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
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
Tuesday, November 9, 2010
Motivation
Tuesday, November 9, 2010
• 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) < ∞
Tuesday, November 9, 2010
• 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
Tuesday, November 9, 2010
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)
Tuesday, November 9, 2010
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
Tuesday, November 9, 2010
• 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.)
Tuesday, November 9, 2010
• 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.
Tuesday, November 9, 2010
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+:
Tuesday, November 9, 2010
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,∞) =∞
Tuesday, November 9, 2010
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) <∞
Tuesday, November 9, 2010
• 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
Tuesday, November 9, 2010
• 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
Tuesday, November 9, 2010
• 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
Tuesday, November 9, 2010
• 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
Tuesday, November 9, 2010
• (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):
Tuesday, November 9, 2010
• (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
Tuesday, November 9, 2010
• (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
�
Tuesday, November 9, 2010
Density plot of stable and tempered stable subordinators
α = .4α = .1α = .5α = .6
Tuesday, November 9, 2010
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| <∞�
Tuesday, November 9, 2010
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])
Tuesday, November 9, 2010
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
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
Tempered Stable vs Stable Subordinators:
Tempered Stable Subordinator
Stable Subordinator
Link
Tuesday, November 9, 2010
Tempered Stable vs Stable Subordinators:
Tempered Stable Subordinator
Stable Subordinator
Link
• Equivalence of Lévy measures:
dΛTSSα (s) =
1α
e−sdΛSSα (s)
Tuesday, November 9, 2010
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)
Tuesday, November 9, 2010
• 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)
Tuesday, November 9, 2010
• 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=⇒
Tuesday, November 9, 2010
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) < ∞
Tuesday, November 9, 2010
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
Tuesday, November 9, 2010
• 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]
Tuesday, November 9, 2010
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
Tuesday, November 9, 2010
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]
Tuesday, November 9, 2010
Behavior
Tuesday, November 9, 2010
Behavior
∞-∞ 0
Tuesday, November 9, 2010
Behavior
∞-∞ 0
≈ 2α-stable distribution
( )
Tuesday, November 9, 2010
Behavior
∞-∞ 0
≈ (TS)-distribution on tails
≈ 2α-stable distribution
( )
Tuesday, November 9, 2010
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
• 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
• 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
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
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
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
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
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
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
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).
�
Tuesday, November 9, 2010
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}
Tuesday, November 9, 2010
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
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
• 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
• 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
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
• 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
• 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
Theorem (conjecture!)
The following convergence is true:
(ZMTSα , XMTS
α ) L−→(Z,XNIG) α→ 0α→ 0
Tuesday, November 9, 2010