FINE REGULARITY OF ) L Séminaire Cristolien d’Analyse ... · Introduction 2-microlocal analysis...

FINE REGULARITY OF (FRACTIONAL) LÉVYPROCESSES

Séminaire Cristolien d’Analyse Multifractale

Paul Balança

Ph.D. student under the supervision of Erick HerbinMAS, École Centrale Paris

May 21, 2013

Paul Balança 1 / 50 Regularity of (fractional) Lévy processes

1 IntroductionMultifractal spectrum of Lévy processes2-microlocal analysis

2 Multifractal and 2-microlocal analysis on Lévy processesMain resultProof (main ideas)Remarks & corollary

3 Regularity of linear fractional stable motionDefinitionRegularity and multifractal spectrum

Introduction

Pointwise Hölder exponent

Definition

A function f belongs to Cαt if

∀u ∈ B(t,ρ); |f(u)− Pt(u)| ≤ C|t− u|α

where Pt is a polynomial of degree less than α.The pointwise Hölder exponent is defined by

αf ,t = sup{α≥ 0 : f ∈ Cαt }.

Introduction

Multifractal analysis

Definition (Multifractal spectrum)

Geometrical description of level sets of the pointwise exponent(iso-Hölder sets),

∀h ∈ R+; Eh = {t ∈ R+ : αf ,t = h}.

The multifractal spectrum is defined by

∀h ∈ R+; df (h, V) = dimH(Eh ∩ V).

where V ∈ O is a non-empty open set.

Introduction Multifractal spectrum of Lévy processes

Lévy processes

Definition

A Lévy process (Xt)t∈R+ satisfies

X0 = 0 a.s;

X has independent increments, i.e. for any t1 ≤ · · · ≤ tn ∈ R+

Xt2 − Xt1 , . . . , Xtn − Xtn−1 are independent;

X has stationary increments, i.e.

∀s≤ t ∈ R+; Xt− XsL= Xt−s.

A Lévy process X has càdlàg sample paths (right continuous withleft limits).

Lévy processes

Theorem (Lévy-Khintchine representation)

The law of (Xt)t∈R+ is characterized by E[eiλXt] = e−tψ(λ), where

∀λ ∈ R; ψ(λ) = iaλ+σ2λ2

2+

∫

R

�

1− eiλx + iλx1|x|≤1

π(dx).

π(dx) is a Lévy measure, i.e.∫

R

�

1∧ x2�

π(dx)

Lévy processes

Theorem (Lévy-Ito decomposition)

A Lévy process (Xt)t∈R+ may be decomposed into the sum of threeindependent processes B, N and Y, where

B is a Brownian motion with drift ∼ iaλ+ σ2λ2

2;

N is compound Poisson process such that |∆Nt| ≥ 1 for allt ∈ R+;

∼∫

|x|>1

�

1− eiλx

π(dx).

Y is pure jump square integrable martingale such that|∆Yt| := |Yt− Yt−| ≤ 1.

∼∫

|x|≤1

�

1− eiλx + iλx

π(dx).

Alpha-stable Lévy processes

Example

Pure jump Lévy process whose Lévy measure π is defined by

π(dx) = |x|−α−1dx where α ∈ (0,2).

Assumption on the Lévy process

In the sequel, we assume that X has no Brownian and driftcomponents, i.e.

a= 0 and σ = 0.

Furthermore, we suppose that π(R) = +∞.

Definition (Blumenthal-Getoor exponent)

The Blumenthal-Getoor exponent is defined by

β = inf�

γ≥ 0 :∫

R

�

1∧ |x|γ�

π(dx)

Assumption on the Lévy process

In the sequel, we assume that X has no Brownian and driftcomponents, i.e.

a= 0 and σ = 0.

Furthermore, we suppose that π(R) = +∞.

Definition (Blumenthal-Getoor exponent)

The Blumenthal-Getoor exponent is defined by

β = inf�

γ≥ 0 :∫

R

�

1∧ |x|γ�

π(dx)

Multifractal spectrum of Lévy processes

Why ?

Pruitt (1981) has proved that

∀t ∈ R+ almost surely; αX,t =1

β.

Can we invert "∀t ∈ R+" and "a.s.", similarly to the Brownianmotion ?

The pointwise regularity of Lévy processes fluctuates randomlyfrom point to point (e.g. jumps).

Why ?

Pruitt (1981) has proved that

∀t ∈ R+ almost surely; αX,t =1

β.

Can we invert "∀t ∈ R+" and "a.s.", similarly to the Brownianmotion ?

The pointwise regularity of Lévy processes fluctuates randomlyfrom point to point (e.g. jumps).

Theorem (S. Jaffard(1999))

Let X be a Lévy process. Then, almost surely

∀h ∈ R+ ∀V ∈ O ; dX(h, V) =

(

βh if h ∈ [0,1/β]−∞ otherwise.

h0 1/β

1

−∞

dX(h)

How this last result can be extended?

Hausdorff g-measure, where g is gauge function⇒ A. Durand(2009);

Lévy fields (multiparameter setting)⇒ A. Durand and S.Jaffard (2011);

2-microlocal analysis...

Introduction 2-microlocal analysis

2-microlocal analysis

Introduced by J.M. Bony (1986) in the context PDE.Leads to the definition of 2-microlocal spaces which extendclassic Hölder spaces Cαt . Different characterizations exist:

Fourier domain, using the Littlewood-Paley decomposition;Wavelet domain (S. Jaffard (1991));Time domain (K. Kolwankar, S. Seuret and J. Levy-Vehel(2003))

Definition (2-microlocal spaces)

Let s′ ≤ 0 and σ ∈ (0,1) such that σ− s′ /∈ N. A function f belongsto the 2-microlocal space Cσ,s

′

t if for all u, v ∈ B(t,ρ),�

�

f(u)− Pt(u)�

−�

(f(v)− Pt(v)�

�

�≤ C|u− v|σ�

|u− t|+ |v− t|�−s′ ,

where Pt is a polynomial.

Introduced by J.M. Bony (1986) in the context PDE.Leads to the definition of 2-microlocal spaces which extendclassic Hölder spaces Cαt . Different characterizations exist:

Fourier domain, using the Littlewood-Paley decomposition;Wavelet domain (S. Jaffard (1991));Time domain (K. Kolwankar, S. Seuret and J. Levy-Vehel(2003))

Let s′ ≤ 0 and σ ∈ (0,1) such that σ− s′ /∈ N. A function f belongsto the 2-microlocal space Cσ,s

′

t if for all u, v ∈ B(t,ρ),�

�

f(u)− Pt(u)�

−�

(f(v)− Pt(v)�

�

�≤ C|u− v|σ�

|u− t|+ |v− t|�−s′ ,

where Pt is a polynomial.

Stability of 2-microlocal spaces

Let t> 0 and f be a function. Then,

∀α > 0; f ∈ Cσ,s′

t ⇐⇒ Iα+f ∈ C

σ+α,s′t

where Iα+f denotes the fractional integral of order α, i.e.

Iα+f(t) =∫ t

0(t− u)α−1+ f(u)du.

The characterization of Cσ,s′

t can be extended to any σ ∈ R \ Z ands′ ≤ 0 such that σ− s′ /∈ N using

the definition with σ ∈ (0, 1);the stability under integration/derivation.

Stability of 2-microlocal spaces

Let t> 0 and f be a function. Then,

∀α > 0; f ∈ Cσ,s′

t ⇐⇒ Iα+f ∈ C

σ+α,s′t

where Iα+f denotes the fractional integral of order α, i.e.

Iα+f(t) =∫ t

0(t− u)α−1+ f(u)du.

The characterization of Cσ,s′

t can be extended to any σ ∈ R \ Z ands′ ≤ 0 such that σ− s′ /∈ N using

the definition with σ ∈ (0, 1);the stability under integration/derivation.

2-microlocal frontier

Hölder space =⇒ pointwise Hölder exponent;2-microlocal space =⇒ 2-microlocal frontier.

Definition (2-microlocal frontier)

The 2-microlocal frontier is defined by

∀s′ ∈ R; σf ,t(s′) = sup�

σ ∈ R : f ∈ Cσ,s′

t

.

Properties

σf ,t is a concave non-decreasing function;

σf ,t has left and right derivatives between 0 and 1;

eαf ,t = σf ,t(0);

αf ,t =− inf{s′ : σf ,t(s′)≥ 0}, when ω(h) = O (1/|log(h)|).

Hölder space =⇒ pointwise Hölder exponent;2-microlocal space =⇒ 2-microlocal frontier.

Definition (2-microlocal frontier)

The 2-microlocal frontier is defined by

∀s′ ∈ R; σf ,t(s′) = sup�

σ ∈ R : f ∈ Cσ,s′

t

.

Properties

σf ,t is a concave non-decreasing function;

σf ,t has left and right derivatives between 0 and 1;

eαf ,t = σf ,t(0);

αf ,t =− inf{s′ : σf ,t(s′)≥ 0}, when ω(h) = O (1/|log(h)|).

s′

σ

0−1−2

1

−3

2−

1

2

σf,t(s′)

α̃f,t

−αf,t

s′

σ

0−1−2

1

−3

2−

1

2

σf,t(s′)

σIαf,t(s′)

α

Stochastic 2-microlocal analysis

Use of 2-microlocal analysis to characterize the regularity ofstochastic processes:

E. Herbin and J. Levy-Vehel (2009) : Gaussian processes andWiener integral;

P. Balança and E. Herbin (2012) : Martingales and stochasticintegral.

Stochastic 2-microlocal analysis

Example (Brownian motion)

Almost surely for all t ∈ R+,

∀s′ ∈ R; σB,t(s′) =�1

2+ s′�

∧1

2.

s′

σ

0−1 − 12

σB,t

1

2

Multifractal and 2-microlocal analysis on Lévy processes

2-microlocal frontier and multifractal spectrum

What is known...

Let X be a Lévy process. We observe that

The set of jumps is dense in R+, meaning that for any t ∈ R+

eαX,t = 0=⇒∀s′ ∈ R; σX,t(s′)≤ 0.

Since the 2-microlocal frontier is concave with left and rightderivatives in [0, 1], for all t ∈ R

∀s′ ∈ R; σX,t(s′)≥ (αX,t+ s′)∧ 0.

What is known...

Let X be a Lévy process. We observe that

The set of jumps is dense in R+, meaning that for any t ∈ R+

eαX,t = 0=⇒∀s′ ∈ R; σX,t(s′)≤ 0.

Since the 2-microlocal frontier is concave with left and rightderivatives in [0, 1], for all t ∈ R

∀s′ ∈ R; σX,t(s′)≥ (αX,t+ s′)∧ 0.

Combine 2-microlocal and multifractal analysis

Hence, a candidate for the 2-microlocal frontier of a Lévy processcould be

∀s′ ∈ R; σX,t(s′) = (αX,t+ s′)∧ 0.

Since Eh = {t ∈ R+ : αX,t = h}, it seems natural to investigate thefollowing dichotomy

eEh =�

t ∈ Eh : ∀s′ ∈ R; σX,t(s′) = (h+ s′)∧ 0

andbEh = Eh \ eEh.

Combine 2-microlocal and multifractal analysis

Hence, a candidate for the 2-microlocal frontier of a Lévy processcould be

∀s′ ∈ R; σX,t(s′) = (αX,t+ s′)∧ 0.

Since Eh = {t ∈ R+ : αX,t = h}, it seems natural to investigate thefollowing dichotomy

eEh =�

t ∈ Eh : ∀s′ ∈ R; σX,t(s′) = (h+ s′)∧ 0

andbEh = Eh \ eEh.

Multifractal and 2-microlocal analysis on Lévy processes Main result

2-microlocal regularity of Lévy processes

Theorem

A Lévy process X almost surely satisfies

∀V ∈ O ; dimH(eEh ∩ V) =

(

βh if h ∈ [0,1/β];−∞ if h ∈ (1/β ,+∞].

Furthermore,

dimH(bEh)≤

(

2βh− 1< βh if h ∈ (1/2β , 1/β);−∞ if h ∈ [0,1/2β]∪ [1/β ,+∞].

Theorem

A Lévy process X almost surely satisfies

∀V ∈ O ; dimH(eEh ∩ V) =

(

βh if h ∈ [0,1/β];−∞ if h ∈ (1/β ,+∞].

Furthermore,

dimH(bEh)≤

(

2βh− 1< βh if h ∈ (1/2β , 1/β);−∞ if h ∈ [0, 1/2β]∪ [1/β ,+∞].

s′

σ

0−1

−12

1

2− 1

β

˜Eh = {t ∈ R : σX,t(s

′) = (h+ s′) ∧ 0}−h

Remarks

For all h ∈ [0, 1/β], dimH(bEh)< dimH(eEh). Hence, for "most"times, the 2-microlocal frontier is "classic";

Even though sample paths of Lévy processes are notcontinuous, we still have αX,t =− inf{s′ : σX,t(s′)≥ 0}.

Multifractal and 2-microlocal analysis on Lévy processes Proof (main ideas)

Proof: multifractal spectrum

Notations

Xt = lim"→0

�∫

[0,t]×D(",1)x J(ds, dx)− t

∫

D(",1)xπ(dx)

�

,

where for all 0≤ a< b, D(a, b) = {x ∈ Rd : a< |x| ≤ b} and J is aPoisson measure of intensity λ⊗π.For any m ∈ R+, Xm denotes

Xmt = lim"→0

�∫

[0,t]×D(",2−m)x J(ds, dx)− t

∫

D(",2−m)xπ(dx)

�

,

i.e. the process where jumps of size greater than 2−m are removed.

Notations

Xt = lim"→0

�∫

[0,t]×D(",1)x J(ds, dx)− t

∫

D(",1)xπ(dx)

�

,

where for all 0≤ a< b, D(a, b) = {x ∈ Rd : a< |x| ≤ b} and J is aPoisson measure of intensity λ⊗π.For any m ∈ R+, Xm denotes

Xmt = lim"→0

�∫

[0,t]×D(",2−m)x J(ds, dx)− t

∫

D(",2−m)xπ(dx)

�

,

i.e. the process where jumps of size greater than 2−m are removed.

For any δ ≥ β and " > 0, let A"δ be

A"δ =⋃

s∈S(ω)|∆Xs|≤"

�

s− |∆Xs|δ, s+ |∆Xs|δ�

.

Then, the random set Aδ is defined by Aδ = limsup"→0+ A"δ.

Let t ∈ Aδ. There exists tn→ t such that |t− tn| ≤ |∆Xtn |δ, i.e.

∀n ∈ N;|Xt− Xtn |

|t− tn|1/δ≥ c1 or

|Xt− Xtn−|

|t− tn|1/δ≥ c1.

Hence αX,t ≤1δ

.

For any δ ≥ β and " > 0, let A"δ be

A"δ =⋃

s∈S(ω)|∆Xs|≤"

�

s− |∆Xs|δ, s+ |∆Xs|δ�

.

Then, the random set Aδ is defined by Aδ = limsup"→0+ A"δ.

Let t ∈ Aδ. There exists tn→ t such that |t− tn| ≤ |∆Xtn |δ, i.e.

∀n ∈ N;|Xt− Xtn |

|t− tn|1/δ≥ c1 or

|Xt− Xtn−|

|t− tn|1/δ≥ c1.

Hence αX,t ≤1δ

.

Conversely, let t /∈ Aδ and s ∈ B(t,ρ). Since t /∈ Aδ, there is no jumpgreater than |t− s|1/δ inside the interval [t− |t− s|, t+ |t− s|].Hence,

|Xs− Xt|= |Xm/δs − Xm/δt |,

where m is such that |t− s|= 2−m.

Lemma

For all δ ≥ β , there exists M ∈ R such that almost surely for allm≥M,

∀u, v ∈ [0, 1], |u− v| ≤ 2−m;�

�Xm/δu − Xm/δv

�

�≤ c2−m/δ.

Hence, |Xs− Xt| ≤ c|t− s|1/δ and αX,t ≥1δ

.

Lemma

∀u, v ∈ [0, 1], |u− v| ≤ 2−m;�

�

�≤ c2−m/δ.

.

Lemma

∀u, v ∈ [0, 1], |u− v| ≤ 2−m;�

�

�≤ c2−m/δ.

.

Therefore, almost surely

∀h> 0; Eh =�

⋂

δ1/h

Aδ

�

\ S

and

E0 =�

⋂

δ>0

Aδ

�

∪ S.

Proof: 2-microlocal frontier

Let t ∈ Eh. We try to find a sufficient condition to have t ∈ eEh, i.e.

∀s′ ∈ R; σX,t(s′) = (h+ s′)∧ 0.

We already know that σX,t(s′)≥ h+ s′.

Since we consider the negative component of the frontier, we haveto study

Ys =

∫ t

0

Xu du,

and exhibits a sequence tn→ t such that

∀n ∈ N; |Yt− Ytn − (t− tn)Xt| ≥ |t− tn|1+h.

∀s′ ∈ R; σX,t(s′) = (h+ s′)∧ 0.

Ys =

∫ t

0

Xu du,

∀s′ ∈ R; σX,t(s′) = (h+ s′)∧ 0.

Ys =

∫ t

0

Xu du,

s′

σ

0−2

−1

1

−1

β

−h

σX,t

σY,t

Since t ∈ Eh, there exists sn→ t such that |t− sn|h ≤ |Xsn − Xt|. Wemay assume that sn ≥ t, Xsn − Xt ≥ 0, ∆Xsn ≥ 0 and settn = sn+ |sn− t|. Then,

Ytn − Ysn − (tn− sn)Xt =∫ tn

sn

Xu du− (tn− sn)Xt

=

∫ tn

sn

(Xu− Xsn)du+ (tn− sn)(Xsn − Xt).

As (tn− sn)(Xsn − Xt)≥ |t− sn|1+h, let find an upper bound for the

first term, therefore proving it is negligible.

Since t ∈ Eh, there exists sn→ t such that |t− sn|h ≤ |Xsn − Xt|. Wemay assume that sn ≥ t, Xsn − Xt ≥ 0, ∆Xsn ≥ 0 and settn = sn+ |sn− t|. Then,

Ytn − Ysn − (tn− sn)Xt =∫ tn

sn

Xu du− (tn− sn)Xt

=

∫ tn

sn

(Xu− Xsn)du+ (tn− sn)(Xsn − Xt).

As (tn− sn)(Xsn − Xt)≥ |t− sn|1+h, let find an upper bound for the

first term, therefore proving it is negligible.

Assumption: for all n ∈ N, there is no other jump of size greaterthan |t− sn|h+" in the interval [sn, tn].

Then, owing to a previous technical lemma,

∀u ∈ [sn, tn]; |Xu− Xsn | ≤ c|t− sn|h+"

and therefore�

�

∫ tn

sn

(Xu− Xsn)du�

�

≤ c|tn− sn| · |t− sn|h+" ≤ c|t− sn|1+h+".

It proves that |Ytn − Ysn − (tn− sn)Xt| ≥ c|t− tn|1+h, which is

sufficient for our purpose.

and therefore�

�

∫ tn

sn

(Xu− Xsn)du�

�

≤ c|tn− sn| · |t− sn|h+" ≤ c|t− sn|1+h+".

and therefore�

�

∫ tn

sn

(Xu− Xsn)du�

�

≤ c|tn− sn| · |t− sn|h+" ≤ c|t− sn|1+h+".

and therefore�

�

∫ tn

sn

(Xu− Xsn)du�

�

≤ c|tn− sn| · |t− sn|h+" ≤ c|t− sn|1+h+".

Let Shn be the set of jump times such that for every u ∈ Shn

2−n ≤ |∆Xu|< 2−n+1 and [u− 2−n/h, u+ 2−n/h]∩ Shn 6= ;

Let Sh denotesSh = limsup

n→+∞Shn.

Owing to the previous slide, if t /∈ Sh−", then t ∈ eEh. Hence,

bEh ⊂⋂

">0

Sh−".

n→+∞Shn.

bEh ⊂⋂

">0

Sh−".

n→+∞Shn.

bEh ⊂⋂

">0

Sh−".

Let finally obtain an upper bound of dimH(bEh).Some basic estimates on Poisson variables show that almost surely

∀n ∈ N; Nn := #(Shn)≤ 2n(2β−1/h).

Hence, a dichotomy appears

If h< 1/(2β), Nn→ 0, and thus bEh = ;.If h> 1/(2β), for any n ∈ N, Shn can be covered by Nn intervalsof size 2−n/h. Then, γ-Hausdorff measure is upper bounded by

∑

n=1

2−γn/h · 2n(2β−1/h) =∑

n=1

2n(2β−(1+γ)/h).

The series converges when γ > 2βh− 1, proving thatdimH(bEh)≤ 2βh− 1< βh.

∀n ∈ N; Nn := #(Shn)≤ 2n(2β−1/h).

∑

n=1

2−γn/h · 2n(2β−1/h) =∑

n=1

2n(2β−(1+γ)/h).

∀n ∈ N; Nn := #(Shn)≤ 2n(2β−1/h).

∑

n=1

2−γn/h · 2n(2β−1/h) =∑

n=1

2n(2β−(1+γ)/h).

∀n ∈ N; Nn := #(Shn)≤ 2n(2β−1/h).

∑

n=1

2−γn/h · 2n(2β−1/h) =∑

n=1

2n(2β−(1+γ)/h).

To conclude, for any h ∈ R+,

Eh = eEh ∪ bEh and dimH(bEh)< dimH(Eh).

Therefore dimH(eEh) = βh.

Multifractal and 2-microlocal analysis on Lévy processes Remarks & corollary

Remarks on the unusual 2-microlocal behaviour

Different situations occur for (bEh)h∈R:

If the Lévy measure satisfies π((−∞, 0)) = 0, then

a.s. ∀h ∈ R+; bEh = ;.

For an β ∈ (0,1) and h ∈ (1/2β , 1/β), there exists a Lévymeasure πh such that

a.s. bEh 6= ;.

Idea: Construct times t at which neighbourhood’s jumpsalways compensate each other.

a.s. ∀h ∈ R+; bEh = ;.

a.s. bEh 6= ;.

a.s. ∀h ∈ R+; bEh = ;.

a.s. bEh 6= ;.

Another approach

Let set σ ∈ [−1,0]. As an extension of the multifractal spectrum,it seems also natural to study the collections of sets

Eσ,s′ =�

t ∈ R+ : σX,t(s′) = σ

.

s′

σ

0−2

−1

1−1

β

−h

σX,t σ = cte

Corollary

A Lévy process X satisfies almost surely for any σ ∈ [−1,0],

∀V ∈ O ; dimH(Eσ,s′ ∩ V) =

(

βs if s ∈ [0,1/β];−∞ otherwise.

where s= σ− s′. Furthermore, for all s′ ∈ R, Eσ,s′ is empty ifσ > 0.

Remark

The unusual 2-microlocal behaviour captured in the main theoremis not displayed by this weaker corollary.

Corollary

A Lévy process X satisfies almost surely for any σ ∈ [−1,0],

∀V ∈ O ; dimH(Eσ,s′ ∩ V) =

(

βs if s ∈ [0,1/β];−∞ otherwise.

where s= σ− s′. Furthermore, for all s′ ∈ R, Eσ,s′ is empty ifσ > 0.

Remark

The unusual 2-microlocal behaviour captured in the main theoremis not displayed by this weaker corollary.

Proof: corollary

We observe that

∀s ∈ [0,1/β); eEs ⊆ Eσ,s′ ⊆ eEs ∪�

∪h

Regularity of linear fractional stable motion Definition

Linear fractional stable motion

A fine description of the 2-microlocal behaviour of Lévy processesalso happens to interesting for the study of another class ofprocesses: the linear fractional stable motion.

Definition

A LFSM X is defined by the stochastic integral

Xt =

∫

R

n

(t− u)H−1/α+ − (−u)H−1/α+

o

Mα(du),

where α ∈ (0, 2), H ∈ (0,1) and Mα is an α-stable randommeasure.

A fine description of the 2-microlocal behaviour of Lévy processesalso happens to interesting for the study of another class ofprocesses: the linear fractional stable motion.

Definition

A LFSM X is defined by the stochastic integral

Xt =

∫

R

n

(t− u)H−1/α+ − (−u)H−1/α+

o

Mα(du),

where α ∈ (0, 2), H ∈ (0,1) and Mα is an α-stable randommeasure.

Sample path properties

If H > 1/α, sample paths are almost surely Hölder continuouswith H− 1/α≤ αX,t ≤ H and eαX,t = H− 1/α.If H < 1/α, sample paths are nowhere bounded.

Figure: LFSM sample paths with α= 1.5

Regularity of linear fractional stable motion Regularity and multifractal spectrum

Regularity of linear fractional stable motion

Theorem

Let X be a LFSM with α ∈ (1, 2) and H ∈ (0, 1). Then, almost surelyfor all σ ∈

�

H− 1α− 1, H− 1

α

�

,

∀V ∈ O ; dimH(Eσ,s′ ∩ V) =

(

α(s−H) + 1 if s ∈�

H− 1α

, H�

;

−∞ otherwise.

where s= σ− s′. Furthermore, for all s′ ∈ R, Eσ,s′ is empty ifσ > H− 1

α.

Regularity of linear fractional stable motion

s′

σ

0−1

−1

2

dimH(Eσ,s′) = α(s−H) + 1

1

2

1

2

−H + 1α

H −1

α

−H

s = σ − s′

(a) Continuous sample paths: H = 56

and α= 32

s′

σ

0−1

−1

2

dimH(Eσ,s′) = α(s−H) + 1

1

2

1

2

H −1

α

−1

2−

1

α

s = σ − s′

(b) Unbounded sample paths: H = 12

and α= 32

Figure: Domains of admissible 2-microlocal frontiers for the LFSM

Multifractal spectrum of LFSM

Corollary

Let X be a LFSM with α ∈ (1, 2) and H > 1/α. Then, itsmultifractal spectrum is equal to

∀V ∈ O ; dX(h, V) =

(

α(h−H) + 1 if h ∈�

H− 1α

, H�

;

−∞ otherwise.

h0 H

1

−∞

dX(h)

H − 1/α

Proof: regularity of LFSM

Let H ∈ (0,1). We exhibit an alternative representation of LFSMwhich displays a fractional integral.For all t ∈ R,

Xta.s.=

CH

∫

R

Lun

(t− u)H−1/α−1+ − (−u)H−1/α−1+

o

du if H ∈� 1α

, 1�

;

Lt if H =1α

CH

∫

R

n

(Lu− Lt)(t− u)H−1/α−1+ − Lu(−u)

H−1/α−1+

o

du if H ∈�

0, 1α

�

,

(1)where CH = H− 1/α and L is an α-stable Lévy process defined by

∀t ∈ R+ Lt =Mα([0, t]) and ∀t ∈ R− Lt =−Mα([t, 0]).

If H ∈� 1α

, 1�

,

Xt = CH

∫ t

0

Lu(t− u)H−1/α−1+ du+ Zt,

where the process Z has C∞ sample paths.Hence, owing to properties of the 2-microlocal frontier,

∀s′ ∈ R; σX,t(s′) = σL,t(s′) +H−1

α,

and our main result on Lévy processes induces the expectedequality.

If H ∈� 1α

, 1�

,

Xt = CH

∫ t

0

Lu(t− u)H−1/α−1+ du+ Zt,

where the process Z has C∞ sample paths.Hence, owing to properties of the 2-microlocal frontier,

∀s′ ∈ R; σX,t(s′) = σL,t(s′) +H−1

α,

and our main result on Lévy processes induces the expectedequality.

If H ∈�

0, 1α

�

, we proceed similarly, observing that

Xt =d

dt

∫ t

b

Lu(t− u)H−1/α du+ eZt,

where the first term now corresponds to a fractionalderivative of L.

Some last remarks...

Fractional Lévy processes

A similar result exists on the class of fractional Lévy processes,defined by

Xt =1

Γ(d+ 1)

∫

R

n

(t− u)d+− (−u)d+

o

L(du),

where d ∈ (0,1/2) and L is a Lévy process such that E[L(1)] = 0and E[L(1)2]

Linear multifractional stable motion

The localized spectrum of singularity of the LMSM, defined by

Xt =

∫

R

n

(t− u)H(t)−1/α+ − (−u)H(t)−1/α+

o

Mα(du),

can also be similarly obtained.

Pending questions

Can we obtain the exact Hausdorff dimension of bEh? It mightrely on other coefficients derived from the Lévy measure.

Can we describe the regularity of the linear fractional stablemotion when α ∈ (0, 1], i.e. when sample paths are alwaysnowhere bounded?

Pending questions

Can we obtain the exact Hausdorff dimension of bEh? It mightrely on other coefficients derived from the Lévy measure.

Can we describe the regularity of the linear fractional stablemotion when α ∈ (0, 1], i.e. when sample paths are alwaysnowhere bounded?

Thank you for your attention !

Questions ?

IntroductionMultifractal spectrum of Lévy processes2-microlocal analysis

Multifractal and 2-microlocal analysis on Lévy processesMain resultProof (main ideas)Remarks & corollary

Regularity of linear fractional stable motionDefinitionRegularity and multifractal spectrum

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