Tempered Stable Process - UMacee.uma.pt/luis/page11/socont10.pdfJosé Luís Silva CCM, University of...
Transcript of Tempered Stable Process - UMacee.uma.pt/luis/page11/socont10.pdfJosé Luís Silva CCM, University of...
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José Luís Silva
CCM, University of Madeira, Portugal
Joint work: M. Erraoui, University Cadi Ayyad, Marrakech, Maroc
α-Continuity of SDEs driven by α-Tempered Stable Process
Tuesday, November 9, 2010
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1. Basics: Lévy processes
Contents
3. Convergence and (UT) condition
4. Stability of SDEs driven by Lévy processes
2. Examples of Lévy ProcessesSubordinatorsFinite variation pathsInfinite variation paths
GammaSSTSS
TSMTS
NIG
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Motivation
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• We consider a class ℒ of Borel measures Λ on ℝ satisfying the following conditions:
Basics on Lévy Processes
Λ({0}) = 0�
R(s2 ∧ 1) dΛ(s) < ∞
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• We consider a class ℒ of Borel measures Λ on ℝ satisfying the following conditions:
Basics on Lévy Processes
Λ({0}) = 0�
R(s2 ∧ 1) dΛ(s) < ∞
• By the Lévy-Kintchine formula, all infinitely divisible distributions FΛ are described via their characteristic function
φΛ(u) =�
Reiux dFΛ(x) = eΨΛ(u), u ∈ R
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where the characteristic exponent ΨΛ is given as
Lévy Processes (cont.)
ℝ ≥ 0ΨΛ(u) = ibu− 1
2cu2 +
�
R(eius − 1− ius11{|s|<1}) dΛ(s)
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where the characteristic exponent ΨΛ is given as
Lévy Processes (cont.)
ℝ ≥ 0ΨΛ(u) = ibu− 1
2cu2 +
�
R(eius − 1− ius11{|s|<1}) dΛ(s)
• A Lévy process X={X(t), t ∈ [0,1]} has the property:
where Ψ(u) is the characteristic exponent of X(1) which has an infinitely divisible distribution.
E(eiuX(t)) = etΨ(u), t ∈ [0, 1], u ∈ R
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• Thus, any infinitely divisible distribution FΛ, generates in a natural way a Lévy process X by setting the law of X(1), L(X(1)) = FΛ.
• The three quantities (b,c,Λ) determine the law L(X(1)).
Lévy Processes (cont.)
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• Thus, any infinitely divisible distribution FΛ, generates in a natural way a Lévy process X by setting the law of X(1), L(X(1)) = FΛ.
• The three quantities (b,c,Λ) determine the law L(X(1)).
Lévy Processes (cont.)
• The measure Λ is called the Lévy measure whereas (b,c,Λ) is called Lévy-Khintchine triplet.
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Examples of Lévy Processes1. Subordinators:
A subordinator is a one-dimensional increasing Lévy process starting from 0.
• We consider a subclass of ℒ of measures Λ supported on ℝ+:
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Examples of Lévy Processes1. Subordinators:
A subordinator is a one-dimensional increasing Lévy process starting from 0.
• We consider a subclass of ℒ of measures Λ supported on ℝ+:
It implies that the process X has infinite activity, i.e., a lmost a l l paths have infinitely many jumps along any finite time interval.
Λ(0,∞) =∞
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Examples of Lévy Processes1. Subordinators:
A subordinator is a one-dimensional increasing Lévy process starting from 0.
• We consider a subclass of ℒ of measures Λ supported on ℝ+:
It implies that the process X has infinite activity, i.e., a lmost a l l paths have infinitely many jumps along any finite time interval.
Λ(0,∞) =∞
Almost all paths of X have finite variation.
� 1
0s dΛ(s) <∞
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• Consider the Lévy measure Λγ with density w.r.t. the Lebesgue measure:
Examples of Lévy Processes (cont.) 1.1 Gamma process:
dΛγ(s) :=e−s
s11{s>0} ds
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• Consider the Lévy measure Λγ with density w.r.t. the Lebesgue measure:
Examples of Lévy Processes (cont.) 1.1 Gamma process:
dΛγ(s) :=e−s
s11{s>0} ds
• The corresponding process Xγ (gamma process) has Laplace transform of form
The law of Xγ (1)
Eµγ
�e−uXγ(t)
�= exp(−t log(1 + u)) =
1(1 + u)t
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• Let α ∈ (0,1) and the Lévy measure:
Examples of Lévy Processes (cont.) 1.2 Stable Subordinator (SS):
ΛSSα
• The corresponding process (stable subordinator) has Laplace transform:
XSSα
EµSSα
�e−uXSS
α (t)�
= exp(−tuα), t ∈ [0, 1]
dΛSSα (s) :=
α
Γ(1− α)1
s1+α11{s>0} ds
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• Let α ∈ (0,1) and the Lévy measure:
Examples of Lévy Processes (cont.) 1.2 Stable Subordinator (SS):
ΛSSα
• The corresponding process (stable subordinator) has Laplace transform:
XSSα
The law of XSSα (1)
EµSSα
�e−uXSS
α (t)�
= exp(−tuα), t ∈ [0, 1]
dΛSSα (s) :=
α
Γ(1− α)1
s1+α11{s>0} ds
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• (TSS) is obtained by taking a stable subordinator and multiplying the Lévy measure by an exponential function, i.e., an exponentially tempered version of (SS)
Examples of Lévy Processes (cont.) 1.3 Tempered Stable Subordinator (TSS):
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• (TSS) is obtained by taking a stable subordinator and multiplying the Lévy measure by an exponential function, i.e., an exponentially tempered version of (SS)
Examples of Lévy Processes (cont.) 1.3 Tempered Stable Subordinator (TSS):
ΛTSSα• Lévy measure
dΛTSSα (s) =
1Γ(1− α)
e−s
s1+α11{s>0} ds
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• (TSS) is obtained by taking a stable subordinator and multiplying the Lévy measure by an exponential function, i.e., an exponentially tempered version of (SS)
Examples of Lévy Processes (cont.) 1.3 Tempered Stable Subordinator (TSS):
ΛTSSα• Lévy measure
dΛTSSα (s) =
1Γ(1− α)
e−s
s1+α11{s>0} ds
XTSSα• Laplace transform of
EµT SSα
�e−uXT SS
α (t)�
= exp�−t
1− (1 + u)α
α
�
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Density plot of stable and tempered stable subordinators
α = .4α = .1α = .5α = .6
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1. Real linear space of all finite discrete measures in [0,1]
Concrete realization of a subordinator: Tsilevich-Vershik-Yor’01
D =�
η =�
ziδxi , xi ∈ [0, 1], zi ∈ R+,�
|zi| <∞�
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1. Real linear space of all finite discrete measures in [0,1]
Concrete realization of a subordinator: Tsilevich-Vershik-Yor’01
D =�
η =�
ziδxi , xi ∈ [0, 1], zi ∈ R+,�
|zi| <∞�
2. Coordinate process X on D; t ∈ [0,1]
Filtration: Ft := σ(X(s), s ≤ t)
X(t) : D −→ R+, η �→ X(t)(η) := η([0, 1])
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In particular, for f (s) = u1[0,t](s), u > 0, t ∈ (0,1] the Laplace transform of X(t) is
3. Law: let Λ be a Lévy measure satisfying the conditions and µΛ a probability measure on (D, F1) with
EµΛ
��−
� 1
0f(t)dη(t)
��= exp
�� 1
0log(ψΛ(f(t))) dt
�
=⇒Tuesday, November 9, 2010
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In particular, for f (s) = u1[0,t](s), u > 0, t ∈ (0,1] the Laplace transform of X(t) is
3. Law: let Λ be a Lévy measure satisfying the conditions and µΛ a probability measure on (D, F1) with
EµΛ
��−
� 1
0f(t)dη(t)
��= exp
�� 1
0log(ψΛ(f(t))) dt
�
X(t) is a subordinator=⇒
EµΛ(e−uX(t)) = exp(t log(ψΛ(u))), t ∈ [0, 1]
=⇒Tuesday, November 9, 2010
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Tempered Stable vs Stable Subordinators:
Tempered Stable Subordinator
Stable Subordinator
Link
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Tempered Stable vs Stable Subordinators:
Tempered Stable Subordinator
Stable Subordinator
Link
• Equivalence of Lévy measures:
dΛTSSα (s) =
1α
e−sdΛSSα (s)
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Tempered Stable vs Stable Subordinators:
Tempered Stable Subordinator
Stable Subordinator
Link
L(XTSSα )
• Then it follows from, e.g., K. Saito book, that
L(XSSα )Equivalent
• Equivalence of Lévy measures:
dΛTSSα (s) =
1α
e−sdΛSSα (s)
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• From Tsilevich-Vershik-Yor’01:
Let Xα be a process such that the law µα := L(Xα(1)) is equivalent to with densityµSS
α
dµα
dµSSα
(η) =exp
�− α−1/αX(1)(η)
�
EµSSα
(%)
= eα−1e−α−1/αX(1)(η)
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• From Tsilevich-Vershik-Yor’01:
Let Xα be a process such that the law µα := L(Xα(1)) is equivalent to with densityµSS
α
dµα
dµSSα
(η) =exp
�− α−1/αX(1)(η)
�
EµSSα
(%)
= eα−1e−α−1/αX(1)(η)
L(XTSSα )
L�
1α
Xα
� WeaklyGamma measure=⇒
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We are interested here in the subclass of ℒ of measures Λ on ℝ satisfying
Examples of Lévy Processes (cont.)
2.1 Tempered Stable process (TS):2. Finite variation paths
Λ(R) = ∞,�
|s|<1|s|dΛ(s) < ∞
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Examples of Lévy Processes (cont.) [Finite variation paths]
Take a symmetric α-stable distribution and multiply its Lévy measure by an exponential in each side
dΛTSα (s) =
�e−|s|
|s|1+α11{s<0} +
e−s
|s|1+α11{s>0}
�ds
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• The characteristic exponent: u ∈ ℝ
Examples of Lévy Processes (cont.) [Finite variation paths]
Take a symmetric α-stable distribution and multiply its Lévy measure by an exponential in each side
dΛTSα (s) =
�e−|s|
|s|1+α11{s<0} +
e−s
|s|1+α11{s>0}
�ds
XTSαThe Corresponding process is denoted by
ΨΛT Sα
(u) = Γ(−α)[(1− iu)α + (1 + iu)α − 2]
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Examples of Lévy Processes (cont.) 2.2 Modified Tempered Stable process (MTS):
• Definition: Lévy measureBessel function 2nd kind
dΛMTSα (s) =
1π
�Kα+ 1
2(|s|)
|s|α+ 12
11{s<0} + (Kα+ 1
2(s)
sα+ 12
11{s>0}
�ds
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Examples of Lévy Processes (cont.) 2.2 Modified Tempered Stable process (MTS):
• Definition: Lévy measureBessel function 2nd kind
dΛMTSα (s) =
1π
�Kα+ 1
2(|s|)
|s|α+ 12
11{s<0} + (Kα+ 1
2(s)
sα+ 12
11{s>0}
�ds
• The characteristic exponent: u ∈ ℝ
ΨΛMT Sα
(u) =1√π
2−α− 12 Γ(−α)[(1 + u2)α − 1]
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Behavior
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Behavior
∞-∞ 0
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Behavior
∞-∞ 0
≈ 2α-stable distribution
( )
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Behavior
∞-∞ 0
≈ (TS)-distribution on tails
≈ 2α-stable distribution
( )
Tuesday, November 9, 2010
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Examples of Lévy Processes (cont.)
Normal Inverse Gaussian process (NIG)
Subclass of ℒ of measures Λ on ℝ satisfying
3. Infinite variation paths
�
|s|≤1|s| dΛ(s) =∞
Tuesday, November 9, 2010
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• Let {XNIG(t), t ∈ [0,1]} be a Lévy process with Lévy measure given by
Examples of Lévy Processes (cont.)
Normal Inverse Gaussian process (NIG)
Subclass of ℒ of measures Λ on ℝ satisfying
3. Infinite variation paths
�
|s|≤1|s| dΛ(s) =∞
Tuesday, November 9, 2010
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• Let {XNIG(t), t ∈ [0,1]} be a Lévy process with Lévy measure given by
Examples of Lévy Processes (cont.)
Normal Inverse Gaussian process (NIG)
Subclass of ℒ of measures Λ on ℝ satisfying
3. Infinite variation paths
�
|s|≤1|s| dΛ(s) =∞
dΛNIG(s) =K1(|s|)
π|s| ds
ΨΛNIG(u) =�1−
�1 + u2
�, u ∈ R
Tuesday, November 9, 2010
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Convergence and UT condition1. Convergence
• Convergence in the Skorohod scape: (D[0,1],J1) of the families and XTSS
α XMTSα
Lemma
We have the following weak convergence:
(i) L(XTSSα (1)) −→ L(Xγ(1)), α→ 0
(ii) L(XMTSα (1)) −→ L(XNIG(1)), α→ 1/2
Tuesday, November 9, 2010
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Proof.(i) Tsilevich-Vershik-Yor’01
(ii) It resumes to show that
ΨΛNIG(u) =�1−
�1 + u2
�α −→ 1
2
=⇒ XMTSα (1) w−→ XNIG(1)
Tuesday, November 9, 2010
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Proof.(i) Tsilevich-Vershik-Yor’01
(ii) It resumes to show that
ΨΛNIG(u) =�1−
�1 + u2
�
ΨΛMT Sα
(u) =1√π
2−α− 12 Γ(−α)[(1 + u2)α − 1]
α −→ 12
=⇒ XMTSα (1) w−→ XNIG(1)
Tuesday, November 9, 2010
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We have the following weak convergence on (D[0,1],D)Propositon
(i) XTSSα
L−→ Xγ , α→ 0
(ii) XMTSα
L−→ XNIG, α→ 1/2
Tuesday, November 9, 2010
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We have the following weak convergence on (D[0,1],D)Propositon
(i) XTSSα
L−→ Xγ , α→ 0
(ii) XMTSα
L−→ XNIG, α→ 1/2
Since Lévy processes are semimartingales with stationary independent increments, then it follows from Jacod-Shiryaev that the convergence of the marginal laws
is equivalent to the weak convergence of processes
Proof.
L(XTSSα (1)) and L(XMTS
α (1))
XTSSα and XMTS
α in D[0, 1].
and
and inTuesday, November 9, 2010
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DefinitonA sequence {Zn, n ∈ ℕ} of semimartingales satisfies the (UT) condition if the sequence of real-valued random variables
tight, i.e., it is almost inside of a compact.
Zn = Var (An,a) (1) + �Mn,a,Mn,a� (1)
+�
s≤1
|∆Zn(s)| 11{|∆Zn(s)|>a}
Tuesday, November 9, 2010
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DefinitonA sequence {Zn, n ∈ ℕ} of semimartingales satisfies the (UT) condition if the sequence of real-valued random variables
tight, i.e., it is almost inside of a compact.
Zn = Var (An,a) (1) + �Mn,a,Mn,a� (1)
+�
s≤1
|∆Zn(s)| 11{|∆Zn(s)|>a}
Mémin-Słomiński’91: Assume that the sequence {L(Zn), n ∈ ℕ} converges weakly in D[0,1], Then the (UT) condition is equivalent to the boundedness in probability of the sequence Var(An,a)(1).
�
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Every Lévy process may be decomposed as (e.g. Applebaum’09)
Z(t) = tE(R(1)) + R0(t) +�
s≤t
�Z(s)11{|�z(s)|>1}
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Every Lévy process may be decomposed as (e.g. Applebaum’09)
càdlàg centred square-integrable martingale with bounded jumps by 1
Z(t) = tE(R(1)) + R0(t) +�
s≤t
�Z(s)11{|�z(s)|>1}
Tuesday, November 9, 2010
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Every Lévy process may be decomposed as (e.g. Applebaum’09)
càdlàg centred square-integrable martingale with bounded jumps by 1
Theorem
1. The family satisfies the (UT) condition
2. The family does not satifies de (UT) condition
�XTSS
α , α ∈ (0, 1/2)�
�XMTS
α , α ∈ (0, 1/2)�
Z(t) = tE(R(1)) + R0(t) +�
s≤t
�Z(s)11{|�z(s)|>1}
Tuesday, November 9, 2010
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• We consider the following SDEs
Stability of SDEs driven by Lévy processes
dY TSSα (t) = aα(Y TSS
α (t))dXTSSα (t), Y TSS
α (0) = 0
dY (t) = a(Y (t))dXγ(t), Y (0) = 0
Tuesday, November 9, 2010
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• We consider the following SDEs
Stability of SDEs driven by Lévy processes
• Assumptions:
(H.1) aα : ℝ+ → (0,∞) continuous s.t. | aα(x)| ≤ K(1+|x|) for all α ∈ (1,1/2).
(H.2) The family aα conv. Unif. to a on each compact set in ℝ+
dY TSSα (t) = aα(Y TSS
α (t))dXTSSα (t), Y TSS
α (0) = 0
dY (t) = a(Y (t))dXγ(t), Y (0) = 0
Tuesday, November 9, 2010
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TheoremUnder (H.1), (H.2) we have:
1. The family of processes is tight
2. The SDE
admits a solution Y.
3. If Y is the unique solution, then
(Y TSSα , XTSS
α )
(Y TSSα , XTSS
α ) L−→(Y, Xγ) α→ 0
dY (t) = a(Y (t))dXγ(t), Y (0) = 0
Tuesday, November 9, 2010
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• The NIG case:
dZMTSα (t) = b(ZMTS
α (t))dXMTSα (t), ZMTS
α (0) = 0
dZ(t) = b(Z(t))◦dXNIG(t), Z(0) = 0
Tuesday, November 9, 2010
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• The NIG case:
• Assumptions:
(H.3) b : ℝ → ℝ is of classe C2 with bounded derivatives.
• Existence of solution, e.g., Protter’05
dZMTSα (t) = b(ZMTS
α (t))dXMTSα (t), ZMTS
α (0) = 0
dZ(t) = b(Z(t))◦dXNIG(t), Z(0) = 0
Tuesday, November 9, 2010
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Theorem (conjecture!)
The following convergence is true:
(ZMTSα , XMTS
α ) L−→(Z,XNIG) α→ 0α→ 0
Tuesday, November 9, 2010