Fractional Brownian motions

The concept of a fractional Brownian motion was first proposed by Mandelbrot & van Ness [76] and is formally the convolution of Wiener increments with a power-law kernel. More abstractly we formulate

Definition 2.41 (Fractional Brownian motion). A real-valued Gaussian process

B

^H

= {B

^H

(t)}

_t∈R defined on a probability space

(Ω,

F

,

P) is called a fractional Brownian motion with Hurst parameter

0 <

H

< 1

, if

(i)

B

^H is centered, (ii) E

[B

^H

(t)B

^H

(s)] =

^c₂

|t|

^sH

+ |s|

^2H

− |t − s|

^2H

, where

c > 0

and

t, s ∈

R^.

Observe that in caseH

=

¹₂ Definition 2.41 coincides with the definition of a Wiener process.

Unlike a Wiener process, a fractional Brownian motion with Hurst parameterH

6=

¹₂ is neither a martingale, nor a semi-martingale, nor Markovian.

It follows directly from Definition 2.41, that a fractional Brownian motion is a process with sta-tionary increments according to Definition 2.2. The first condition holds since the expectation operatorEis linear and

B

^His centered, because then for all

t, s ∈

R^{it is}

E

[B

^H

(t) − B

^H

(s)] = 0 =

E

[B

^H

(t − s) − B

^H

(0)].

The second condition of Definition 2.2 is also satisfied. This can be seen by the very elemen-tary computation

D

₃

(t; u, v) =

E

(B

^H

(u) − B

^H

(t))(B

^H

(v) − B

^H

(t))

=

E

B

^H

(u)B

^H

(v) − B

^H

(u)B

^H

(t) − B

^H

(t)B

^H

(v) + B

^H

(t)B

^H

(t)

= c 2

|u − t|

^2H

+ |v − t|

^2H

− |u − v|

^2H

=

E

B

^H

(u − t)B

^H

(v − t)

= D

2

(u − t, v − t).

Moreover, Definition 2.41(ii) yields

D(τ ) = c|t|

^2H which can be written in the form (2.11), where

φ(λ) = c

1

|λ|

^2H+1

, c

₁

= c 4

R∞

0

(1 − cos λ)λ

^−2H−1

dλ = cΓ(2

H

+ 1) sin(π

H

)

2π .

(2.28)

Thus the spectrum of

B

^Hincorporates all

λ ∈

Rand in the fashion of Example 2.6 one verifies that the process

B

^H satisfies Hypotheses(

φ

)and(

φ

₀)(see page 24) if and only if

γ = γ

₀

= 2

H

+ 1

and

θ = 0

. Following Remark 2.5, we outline that

γ

0

= 2

H

+ 1

indicates the presence of long-range dependence in all the cases where H

>

¹₂ and

θ = 0

signals that fractional Brownian motions are not appropriate to study intermittency effects. We summarize the properties of a fractional Brownian motion in the following corollary.

Corollary 2.42. Let

B

^H be a fractional Brownian motion with Hurst parameter H

∈ (0, 1)

. Then the following are true.

(i)

B

^H is mean-square continuous and has continuous paths almost sure.

(ii) The trajectories of

B

^Hare locally Hölder-continuous of any order strictly less then H.

(iii) The trajectories of

B

^Hare almost surely nowhere mean-square differentiable.

(iv) Let

T > 0

,

0 < p < ∞

,

2 ≤ q < ∞

and

0 < σ <

H. Then

B

^H

∈

₀

W

^σ_p

([0, T ]; L

q

(Ω))

. (v) Let

T > 0

,

0 < p < ∞

,

1 < q ≤ 2

and H

≤ σ < 1

. Then

B

^H

6∈

₀

W

^σ_p

([0, T ]; L

_q

(Ω))

.

Proof. Assertions (i)-(iii) follow from Theorem 2.18, while (iv) and (v) are consequences of Theorem 2.21.

The paths of

B

^Hget less zigzagged asHgoes from

0

to

1

. On this basis, one typically classifies fractional Brownian motions into antipersistent (in case

0 <

H

<

¹₂), chaotic (in case H

=

1

2) and persistent (in case ¹₂

<

H

< 1

). This can be loosely explained by considering the covariance of two consecutive increments. When

0 <

H

<

¹₂, the increments of

B

^H tend to have opposite signs. On the other hand, in case¹₂

<

H

< 1

, the correlation of two consecutive increments is strictly positive.

The following figures were generated with the aid of a Wolfram Demonstration Project con-tributed by R. E. Maeder.¹

Figure 2.1. A sample path of a fractional Brownian motion with Hurst parameter H= 0.2.

Figure 2.2. A sample path of a fractional Brownian motion with Hurst parameter H= 0.5.

Figure 2.3. A sample path of fractional Brownian motion with Hurst parameter H= 0.9.

1Seehttp://demonstrations.wolfram.com/OneDimensionalFractionalBrownianMotion/.

Regarding the fractional dimension of the graphs of it due to Falconer [43, Theorem 16.7]

that with probability

1

, the Hausdorff and box dimension of the graph

(t, B

^H

(t))

0≤t≤1 equal

2 −

H. Thus, ifHis close to zero, the process

B

^H zigzags so much that the dimension of the graph

(t, B

^H

(t))

0≤t≤1 is close to the dimension

2

of the unit square. On the other hand

B

^H is an index-Hrandom field (cf. Angulo et al. [5]) so that the Hausdorff dimension of its image

{B

^H

(t) : t ∈ [0, 1]}

equals

1

a.s. for everyH

∈ (0, 1)

.

Concerning stochastic integration we can easily reproduce the results of, e.g., Pipiras & Taqqu [86], Sp. & Wilke [104] and Biagini et al. [17].

Corollary 2.43. Let

B

^H be a fractional Brownian motion with Hurst parameter

0 <

H

< 1

. Then for all

f, g ∈ H ˙

1 2−H

2

(

R

)

we have the isometry E

Z

R

f (τ )dB

^H

(τ )

Z

R

g(τ )dB

^H

(τ )

= c

₁ Z

R

(Ff )(λ)(Fg)(λ)|λ|

^1−2H

dλ

with the constant

c

1 from (2.28).

Proof. The claim is immediate by Theorem 2.30.

Note, that by Plancherel’s Theorem (cf. Theorem A.1) our result also covers the well-known Itô-isometry in caseH

=

¹₂. Focusing our multiplier results from Chapter 2 we denote

ζ(t, x, ω) :=

∞

X

k=1

b

_k

(t, x)B

^H_k

(t, ω),

where

(B

_k^H

)

k∈Nare entirely independent fractional Brownian motions with Hurst parameter

0 <

H

< 1

and the scalar functions

b

i

∈ L

2

(J ; L

2

(∂G))

,

i ∈

N, are supposed to be determinis-tic. Denoting

b := (b

i

)

_i∈N, we have in this particular situation

Corollary 2.44. Let

s ≥ 0

and

G ⊂

R^N be a domain with boundary of class

C

^[s]+1. Then

b ∈ L

_2,H

(J ;

₀

W

^s₂

(∂G; `

₂

)) ⇐⇒ ζ ∈ L

₂

(J ;

₀

W

^s₂

(∂G; L

₂

(Ω))).

as a result of Theorem 2.22 and also by Theorem 2.25

Corollary 2.45. Let

G ⊂

R^N be a domain with boundary of class

C

¹ and

0 < σ <

H. Then

b ∈

₀

W

_2,^σ_H

(J ; L

₂

(∂G; `

₂

)) = ⇒ ζ ∈

₀

W

^σ₂

(J; L

₂

(∂G; L

₂

(Ω))).

If, in addition,

(b(s, x)|b(t, x))

_`2

≥ 0

for every

s, t ∈ J

,

x ∈ ∂G

, then the converse is also true.

Interpreting the spectral density (2.28) tempts to differentiate

B

^H and claim that the frac-tional noise

B ˙

^H has a spectral density proportional to

λ

^1−2H which suggests that in case H

>

¹₂, there is infinite energy at high frequencies and coincides with the known fact that white noise (H

=

¹₂) has a flat power spectrum. Thus, for H

∈ [

¹₂

, 1)

fractional white noise (also called

1/f

^α-noise with

0 ≤ α = 1 − 2

H

< 1

) interpolates between white noise (

1/f

⁰ -noise) and pink noise (

1/f

¹-noise), which is a signal with a frequency spectrum such that the power spectral density is proportional to the reciprocal of the frequency. The following fig-ures were generated with the aid of a Matlab routine contributed by the SAMP group, based at the Department of Engineering Science, Oxford University.²

Figure 2.4.Sample of1/f⁰-noise.

Figure 2.5.Sample of1/f^1/2-noise.

2Seehttp://www.eng.ox.ac.uk/samp/powernoise_soft.html.

Figure 2.6.Sample of1/f¹-noise.

By having power at all frequencies, the total energy of a

1/f

^α-noise is infinite, so it is apparent that such a signal only exists as a theoretical model. In practice, approximations are used where the spectral density decays rapidly for high frequencies. However,

1/f

^α-noise occurs widely and has applications in a large number of fields, so for instance:

• White noise is used by some emergency sirens, due to its ability to cut through back-ground noise, which makes it easier to locate.

• White noise is used extensively in audio synthesis, typically to recreate percussive in-struments such as cymbals which have high noise content in their frequency domain.

• The sound of a waterfall results from the collision of drops among themselves or with the water surface. The smaller the drops the more efficient is air friction, so that small drops are stronger decelerated. At the impact the smaller drops are consequently slower than bigger ones, such that they contribute only faintly high frequency sound fractures.

Therefrom the sound of a waterfall is approximately pink noise.

• The human auditory system, which processes frequencies in a roughly logarithmic fash-ion, does not perceive them with equal sensitivity; signals in the 24-kHz octave sound loudest, and the loudness of other frequencies drops increasingly. While white noise is de facto equitable loud in every bandwidth, people sense pink noise having this feature rather than white noise.

Getting more and more general we are now accomplish to fractional Riesz-Bessel motions.

Im Dokument Regularity and integration theory for a class of stochastic processes with applications to parabolic problems (Seite 56-62)