fine regularity of ) l séminaire cristolien d’analyse ... · introduction 2-microlocal analysis...
TRANSCRIPT
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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
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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
Paul Balança 2 / 50 Regularity of (fractional) Lévy processes
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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 }.
Paul Balança 3 / 50 Regularity of (fractional) Lévy processes
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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.
Paul Balança 4 / 50 Regularity of (fractional) Lévy processes
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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).
Paul Balança 5 / 50 Regularity of (fractional) Lévy processes
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Introduction Multifractal spectrum of Lévy processes
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)
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Introduction Multifractal spectrum of Lévy processes
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).
Paul Balança 7 / 50 Regularity of (fractional) Lévy processes
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Introduction Multifractal spectrum of Lévy processes
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).
Paul Balança 8 / 50 Regularity of (fractional) Lévy processes
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Introduction Multifractal spectrum of Lévy processes
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)
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Introduction Multifractal spectrum of Lévy processes
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)
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Introduction Multifractal spectrum of Lévy processes
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).
Paul Balança 10 / 50 Regularity of (fractional) Lévy processes
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Introduction Multifractal spectrum of Lévy processes
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).
Paul Balança 10 / 50 Regularity of (fractional) Lévy processes
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Introduction Multifractal spectrum of Lévy processes
Multifractal spectrum of Lévy processes
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)
Paul Balança 11 / 50 Regularity of (fractional) Lévy processes
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Introduction Multifractal spectrum of Lévy processes
Multifractal spectrum of Lévy processes
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...
Paul Balança 12 / 50 Regularity of (fractional) Lévy processes
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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.
Paul Balança 13 / 50 Regularity of (fractional) Lévy processes
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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.
Paul Balança 13 / 50 Regularity of (fractional) Lévy processes
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Introduction 2-microlocal analysis
2-microlocal analysis
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.
Definition (2-microlocal spaces)
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.
Paul Balança 14 / 50 Regularity of (fractional) Lévy processes
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Introduction 2-microlocal analysis
2-microlocal analysis
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.
Definition (2-microlocal spaces)
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.
Paul Balança 14 / 50 Regularity of (fractional) Lévy processes
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Introduction 2-microlocal analysis
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)|).
Paul Balança 15 / 50 Regularity of (fractional) Lévy processes
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Introduction 2-microlocal analysis
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)|).
Paul Balança 15 / 50 Regularity of (fractional) Lévy processes
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Introduction 2-microlocal analysis
2-microlocal frontier
s′
σ
0−1−2
1
−3
2−
1
2
σf,t(s′)
α̃f,t
−αf,t
Paul Balança 16 / 50 Regularity of (fractional) Lévy processes
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Introduction 2-microlocal analysis
2-microlocal frontier
s′
σ
0−1−2
1
−3
2−
1
2
σf,t(s′)
σIαf,t(s′)
α
Paul Balança 17 / 50 Regularity of (fractional) Lévy processes
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Introduction 2-microlocal analysis
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.
Paul Balança 18 / 50 Regularity of (fractional) Lévy processes
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Introduction 2-microlocal analysis
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
Paul Balança 19 / 50 Regularity of (fractional) Lévy processes
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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.
Paul Balança 20 / 50 Regularity of (fractional) Lévy processes
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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.
Paul Balança 20 / 50 Regularity of (fractional) Lévy processes
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Multifractal and 2-microlocal analysis on Lévy processes
2-microlocal frontier and multifractal spectrum
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.
Paul Balança 21 / 50 Regularity of (fractional) Lévy processes
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Multifractal and 2-microlocal analysis on Lévy processes
2-microlocal frontier and multifractal spectrum
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.
Paul Balança 21 / 50 Regularity of (fractional) Lévy processes
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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 ,+∞].
Paul Balança 22 / 50 Regularity of (fractional) Lévy processes
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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 ,+∞].
Paul Balança 22 / 50 Regularity of (fractional) Lévy processes
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Multifractal and 2-microlocal analysis on Lévy processes Main result
2-microlocal regularity of Lévy processes
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}.
Paul Balança 23 / 50 Regularity of (fractional) Lévy processes
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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.
Paul Balança 24 / 50 Regularity of (fractional) Lévy processes
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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.
Paul Balança 24 / 50 Regularity of (fractional) Lévy processes
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Multifractal and 2-microlocal analysis on Lévy processes Proof (main ideas)
Proof: multifractal spectrum
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
.
Paul Balança 25 / 50 Regularity of (fractional) Lévy processes
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Multifractal and 2-microlocal analysis on Lévy processes Proof (main ideas)
Proof: multifractal spectrum
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
.
Paul Balança 25 / 50 Regularity of (fractional) Lévy processes
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Multifractal and 2-microlocal analysis on Lévy processes Proof (main ideas)
Proof: multifractal spectrum
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
.
Paul Balança 26 / 50 Regularity of (fractional) Lévy processes
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Multifractal and 2-microlocal analysis on Lévy processes Proof (main ideas)
Proof: multifractal spectrum
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
.
Paul Balança 26 / 50 Regularity of (fractional) Lévy processes
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Multifractal and 2-microlocal analysis on Lévy processes Proof (main ideas)
Proof: multifractal spectrum
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
.
Paul Balança 26 / 50 Regularity of (fractional) Lévy processes
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Multifractal and 2-microlocal analysis on Lévy processes Proof (main ideas)
Proof: multifractal spectrum
Therefore, almost surely
∀h> 0; Eh =�
⋂
δ1/h
Aδ
�
\ S
and
E0 =�
⋂
δ>0
Aδ
�
∪ S.
Paul Balança 27 / 50 Regularity of (fractional) Lévy processes
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Multifractal and 2-microlocal analysis on Lévy processes Proof (main ideas)
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.
Paul Balança 28 / 50 Regularity of (fractional) Lévy processes
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Multifractal and 2-microlocal analysis on Lévy processes Proof (main ideas)
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.
Paul Balança 28 / 50 Regularity of (fractional) Lévy processes
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Multifractal and 2-microlocal analysis on Lévy processes Proof (main ideas)
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.
Paul Balança 28 / 50 Regularity of (fractional) Lévy processes
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Multifractal and 2-microlocal analysis on Lévy processes Proof (main ideas)
Proof: 2-microlocal frontier
s′
σ
0−2
−1
1
1
−1
β
−h
σX,t
σY,t
Paul Balança 29 / 50 Regularity of (fractional) Lévy processes
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Multifractal and 2-microlocal analysis on Lévy processes Proof (main ideas)
Proof: 2-microlocal frontier
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.
Paul Balança 30 / 50 Regularity of (fractional) Lévy processes
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Multifractal and 2-microlocal analysis on Lévy processes Proof (main ideas)
Proof: 2-microlocal frontier
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.
Paul Balança 30 / 50 Regularity of (fractional) Lévy processes
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Multifractal and 2-microlocal analysis on Lévy processes Proof (main ideas)
Proof: 2-microlocal frontier
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.
Paul Balança 31 / 50 Regularity of (fractional) Lévy processes
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Multifractal and 2-microlocal analysis on Lévy processes Proof (main ideas)
Proof: 2-microlocal frontier
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.
Paul Balança 31 / 50 Regularity of (fractional) Lévy processes
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Multifractal and 2-microlocal analysis on Lévy processes Proof (main ideas)
Proof: 2-microlocal frontier
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.
Paul Balança 31 / 50 Regularity of (fractional) Lévy processes
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Multifractal and 2-microlocal analysis on Lévy processes Proof (main ideas)
Proof: 2-microlocal frontier
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.
Paul Balança 31 / 50 Regularity of (fractional) Lévy processes
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Multifractal and 2-microlocal analysis on Lévy processes Proof (main ideas)
Proof: 2-microlocal frontier
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−".
Paul Balança 32 / 50 Regularity of (fractional) Lévy processes
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Multifractal and 2-microlocal analysis on Lévy processes Proof (main ideas)
Proof: 2-microlocal frontier
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−".
Paul Balança 32 / 50 Regularity of (fractional) Lévy processes
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Multifractal and 2-microlocal analysis on Lévy processes Proof (main ideas)
Proof: 2-microlocal frontier
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−".
Paul Balança 32 / 50 Regularity of (fractional) Lévy processes
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Multifractal and 2-microlocal analysis on Lévy processes Proof (main ideas)
Proof: 2-microlocal frontier
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.
Paul Balança 33 / 50 Regularity of (fractional) Lévy processes
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Multifractal and 2-microlocal analysis on Lévy processes Proof (main ideas)
Proof: 2-microlocal frontier
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.
Paul Balança 33 / 50 Regularity of (fractional) Lévy processes
-
Multifractal and 2-microlocal analysis on Lévy processes Proof (main ideas)
Proof: 2-microlocal frontier
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.
Paul Balança 33 / 50 Regularity of (fractional) Lévy processes
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Multifractal and 2-microlocal analysis on Lévy processes Proof (main ideas)
Proof: 2-microlocal frontier
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.
Paul Balança 33 / 50 Regularity of (fractional) Lévy processes
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Multifractal and 2-microlocal analysis on Lévy processes Proof (main ideas)
Proof: 2-microlocal frontier
To conclude, for any h ∈ R+,
Eh = eEh ∪ bEh and dimH(bEh)< dimH(Eh).
Therefore dimH(eEh) = βh.
Paul Balança 34 / 50 Regularity of (fractional) Lévy processes
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Multifractal and 2-microlocal analysis on Lévy processes Remarks & corollary
2-microlocal regularity of Lévy processes
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.
Paul Balança 35 / 50 Regularity of (fractional) Lévy processes
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Multifractal and 2-microlocal analysis on Lévy processes Remarks & corollary
2-microlocal regularity of Lévy processes
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.
Paul Balança 35 / 50 Regularity of (fractional) Lévy processes
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Multifractal and 2-microlocal analysis on Lévy processes Remarks & corollary
2-microlocal regularity of Lévy processes
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.
Paul Balança 35 / 50 Regularity of (fractional) Lévy processes
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Multifractal and 2-microlocal analysis on Lévy processes Remarks & corollary
2-microlocal regularity of Lévy processes
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
Paul Balança 36 / 50 Regularity of (fractional) Lévy processes
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Multifractal and 2-microlocal analysis on Lévy processes Remarks & corollary
2-microlocal regularity of Lévy processes
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.
Paul Balança 37 / 50 Regularity of (fractional) Lévy processes
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Multifractal and 2-microlocal analysis on Lévy processes Remarks & corollary
2-microlocal regularity of Lévy processes
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.
Paul Balança 37 / 50 Regularity of (fractional) Lévy processes
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Multifractal and 2-microlocal analysis on Lévy processes Remarks & 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.
Paul Balança 39 / 50 Regularity of (fractional) Lévy processes
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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.
Paul Balança 39 / 50 Regularity of (fractional) Lévy processes
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Regularity of linear fractional stable motion Definition
Linear fractional stable motion
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
Paul Balança 40 / 50 Regularity of (fractional) Lévy processes
-
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.
Paul Balança 41 / 50 Regularity of (fractional) Lévy processes
-
Regularity of linear fractional stable motion Regularity and multifractal spectrum
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
Paul Balança 42 / 50 Regularity of (fractional) Lévy processes
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Regularity of linear fractional stable motion Regularity and multifractal spectrum
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
Paul Balança 43 / 50 Regularity of (fractional) Lévy processes
-
Regularity of linear fractional stable motion Regularity and multifractal spectrum
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]).
Paul Balança 44 / 50 Regularity of (fractional) Lévy processes
-
Regularity of linear fractional stable motion Regularity and multifractal spectrum
Proof: regularity of LFSM
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.
Paul Balança 45 / 50 Regularity of (fractional) Lévy processes
-
Regularity of linear fractional stable motion Regularity and multifractal spectrum
Proof: regularity of LFSM
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.
Paul Balança 45 / 50 Regularity of (fractional) Lévy processes
-
Regularity of linear fractional stable motion Regularity and multifractal spectrum
Proof: regularity of LFSM
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.
Paul Balança 46 / 50 Regularity of (fractional) Lévy processes
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Regularity of linear fractional stable motion Regularity and multifractal spectrum
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]
-
Regularity of linear fractional stable motion Regularity and multifractal spectrum
Some last remarks...
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.
Paul Balança 48 / 50 Regularity of (fractional) Lévy processes
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Regularity of linear fractional stable motion Regularity and multifractal spectrum
Some last remarks...
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?
Paul Balança 49 / 50 Regularity of (fractional) Lévy processes
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Regularity of linear fractional stable motion Regularity and multifractal spectrum
Some last remarks...
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?
Paul Balança 49 / 50 Regularity of (fractional) Lévy processes
-
Regularity of linear fractional stable motion Regularity and multifractal spectrum
Thank you for your attention !
Questions ?
Paul Balança 50 / 50 Regularity of (fractional) Lévy processes
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