tempered stable process - umacee.uma.pt/luis/page11/socont10.pdfjosé luís silva ccm, university of...

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


Motivation


• 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) < ∞


• 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


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)


where the characteristic exponent ΨΛ is given as


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


• 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)).



• 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)).


• The measure Λ is called the Lévy measure whereas (b,c,Λ) is called Lévy-Khintchine triplet.


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,∞) =∞





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) <∞


• 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


• 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


• 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


• 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


• (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




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

�


Density plot of stable and tempered stable subordinators

α = .4α = .1α = .5α = .6


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| <∞�


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])


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




Stable Subordinator

Link

• Equivalence of Lévy measures:

dΛTSSα (s) =

1α

e−sdΛSSα (s)




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)


• 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)


• 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=⇒


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) < ∞


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


• 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]


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


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]


Behavior


Behavior

∞-∞ 0


Behavior

∞-∞ 0

≈ 2α-stable distribution

( )


Behavior

∞-∞ 0

≈ (TS)-distribution on tails

≈ 2α-stable distribution

( )



Normal Inverse Gaussian process (NIG)

Subclass of ℒ of measures Λ on ℝ satisfying

3. Infinite variation paths

�

|s|≤1|s| dΛ(s) =∞


• Let {XNIG(t), t ∈ [0,1]} be a Lévy process with Lévy measure given by





�

|s|≤1|s| dΛ(s) =∞


• Let {XNIG(t), t ∈ [0,1]} be a Lévy process with Lévy measure given by





�

|s|≤1|s| dΛ(s) =∞

dΛNIG(s) =K1(|s|)

π|s| ds

ΨΛNIG(u) =�1−

�1 + u2

�, u ∈ R


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


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)


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)


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


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}


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).

�


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}



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}



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}


• 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


• 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


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


• 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


• 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


Theorem (conjecture!)

The following convergence is true:

(ZMTSα , XMTS

α ) L−→(Z,XNIG) α→ 0α→ 0


tempered stable process - umacee.uma.pt/luis/page11/socont10.pdfjosé luís silva ccm, university of...

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