelbrot-van Ness representation

Bt = CH

t

∞{(t − s)H−1/2 − (−s)H−1/2}dWs +

t

0 (t − s)H−1/2dWs

• .

Here W is a Brownian motion.

Jorge A. León (Cinvestav–IPN) FBM Roscoff 2010 31 / 62

Representation of fBm on an finite interval

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

Jorge A. León (Cinvestav–IPN) FBM Roscoff 2010 32 / 62

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

0 KH(t, s)dWs,

Jorge A. León (Cinvestav–IPN) FBM Roscoff 2010 33 / 62

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

0 KH(t, s)dWs,

where For H > 1/2, KH(t, s) = cHs

1 2 −H

t

s (u − s)H− 3

2uH− 1 2du

s < t.

Jorge A. León (Cinvestav–IPN) FBM Roscoff 2010 34 / 62

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

0 KH(t, s)dWs,

where For H > 1/2, KH(t, s) = cHs

1 2 −H

t

s (u − s)H− 3

2uH− 1 2du,

s < t. For H < 1/2, KH(t, s) = cH

t

s

H− 1

2 (t − s)H− 1 2

−(H − 1 2)s

1 2 −H

t

s uH− 3

2(u − s)H− 1 2du

• ,

s < t.

Jorge A. León (Cinvestav–IPN) FBM Roscoff 2010 35 / 62

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

Wiener integrals

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

n

• i=0

aiI(tj,tj+1].

Jorge A. León (Cinvestav–IPN) FBM Roscoff 2010 37 / 62

Wiener integrals

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

n

• i=0

aiI(tj,tj+1]. The Wiener integral with respect to B I(f ) =

n

• i=0

ai(Bti+1 − Bti) and the space L(B) = {X ∈ L2(Ω) : X = L2(Ω) − lim

n→∞ I(fn), for some {fn} ⊂ E}.

Jorge A. León (Cinvestav–IPN) FBM Roscoff 2010 38 / 62

Wiener integrals

L(B) = {X ∈ L2(Ω) : X = L2(Ω) − lim

n→∞ 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

Wiener integrals

L(B) = {X ∈ L2(Ω) : X = L2(Ω) − lim

n→∞ 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

Wiener integrals

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. Remarks a) For H < 1/2, H = {f ∈ L2([0, T]) : f (s) = cHs

1 2 −H(I 1 2 −H

T− uH− 1

2φf (u))(s)

for some φf ∈ L2} with (Iα

T−g)(s) = 1 Γ(α)

T

s (x − s)α−1g(x)dx.

Jorge A. León (Cinvestav–IPN) FBM Roscoff 2010 41 / 62

Wiener integrals

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. Remarks a) For H < 1/2, H = {f ∈ L2([0, T]) : f (s) = cHs

1 2 −H(I 1 2 −H

T− uH− 1

2φf (u))(s)

for some φf ∈ L2} with the inner product (f , g) = (φf , φg)L2([0,T]).

Jorge A. León (Cinvestav–IPN) FBM Roscoff 2010 42 / 62

Wiener integrals

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. Remarks 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 Ff ∗(x)Fg∗(x)|x|1−2Hdx.

Jorge A. León (Cinvestav–IPN) FBM Roscoff 2010 43 / 62

Wiener integrals

a) For H < 1/2, H = {f ∈ L2([0, T]) : f (s) = cHs

1 2 −H(I 1 2 −H

T− 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 Ff ∗(x)Fg∗(x)|x|1−2Hdx.

c) W s,2(R) = {f ∈ S : (1 + |x|2)s/2Ff (x) ∈ L2(R)}.

Jorge A. León (Cinvestav–IPN) FBM Roscoff 2010 44 / 62

Representation of Wiener integrals

Moreover, there exists an isometry K ∗

H : H → L2([0, T]) and a

Brownian motion W such that :

1

I(f ) =

T

0 (K ∗ Hf )(s)dWs.

Jorge A. León (Cinvestav–IPN) FBM Roscoff 2010 45 / 62

Representation of Wiener integrals

Moreover, there exists an isometry K ∗

H : H → L2([0, T]) and a

Brownian motion W such that :

1

I(f ) =

T

0 (K ∗ Hf )(s)dWs.

2

K ∗

HI[0,t] = KH(t, ·) with

KH(t, s) = cHs

1 2 −H

t

s (u −s)H− 3

2uH− 1 2du,

s < t and H > 1/2 and KH(t, s) = cH

t

s

H− 1

2 (t − s)H− 1 2

−(H − 1 2)s

1 2 −H

t

s uH− 3

2(u − s)H− 1 2du

• ,

H < 1/2.

Jorge A. León (Cinvestav–IPN) FBM Roscoff 2010 46 / 62

Representation of Wiener integrals

Moreover, there exists an isometry K ∗

H : H → L2([0, T]) and a

Brownian motion W such that :

1

I(φ) =

T

0 (K ∗ Hφ)(s)dWs.

2

K ∗

HI[0,t] = KH(t, ·) .

3

For H < 1/2, K ∗

Hf = φf ,

with f (s) = cHs

1 2 −H(I 1 2 −H

T− uH− 1

2φf (u))(s), s ∈ [0, T]. Jorge A. León (Cinvestav–IPN) FBM Roscoff 2010 47 / 62

Representation of Wiener integrals

Moreover, there exists an isometry K ∗

H : H → L2([0, T]) and a

Brownian motion W such that :

1

I(φ) =

T

0 (K ∗ Hφ)(s)dWs.

2

K ∗

HI[0,t] = KH(t, ·) .

3

For H < 1/2, K ∗

Hf = φf ,

with f (s) = cHs

1 2 −H(I 1 2 −H

T− uH− 1

2φf (u))(s), s ∈ [0, T]. 4

For H > 1/2 and φ ∈ E, (K ∗

Hφ)(s) = cHs1/2−H(I H− 1

2

T− uh− 1

2φ(u))(s), s ∈ [0, T]. Jorge A. León (Cinvestav–IPN) FBM Roscoff 2010 48 / 62

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

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.

Jorge A. León (Cinvestav–IPN) FBM Roscoff 2010 50 / 62

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 is

given by DF =

n

• i=1

∂f ∂xi (B(φ1), . . . , B(φn))φi.

Jorge A. León (Cinvestav–IPN) FBM Roscoff 2010 51 / 62

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 is

given by DF =

n

• i=1

∂f ∂xi (B(φ1), . . . , B(φn))φi. The operator D is closable from L2(Ω) into L2(Ω; H).

Jorge A. León (Cinvestav–IPN) FBM Roscoff 2010 52 / 62

Divergence operator

The divergence operator δ is the adjoint of D. It is defined by the duality relation E (Fδ(u)) = E (DF, uH) , F ∈ S, u ∈ L2(Ω, H).

Jorge A. León (Cinvestav–IPN) FBM Roscoff 2010 53 / 62

Transfer principle

Let W be the Brownian motion such that Bt =

t

0 KH(t, s)dWs

t ∈ [0, T]. Then,

1

DomD=DomDW and K ∗

HDF = DWF.

Jorge A. León (Cinvestav–IPN) FBM Roscoff 2010 54 / 62

Transfer principle

Let W be the Brownian motion such that Bt =

t

0 KH(t, s)dWs

t ∈ [0, T]. Then,

1

DomD=DomDW and K ∗

HDF = DWF.

2

φ ∈Domδ if and only if K ∗

Hφ ∈DomδW and

δ(φ) = δW(K ∗

Hφ).

Jorge A. León (Cinvestav–IPN) FBM Roscoff 2010 55 / 62

Transfer principle

1

DomD=DomDW and K ∗

HDF = DWF.

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 the duality relation E (Fδ(u)) = E (DF, uH) , F ∈ S, u ∈ L2(Ω, H). Remark For H = 1/2, H = L2([0, T]).

Jorge A. León (Cinvestav–IPN) FBM Roscoff 2010 56 / 62

Transfer principle

1

DomD=DomDW and K ∗

HDF = DWF.

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 the duality relation E (Fδ(u)) = E (DF, uH) , F ∈ S, u ∈ L2(Ω, H). Remark For H = 1/2, H = L2([0, T]). So E

• FδW(u)
• = E

T

0 (DW s F)usds

• ,

F ∈ S, u ∈ L2(Ω × [0, T]).

Jorge A. León (Cinvestav–IPN) FBM Roscoff 2010 57 / 62

Divergence operator

The divergence operator δ is the adjoint of D. It is defined by the duality relation E (Fδ(u)) = E (DF, uH) , F ∈ S, u ∈ L2(Ω, H).

Proposition

Let u ∈Dom δ and F ∈ Dom D such that (Fu) ∈ L2(Ω; H) and (Fδ(u) − DF, uH) ∈ L2(Ω). Then Fδ(u) = δ(Fu)

Jorge A. León (Cinvestav–IPN) FBM Roscoff 2010 58 / 62

Divergence operator

The divergence operator δ is the adjoint of D. It is defined by the duality relation E (Fδ(u)) = E (DF, uH) , F ∈ S, u ∈ L2(Ω, H).

Proposition

Let u ∈Dom δ and F ∈ Dom D such that (Fu) ∈ L2(Ω; H) and (Fδ(u) − DF, uH) ∈ L2(Ω). Then Fδ(u) = δ(Fu) + DF, uH.

Jorge A. León (Cinvestav–IPN) FBM Roscoff 2010 59 / 62

Divergence operator

The divergence operator δ is the adjoint of D. It is defined by the duality relation E (Fδ(u)) = E (DF, uH) , F ∈ S, u ∈ L2(Ω, H).

Proposition

Let u ∈Dom δ and F ∈ Dom D such that (Fu) ∈ L2(Ω; H) and (Fδ(u) − DF, uH) ∈ L2(Ω). Then Fδ(u) = δ(Fu) + DF, uH. Proof : Let G ∈ S, then

Jorge A. León (Cinvestav–IPN) FBM Roscoff 2010 60 / 62

Divergence operator

The divergence operator δ is the adjoint of D. It is defined by the duality relation E (Fδ(u)) = E (DF, uH) , F ∈ S, u ∈ L2(Ω, H).

Proposition

Let u ∈Dom δ and F ∈ Dom D such that (Fu) ∈ L2(Ω; H) and (Fδ(u) − DF, uH) ∈ L2(Ω). Then Fδ(u) = δ(Fu) + DF, uH. Proof : Let G ∈ S, then E (DG, FuH) = E (D(GF), uH − G DF, uH)

Jorge A. León (Cinvestav–IPN) FBM Roscoff 2010 61 / 62

Divergence operator

The divergence operator δ is the adjoint of D. E (Fδ(u)) = E (DF, uH) , F ∈ S, u ∈ L2(Ω, H).

Proposition

Let u ∈Dom δ and F ∈ Dom D such that (Fu) ∈ L2(Ω; H) and (Fδ(u) − DF, uH) ∈ L2(Ω). Then Fδ(u) = δ(Fu) − DF, uH. Proof : Let G ∈ S, then E (DG, FuH) = E (D(GF), uH − G DF, uH) = E (G (Fδ(u) − DF, uH)) .

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