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Some Elementson Lévy Processes

Lucia Jarešová

Econometrics and Operational Research

Charles UniversityFaculty of Mathematics and PhysicsPrague, Czech Republic

8th November 2010

SomeElementson Lévy

Processes

OutlineLiterature

IntroductionBasic DefinitionsPoisson Processetc.

Basic Aspectson LévyProcessesFamousProcessesMain PropertiesExamples

Structure ofLévyProcessesJump ProcessDecompositionof a LévyProcess

Some SamplePathPropertiesRecurrence andtransience

Stock Modelwith JumpsJump Diffusion

Outline

Outline

Literature1 Introduction

Basic DefinitionsPoisson Process etc.

2 Basic Aspects on Lévy ProcessesFamous ProcessesMain PropertiesExamples

3 Structure of Lévy ProcessesJump ProcessDecomposition of a Lévy Process

4 Some Sample Path PropertiesRecurrence and transience

5 Stock Model with JumpsJump Diffusion

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Outline Literature

Jean BertoinSome Elements on Lévy Processes, in Handbook ofStatistics, Vol. 19Elsevier Science2001

Karel JanečekAdvanced Topics in Financial MathematicsStudy material

Paul WilmottPaul Wilmott on Quantitative FinanceWiley2006

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Introduction

Introduction

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Introduction Basic Definitions

Lévy processes

= processes in continuous time with independentand stationary increments

• Important class of Markovprocesses.

• Natural examples of semimartingalesfor which stochastic calculus applies.

• Appeared in physics: problemsin turbulence, laser cooling.

• Important role in mathematicalfinance (heavy tails).

Paul Pierre Lévy

(1886-1971)

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Filtered Probability Space

(Ω,F , (Ft)t≥0,P)

Filtration (Ft)t≥0 fulfills the standard conditions :

• Fs ⊆ Ft for s ≤ t (as times moves forward, we obtainmore and more information).

• Filtration is right-continuous, i.e. Ft = Ft+ =⋂ε>0Ft+ε.

Xt is an Ft-adapted stochastic process, if σ(Xt) ⊆ Ft ,∀t ≥ 0(Xt is Ft-measurable for each t).

Filtration models the flow of public information. Priceprocesses are adapted to this filtration (i.e. the filtrationcontains the observed history of market variables).

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Stopping Time (Markův čas)

Definition 1Suppose a (Ω,F , (Ft)t≥0,P) . A stopping time τ is a randomvariable taking values in [0,∞] and satisfying

[τ ≤ t] ∈ Ft , ∀t ≥ 0.

Properties: [τ = t] ∈ Ft , i.e. the decision to stop at time t isbased on information available at time t.

Definition 2We have an adapted process Xt and a stopping time τ . Thestopped process is defined as

Xt∧τ =

Xt t ≤ τXτ t > τ

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Stopping Time

Examples:Hitting time of a one-sided boundary by a Brownian motion

τa := inft ≥ 0 : Wt = a

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Introduction Poisson Process etc.

Poisson Process: Construction

τ1, τ2, · · · ∼ Exp(λ) exponentially i.i.d. random variables,i.e. pdf f (t) = λe−λt , t ≥ 0, Eτi = 1/λ.Let Sn be the time of the n-th jump

Sn =n∑k=1

τk .

Poisson process with intensity λ is the number of jumps at orbefore time t

Nt =∞∑i=1

I[Si≤t]

• Nt is right-continuous• Nt is not predictable w.r.t. Ft , i.e. not Ft− measurable

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Poisson Process: Basic Properties

Poisson process jumps are of size 1.

Lemma 3The Poisson process Nt with intensity λ > 0 has the Poissondistribution Po(λt)

P[Nt = k] =(λt)k

k!e−λt , k = 0, 1, . . .

Memorylessnes of the exponential distribution⇒ Poisson process is memorylessFor t, s ≥ 0

L(Nt+s − Ns) = L(Nt) = Po(λt)

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Poisson Process: Basic Properties

Mean and variance of the Poisson process:

ENt = λt

Var(Nt) = λt

Theorem 4The compensated Poisson process defined as

Mt = Nt − λt

is a martingale.

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Compound Poisson Process

. . . to allow the jump sizes to be random

ξ1, ξ2, . . . i.i.d. with β = Eξi independent of Poisson process Nt

Compound Poisson process

Yt :=Nt∑i=1

ξi , t ≥ 0.

The compound Poisson processis memoryless, increments areindependent andL(Yt+s − Ys) = L(Yt).

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Compound Poisson Process

Mean of the compound Poisson process:

EYt = E

[Nt∑i=1

ξi

]=∞∑k=0

E

[k∑i=1

ξi∣∣Nt = k

]P [Nt = k] =

= Eξ1

∞∑k=0

kP [Nt = k] = Eξ1ENt = βλt

Theorem 5The compensated compound Poisson process defined as

Mt = Yt − βλt

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Basic Aspects on Lévy Processes

Basic Aspects on LévyProcesses

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Basic Aspects on Lévy Processes Famous Processes

Wiener Process

Definition 6An Ft-adapted stochastic process X = (Xt , t ≥ 0) with valuesin R is said to be a Wiener process, if ∀s, t ≥ 0

1 W0 = 0 almost surely.

2 Independent increments: Wt+s −Wt is independentof Ft .

3 Normal increments: Wt+s −Wt ∼ N(0, s).

4 Sample paths are continuous.

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Poisson Process

Definition 7An Ft-adapted stochastic counting process (Nt , t ≥ 0) withvalues in N is said to be a Poisson process, if ∀s, t ≥ 0

1 N0 = 0 almost surely.

2 Independent increments: Nt+s − Nt is independentof Ft .

3 Stationary increments: Nt+s − Nt has the samedistribution as Ns .

4 No counted occurrences are simultaneous.

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Lévy Process

Definition 8An Ft-adapted stochastic process X = (Xt , t ≥ 0) with valuesin Rd is said to be a Lévy process, if ∀s, t ≥ 0

1 X0 = 0 almost surely.

2 Independent increments: Xt+s − Xt is independentof Ft .

3 Stationary increments: Xt+s − Xt has the samedistribution as Xs .

4 Sample paths are right-continuous and possess leftlimits.

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Càdlàg Process

= everywhere right continuous and has left limitseverywhere

càdlàg: ”continu à droite, limite à gauche”

RCLL: ”right continuous with left limits”

corlol: ”continuous on (the) right, limit on (the) left”

Skorokhod space = the collection of càdlàg functions on agiven domain.

• Lévy process is càdlàg.• Continuous process is càdlàg.

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Basic Aspects on Lévy Processes Main Properties

Markov Property

From the properties of Lévy process we get immediately• Xt+s |Xt = x is independent of Ft , s, t ≥ 0.• L(Xt+s |Xt = x) = L(x + Xs), s, t ≥ 0.

Theorem 9( Markov Property) Let τ be an (Ft)-stopping time, τ <∞a.s. .• Xτ+t |Xτ = x is independent of Fτ , t ≥ 0.• L(Xτ+t |Xτ = t) = L(x + Xt), t ≥ 0.

Often applied to investigate distributions related to firstpassage time (čas prvního průchodu)τB := inft ≥ 0 : Xt ∈ B, B is a Borel set.⇒ τB is a stopping time.

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Infinitely Divisibility

Elementary decomposition of a Process, n ∈ N:

X1 = X 1n

+(

X 2n− X 1

n

)+ · · ·+

(X nn− X (n−1)

n

)⇒ Distributions of a LP are infinitely divisible (can beexpressed as the sum of n i.i.d. variables, n ∈ N).

Characteristic function of an infinitely divisible variable X1 canbe expressed in the form

E(

e i〈λ,X1〉)

= e−Ψ(λ), λ ∈ Rd ,

where 〈·, ·〉 is the scalar product and Ψ : Rd → C isa continuous function with Ψ(0) = 0 known as thecharacteristic exponent of X .

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Law of the Whole Process

Making use of the independence, stationarity of increments andright-continuity of the sample paths we get

E(

e i〈λ,Xt〉)

= e−tΨ(λ), λ ∈ Rd , t ≥ 0.

⇒ the law of the Lévy process is completely determinedby Ψ.

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Lévy-Khintchine formula

Theorem 10A function Ψ : Rd → C is the characteristic exponentof an infinitely divisible distribution if and only if it can beexpressed in the form

Ψ(λ) = −i〈a, λ〉+12

Q(λ)+

+

∫Rd

(1− e i〈λ,x〉 + i〈λ, x〉I[|x |<1]

)Π(dx),

where a ∈ Rd , Q is a positive semi-definite quadratic form onRd , and Π a measure on Rd\0 with

∫(1 ∧ |x |2)Π(dx) <∞

called Lévy measure.Moreover, a, Q and Π are then uniquely determined by Ψ.

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Lévy-Khintchine formula

Note: This formula gives the generic form of characteristicexponents.⇒ Key to understanding the probabilistic structure of Lévyprocesses.

1D-version:E[e iλX1

]= e−Ψ(λ) =

= exp(

iaλ− 12σ2λ2 −

∫R

(1− e iλx + iλx I|x |<1

)Π(dx)

)

Ψ(λ) = −iaλ+12σ2λ2 +

∫R

(1− e iλx + iλx I|x |<1

)Π(dx)

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Basic Aspects on Lévy Processes Examples

Poisson Distribution

X ∼ Po(c), c > 0:

P(X = n) =cn

n!e−c

Characteristic function:

E[e iλX

]=∞∑n=0

e iλncn

n!e−c = exp

(−c(1− e iλ)

)Lévy measure: Π(dx) = cδ1(dx) (δ1 is the Dirac measure at 1)

Associated Lévy process: Poisson process with intensity c

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Normal Distribution

X ∼ N(0, 1):

f (x) =1√2π

e−x22


E[e iλX

]=

1√2π

∫ ∞−∞

e iλxe−x22 = exp

(−λ

2

2

)Lévy measure: Π(dx) = 0dxAssociated Lévy process: Standard Brownian motion

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Cauchy Distribution

X ∼ Cauchy:

f (x) =1

π(1 + x2)


E[e iλX

]=

1π

∫ ∞−∞

e iλxdx(1 + x2)

= exp(−|λ|) =

= exp(− 1π

∫ ∞−∞

(1− e iλx

)x−2dx

)Lévy measure: Π(dx) = π−1x−2dxAssociated Lévy process: Standard Cauchy process

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Gamma Distribution

X ∼ Γ(c , 1), c > 0:

f (x) =xc−1e−x

Γ(c)


E[e iλX

]=

1Γ(c)

∫ ∞−∞

e iλxxc−1e−xdx = (1− iλ)−c =

= exp(−c∫ ∞

0

(1− e iλx

)x−1e−xdx

)Lévy measure: Π(dx) = cI[x>0]x−1e−xdxAssociated Lévy process: Gamma process with shapeparameter c

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Stable Distributions

X ∼ SD(α, β, γ), α ∈ (0, 1) ∪ (1, 2), β ∈ [−1, 1], γ > 0:(α = 1 transform of Cauchy distribution,α = 2 normal distribution)


E[e iλX

]= exp (−γ|λ|α(1− iβ sgn(λ) tan(πα/2)))

Lévy measure: Π(dx) =

c+x−α−1dx x > 0

c−|x |−α−1dx x < 0,

where c+ and c− are two nonnegative real numbers such thatβ = (c+ − c−)/(c+ + c−)

Associated Lévy process: Stable Lévy process with index αand skewness β

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Structure of Lévy Processes

Structure of Lévy Processes

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Structure of Lévy Processes Jump Process

Jumps in Lévy Process

Left-limit of X at time t:

Xt− = lims→t−

Xs

(possible) jump:∆Xt = Xt − Xt−

For any Borel set 0 6= B ⊆ Rd , write

NBt = Cards ∈ (0, t] : ∆Xs ∈ B

for the number of jumps accomplished by X before time t thattake values in B.

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Jumps → Poisson Process

Independence and stationarity of the increments of X

⇓

• NBt has independent and stationary increments• sample paths of NBt are right-continuous and they increase

by jumps of size 1

⇓

NBt is a Poisson process with intensity Λ(B)

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Lévy measure

B1, . . . ,Bn, . . . is a countable partition of B (disjoint sets,union is B)

⇓NB1t , . . . ,N

Bnt , . . . are independent Poisson processes with

intensities Λ(Bi ), i = 1, . . . ,∞

NBt = NB1t + · · ·+ NBnt + . . .

is a Poisson process with intensity

Λ(B) = Λ(B1) + · · ·+ Λ(Bn) + . . .

⇓

Λ is a Borel measure on Rd\0 that gives a finite massto the complement of any neighbourhood of the origin.

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Structure of the Jumps

Theorem 11The jump process ∆X = (∆Xt , t ≥ 0) of a Lévy process X is aPoisson point process valued in Rd , whose characteristicmeasure is the Lévy measure Π.

This means that for every Borel set B at a positive distancefrom the origin, the counting process NB is a Poisson processwith intensity Π(B), and to disjoint Borel sets correspondindependent Possion processes.

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Ex.: Compound Poisson Process

Let Λ be a finite measure on Rd that gives no mass to theorigin.Let (∆t , t ≥ 0) be a Poisson point process with thecharacterisic finite measure Λ.

Compound Poisson process

Yt =∑

0≤s≤t∆s

is a right-continuous step process and by construction its jumpprocess is ∆Yt = ∆t .

Yt is a Lévy process.

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Ex.: Compound Poisson Process

We compute the characteristic function

E(

e i〈λ,Y1〉)

= E

exp

i∑

0≤s≤1

〈λ,∆s〉

=

= exp−∫

Rd

(1− e i〈λ,x〉

)Λ(dx)

It is a special case of the Lévy-Khintchine formula.

Characteristic measure Λ of the jump process is the Lévymeasure.

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Structure of Lévy Processes Decomposition of a Lévy Process

Probabilistic meaning of the LK-Formula

Decomposition of the characteristic exponent Ψ of theLévy-Khintchine formula

Ψ = Ψ(0) + Ψ(1) + Ψ(2) + Ψ(3),

where

Ψ(0)(λ) = −i 〈a, λ〉Ψ(1)(λ) = 1

2 Q(λ)

Ψ(2)(λ) =∫Rd(1− e i〈λ,x〉

)I[|x |≥1]Π(dx)

Ψ(3)(λ) =∫Rd(1− e i〈λ,x〉 + i 〈λ, x〉

)I[|x |<1]Π(dx)

Each Ψ(i) is a characteristic exponent of some Lévyprocess.

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Continuous Part: Ψ(0) and Ψ(1)

Constant driftis a deterministic linear process with characteristic exponent

Ψ(0)(λ) = −i 〈a, λ〉

Brownian componentis a linear transform of a d-dimensional Brownian motion withcharacteristic exponent

Ψ(1)(λ) =12

Q(λ)

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Large Jumps: Ψ(2)

Ψ(2)(λ) =

∫Rd

(1− e i〈λ,x〉

)I[|x |≥1]Π(dx)

is a characteristic exponent of a compound Poisson processwith Lévy measure I[|x |≥1]Π(dx)

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Small Jumps: Ψ(3)

Ψ(3)(λ) =

∫Rd

(1− e i〈λ,x〉 + i 〈λ, x〉

)I[|x |<1]Π(dx)

In the case when ∫Rd|x |I[|x |<1]Π(dx) <∞,

we can re-write

Ψ(3)(λ) = i⟨λ, a′

⟩+

∫Rd

(1− e i〈λ,x〉

)I[|x |<1]Π(dx)

with a′ =∫Rd x I[|x |<1]Π(dx).

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Small Jumps: Ψ(3)

We can consider a Poisson point process ∆(3) withcharacteristic measure I[|x |<1]Π(dx).

The hypothesis∫Rd |x |I[|x |<1]Π(dx) <∞ ensures that the series∑

0≤s≤t |∆(3)s | converges a.s. for every t ≥ 0, and this enables

us to setY (3)t = −a′t +

∑0≤s≤t

∆(3)s .

Y (3)t is a Lévy process with characteristic exponent Ψ(3).

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Lévy-Itô Decomposition

General Lévy Process X can be decomposed as the sumof four independent Lévy processes:

X = Y (0) + Y (1) + Y (2) + Y (3),

where• Y (0) is a constant drift.• Y (1) is linear transform of a Brownian motion.• Y (2) is a compound Poisson process with jumps of size

greater than or equal to 1.• Y (3) is a pure jump process with jumps of size less than 1,

that is obtained as the limit of compensated compoundPoisson processes.

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Jump Diffusion Process

+ +

=

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Some Sample Path Properties

Some Sample Path Properties

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Some Sample Path Properties Recurrence and transience

Definitions

X is a Lévy process with values in Rd

X is recurrent if

lim inft→∞|Xt | = 0a.s.

X is transient if

lim inft→∞|Xt | =∞a.s.

The potential of a Borel set B ⊆ Rd is the expected time spentby the Lévy process in B,

U(B) :=

∫ ∞0

P(Xt ∈ B)dt = E(∫ ∞

0I[Xt∈B]dt

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Some Sample Path Properties Recurrence and transience

Analytic Characterization

Theorem 12For ε > 0, let Bε stand for the open ball in Rd centered at theorigin with radius ε.If U(Bε) <∞ for some ε > 0, then the Lévy process istransient. Otherwise the Lévy process is recurrent.

Theorem 13( Chung and Fuchs test) Let X be a real-valued Lévy processwith finite mean EX1 = µ ∈ R.Then X is transient if µ 6= 0 and recurrent if µ = 0.

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Stock Model with Jumps

Stock Model with Jumps

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Stock Model with Jumps Jump Diffusion

Simple Stock Price with Jumps

We assume the stock price follows the SDE

dS = µSdt + σSdW + (J − 1)Sdq

W is the Brownian motionq is the Poisson process independent of W

dq =

0 with probability 1− λdt1 with probability λdt

When dq = 1, the process jumps from S to JS .

The jump size J is random variable independent of theBrownian motion W and the Poisson process q.

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Stock Model with Jumps Jump Diffusion

Jump Diffusion Models

+

• capture a real phenomenon that is missing from theBlack-Scholes model.

-

• difficulty in parameter estimation• it is hard to find a numerical solution• impossibility of perfect risk-free hedging, only henging ”on

average”

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