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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
2/49 ,
SomeElementson Lévy
Processes
OutlineLiterature
IntroductionBasic DefinitionsPoisson Processetc.
Basic Aspectson LévyProcessesFamousProcessesMain PropertiesExamples
Structure ofLévyProcessesJump ProcessDecompositionof a LévyProcess
Some SamplePathPropertiesRecurrence andtransience
Stock Modelwith JumpsJump Diffusion
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
3/49 ,
SomeElementson Lévy
Processes
OutlineLiterature
IntroductionBasic DefinitionsPoisson Processetc.
Basic Aspectson LévyProcessesFamousProcessesMain PropertiesExamples
Structure ofLévyProcessesJump ProcessDecompositionof a LévyProcess
Some SamplePathPropertiesRecurrence andtransience
Stock Modelwith JumpsJump Diffusion
Introduction
Introduction
4/49 ,
SomeElementson Lévy
Processes
OutlineLiterature
IntroductionBasic DefinitionsPoisson Processetc.
Basic Aspectson LévyProcessesFamousProcessesMain PropertiesExamples
Structure ofLévyProcessesJump ProcessDecompositionof a LévyProcess
Some SamplePathPropertiesRecurrence andtransience
Stock Modelwith JumpsJump Diffusion
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)
5/49 ,
SomeElementson Lévy
Processes
OutlineLiterature
IntroductionBasic DefinitionsPoisson Processetc.
Basic Aspectson LévyProcessesFamousProcessesMain PropertiesExamples
Structure ofLévyProcessesJump ProcessDecompositionof a LévyProcess
Some SamplePathPropertiesRecurrence andtransience
Stock Modelwith JumpsJump Diffusion
Introduction Basic Definitions
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).
6/49 ,
SomeElementson Lévy
Processes
OutlineLiterature
IntroductionBasic DefinitionsPoisson Processetc.
Basic Aspectson LévyProcessesFamousProcessesMain PropertiesExamples
Structure ofLévyProcessesJump ProcessDecompositionof a LévyProcess
Some SamplePathPropertiesRecurrence andtransience
Stock Modelwith JumpsJump Diffusion
Introduction Basic Definitions
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
7/49 ,
SomeElementson Lévy
Processes
OutlineLiterature
IntroductionBasic DefinitionsPoisson Processetc.
Basic Aspectson LévyProcessesFamousProcessesMain PropertiesExamples
Structure ofLévyProcessesJump ProcessDecompositionof a LévyProcess
Some SamplePathPropertiesRecurrence andtransience
Stock Modelwith JumpsJump Diffusion
Introduction Basic Definitions
Stopping Time
Examples:Hitting time of a one-sided boundary by a Brownian motion
τa := inft ≥ 0 : Wt = a
8/49 ,
SomeElementson Lévy
Processes
OutlineLiterature
IntroductionBasic DefinitionsPoisson Processetc.
Basic Aspectson LévyProcessesFamousProcessesMain PropertiesExamples
Structure ofLévyProcessesJump ProcessDecompositionof a LévyProcess
Some SamplePathPropertiesRecurrence andtransience
Stock Modelwith JumpsJump Diffusion
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
9/49 ,
SomeElementson Lévy
Processes
OutlineLiterature
IntroductionBasic DefinitionsPoisson Processetc.
Basic Aspectson LévyProcessesFamousProcessesMain PropertiesExamples
Structure ofLévyProcessesJump ProcessDecompositionof a LévyProcess
Some SamplePathPropertiesRecurrence andtransience
Stock Modelwith JumpsJump Diffusion
Introduction Poisson Process etc.
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)
10/49 ,
SomeElementson Lévy
Processes
OutlineLiterature
IntroductionBasic DefinitionsPoisson Processetc.
Basic Aspectson LévyProcessesFamousProcessesMain PropertiesExamples
Structure ofLévyProcessesJump ProcessDecompositionof a LévyProcess
Some SamplePathPropertiesRecurrence andtransience
Stock Modelwith JumpsJump Diffusion
Introduction Poisson Process etc.
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.
11/49 ,
SomeElementson Lévy
Processes
OutlineLiterature
IntroductionBasic DefinitionsPoisson Processetc.
Basic Aspectson LévyProcessesFamousProcessesMain PropertiesExamples
Structure ofLévyProcessesJump ProcessDecompositionof a LévyProcess
Some SamplePathPropertiesRecurrence andtransience
Stock Modelwith JumpsJump Diffusion
Introduction Poisson Process etc.
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).
12/49 ,
SomeElementson Lévy
Processes
OutlineLiterature
IntroductionBasic DefinitionsPoisson Processetc.
Basic Aspectson LévyProcessesFamousProcessesMain PropertiesExamples
Structure ofLévyProcessesJump ProcessDecompositionof a LévyProcess
Some SamplePathPropertiesRecurrence andtransience
Stock Modelwith JumpsJump Diffusion
Introduction Poisson Process etc.
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
is a martingale.13/49 ,
SomeElementson Lévy
Processes
OutlineLiterature
IntroductionBasic DefinitionsPoisson Processetc.
Basic Aspectson LévyProcessesFamousProcessesMain PropertiesExamples
Structure ofLévyProcessesJump ProcessDecompositionof a LévyProcess
Some SamplePathPropertiesRecurrence andtransience
Stock Modelwith JumpsJump Diffusion
Basic Aspects on Lévy Processes
Basic Aspects on LévyProcesses
14/49 ,
SomeElementson Lévy
Processes
OutlineLiterature
IntroductionBasic DefinitionsPoisson Processetc.
Basic Aspectson LévyProcessesFamousProcessesMain PropertiesExamples
Structure ofLévyProcessesJump ProcessDecompositionof a LévyProcess
Some SamplePathPropertiesRecurrence andtransience
Stock Modelwith JumpsJump Diffusion
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.
15/49 ,
SomeElementson Lévy
Processes
OutlineLiterature
IntroductionBasic DefinitionsPoisson Processetc.
Basic Aspectson LévyProcessesFamousProcessesMain PropertiesExamples
Structure ofLévyProcessesJump ProcessDecompositionof a LévyProcess
Some SamplePathPropertiesRecurrence andtransience
Stock Modelwith JumpsJump Diffusion
Basic Aspects on Lévy Processes Famous Processes
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.
16/49 ,
SomeElementson Lévy
Processes
OutlineLiterature
IntroductionBasic DefinitionsPoisson Processetc.
Basic Aspectson LévyProcessesFamousProcessesMain PropertiesExamples
Structure ofLévyProcessesJump ProcessDecompositionof a LévyProcess
Some SamplePathPropertiesRecurrence andtransience
Stock Modelwith JumpsJump Diffusion
Basic Aspects on Lévy Processes Famous Processes
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.
17/49 ,
SomeElementson Lévy
Processes
OutlineLiterature
IntroductionBasic DefinitionsPoisson Processetc.
Basic Aspectson LévyProcessesFamousProcessesMain PropertiesExamples
Structure ofLévyProcessesJump ProcessDecompositionof a LévyProcess
Some SamplePathPropertiesRecurrence andtransience
Stock Modelwith JumpsJump Diffusion
Basic Aspects on Lévy Processes Famous Processes
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.
18/49 ,
SomeElementson Lévy
Processes
OutlineLiterature
IntroductionBasic DefinitionsPoisson Processetc.
Basic Aspectson LévyProcessesFamousProcessesMain PropertiesExamples
Structure ofLévyProcessesJump ProcessDecompositionof a LévyProcess
Some SamplePathPropertiesRecurrence andtransience
Stock Modelwith JumpsJump Diffusion
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.
19/49 ,
SomeElementson Lévy
Processes
OutlineLiterature
IntroductionBasic DefinitionsPoisson Processetc.
Basic Aspectson LévyProcessesFamousProcessesMain PropertiesExamples
Structure ofLévyProcessesJump ProcessDecompositionof a LévyProcess
Some SamplePathPropertiesRecurrence andtransience
Stock Modelwith JumpsJump Diffusion
Basic Aspects on Lévy Processes Main Properties
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 .
20/49 ,
SomeElementson Lévy
Processes
OutlineLiterature
IntroductionBasic DefinitionsPoisson Processetc.
Basic Aspectson LévyProcessesFamousProcessesMain PropertiesExamples
Structure ofLévyProcessesJump ProcessDecompositionof a LévyProcess
Some SamplePathPropertiesRecurrence andtransience
Stock Modelwith JumpsJump Diffusion
Basic Aspects on Lévy Processes Main Properties
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.
21/49 ,
SomeElementson Lévy
Processes
OutlineLiterature
IntroductionBasic DefinitionsPoisson Processetc.
Basic Aspectson LévyProcessesFamousProcessesMain PropertiesExamples
Structure ofLévyProcessesJump ProcessDecompositionof a LévyProcess
Some SamplePathPropertiesRecurrence andtransience
Stock Modelwith JumpsJump Diffusion
Basic Aspects on Lévy Processes Main Properties
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.
22/49 ,
SomeElementson Lévy
Processes
OutlineLiterature
IntroductionBasic DefinitionsPoisson Processetc.
Basic Aspectson LévyProcessesFamousProcessesMain PropertiesExamples
Structure ofLévyProcessesJump ProcessDecompositionof a LévyProcess
Some SamplePathPropertiesRecurrence andtransience
Stock Modelwith JumpsJump Diffusion
Basic Aspects on Lévy Processes Main Properties
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)
23/49 ,
SomeElementson Lévy
Processes
OutlineLiterature
IntroductionBasic DefinitionsPoisson Processetc.
Basic Aspectson LévyProcessesFamousProcessesMain PropertiesExamples
Structure ofLévyProcessesJump ProcessDecompositionof a LévyProcess
Some SamplePathPropertiesRecurrence andtransience
Stock Modelwith JumpsJump Diffusion
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
24/49 ,
SomeElementson Lévy
Processes
OutlineLiterature
IntroductionBasic DefinitionsPoisson Processetc.
Basic Aspectson LévyProcessesFamousProcessesMain PropertiesExamples
Structure ofLévyProcessesJump ProcessDecompositionof a LévyProcess
Some SamplePathPropertiesRecurrence andtransience
Stock Modelwith JumpsJump Diffusion
Basic Aspects on Lévy Processes Examples
Normal Distribution
X ∼ N(0, 1):
f (x) =1√2π
e−x22
Characteristic function:
E[e iλX
]=
1√2π
∫ ∞−∞
e iλxe−x22 = exp
(−λ
2
2
)Lévy measure: Π(dx) = 0dxAssociated Lévy process: Standard Brownian motion
25/49 ,
SomeElementson Lévy
Processes
OutlineLiterature
IntroductionBasic DefinitionsPoisson Processetc.
Basic Aspectson LévyProcessesFamousProcessesMain PropertiesExamples
Structure ofLévyProcessesJump ProcessDecompositionof a LévyProcess
Some SamplePathPropertiesRecurrence andtransience
Stock Modelwith JumpsJump Diffusion
Basic Aspects on Lévy Processes Examples
Cauchy Distribution
X ∼ Cauchy:
f (x) =1
π(1 + x2)
Characteristic function:
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
26/49 ,
SomeElementson Lévy
Processes
OutlineLiterature
IntroductionBasic DefinitionsPoisson Processetc.
Basic Aspectson LévyProcessesFamousProcessesMain PropertiesExamples
Structure ofLévyProcessesJump ProcessDecompositionof a LévyProcess
Some SamplePathPropertiesRecurrence andtransience
Stock Modelwith JumpsJump Diffusion
Basic Aspects on Lévy Processes Examples
Gamma Distribution
X ∼ Γ(c , 1), c > 0:
f (x) =xc−1e−x
Γ(c)
Characteristic function:
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
27/49 ,
SomeElementson Lévy
Processes
OutlineLiterature
IntroductionBasic DefinitionsPoisson Processetc.
Basic Aspectson LévyProcessesFamousProcessesMain PropertiesExamples
Structure ofLévyProcessesJump ProcessDecompositionof a LévyProcess
Some SamplePathPropertiesRecurrence andtransience
Stock Modelwith JumpsJump Diffusion
Basic Aspects on Lévy Processes Examples
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)
Characteristic function:
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
28/49 ,
SomeElementson Lévy
Processes
OutlineLiterature
IntroductionBasic DefinitionsPoisson Processetc.
Basic Aspectson LévyProcessesFamousProcessesMain PropertiesExamples
Structure ofLévyProcessesJump ProcessDecompositionof a LévyProcess
Some SamplePathPropertiesRecurrence andtransience
Stock Modelwith JumpsJump Diffusion
Structure of Lévy Processes
Structure of Lévy Processes
29/49 ,
SomeElementson Lévy
Processes
OutlineLiterature
IntroductionBasic DefinitionsPoisson Processetc.
Basic Aspectson LévyProcessesFamousProcessesMain PropertiesExamples
Structure ofLévyProcessesJump ProcessDecompositionof a LévyProcess
Some SamplePathPropertiesRecurrence andtransience
Stock Modelwith JumpsJump Diffusion
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.
30/49 ,
SomeElementson Lévy
Processes
OutlineLiterature
IntroductionBasic DefinitionsPoisson Processetc.
Basic Aspectson LévyProcessesFamousProcessesMain PropertiesExamples
Structure ofLévyProcessesJump ProcessDecompositionof a LévyProcess
Some SamplePathPropertiesRecurrence andtransience
Stock Modelwith JumpsJump Diffusion
Structure of Lévy Processes Jump Process
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)
31/49 ,
SomeElementson Lévy
Processes
OutlineLiterature
IntroductionBasic DefinitionsPoisson Processetc.
Basic Aspectson LévyProcessesFamousProcessesMain PropertiesExamples
Structure ofLévyProcessesJump ProcessDecompositionof a LévyProcess
Some SamplePathPropertiesRecurrence andtransience
Stock Modelwith JumpsJump Diffusion
Structure of Lévy Processes Jump Process
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.
32/49 ,
SomeElementson Lévy
Processes
OutlineLiterature
IntroductionBasic DefinitionsPoisson Processetc.
Basic Aspectson LévyProcessesFamousProcessesMain PropertiesExamples
Structure ofLévyProcessesJump ProcessDecompositionof a LévyProcess
Some SamplePathPropertiesRecurrence andtransience
Stock Modelwith JumpsJump Diffusion
Structure of Lévy Processes Jump Process
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.
33/49 ,
SomeElementson Lévy
Processes
OutlineLiterature
IntroductionBasic DefinitionsPoisson Processetc.
Basic Aspectson LévyProcessesFamousProcessesMain PropertiesExamples
Structure ofLévyProcessesJump ProcessDecompositionof a LévyProcess
Some SamplePathPropertiesRecurrence andtransience
Stock Modelwith JumpsJump Diffusion
Structure of Lévy Processes Jump Process
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.
34/49 ,
SomeElementson Lévy
Processes
OutlineLiterature
IntroductionBasic DefinitionsPoisson Processetc.
Basic Aspectson LévyProcessesFamousProcessesMain PropertiesExamples
Structure ofLévyProcessesJump ProcessDecompositionof a LévyProcess
Some SamplePathPropertiesRecurrence andtransience
Stock Modelwith JumpsJump Diffusion
Structure of Lévy Processes Jump Process
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.
35/49 ,
SomeElementson Lévy
Processes
OutlineLiterature
IntroductionBasic DefinitionsPoisson Processetc.
Basic Aspectson LévyProcessesFamousProcessesMain PropertiesExamples
Structure ofLévyProcessesJump ProcessDecompositionof a LévyProcess
Some SamplePathPropertiesRecurrence andtransience
Stock Modelwith JumpsJump Diffusion
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.
36/49 ,
SomeElementson Lévy
Processes
OutlineLiterature
IntroductionBasic DefinitionsPoisson Processetc.
Basic Aspectson LévyProcessesFamousProcessesMain PropertiesExamples
Structure ofLévyProcessesJump ProcessDecompositionof a LévyProcess
Some SamplePathPropertiesRecurrence andtransience
Stock Modelwith JumpsJump Diffusion
Structure of Lévy Processes Decomposition of a Lévy Process
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)
37/49 ,
SomeElementson Lévy
Processes
OutlineLiterature
IntroductionBasic DefinitionsPoisson Processetc.
Basic Aspectson LévyProcessesFamousProcessesMain PropertiesExamples
Structure ofLévyProcessesJump ProcessDecompositionof a LévyProcess
Some SamplePathPropertiesRecurrence andtransience
Stock Modelwith JumpsJump Diffusion
Structure of Lévy Processes Decomposition of a Lévy Process
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)
38/49 ,
SomeElementson Lévy
Processes
OutlineLiterature
IntroductionBasic DefinitionsPoisson Processetc.
Basic Aspectson LévyProcessesFamousProcessesMain PropertiesExamples
Structure ofLévyProcessesJump ProcessDecompositionof a LévyProcess
Some SamplePathPropertiesRecurrence andtransience
Stock Modelwith JumpsJump Diffusion
Structure of Lévy Processes Decomposition of a Lévy Process
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).
39/49 ,
SomeElementson Lévy
Processes
OutlineLiterature
IntroductionBasic DefinitionsPoisson Processetc.
Basic Aspectson LévyProcessesFamousProcessesMain PropertiesExamples
Structure ofLévyProcessesJump ProcessDecompositionof a LévyProcess
Some SamplePathPropertiesRecurrence andtransience
Stock Modelwith JumpsJump Diffusion
Structure of Lévy Processes Decomposition of a Lévy Process
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).
40/49 ,
SomeElementson Lévy
Processes
OutlineLiterature
IntroductionBasic DefinitionsPoisson Processetc.
Basic Aspectson LévyProcessesFamousProcessesMain PropertiesExamples
Structure ofLévyProcessesJump ProcessDecompositionof a LévyProcess
Some SamplePathPropertiesRecurrence andtransience
Stock Modelwith JumpsJump Diffusion
Structure of Lévy Processes Decomposition of a Lévy Process
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.
41/49 ,
SomeElementson Lévy
Processes
OutlineLiterature
IntroductionBasic DefinitionsPoisson Processetc.
Basic Aspectson LévyProcessesFamousProcessesMain PropertiesExamples
Structure ofLévyProcessesJump ProcessDecompositionof a LévyProcess
Some SamplePathPropertiesRecurrence andtransience
Stock Modelwith JumpsJump Diffusion
Structure of Lévy Processes Decomposition of a Lévy Process
Jump Diffusion Process
+ +
=
42/49 ,
SomeElementson Lévy
Processes
OutlineLiterature
IntroductionBasic DefinitionsPoisson Processetc.
Basic Aspectson LévyProcessesFamousProcessesMain PropertiesExamples
Structure ofLévyProcessesJump ProcessDecompositionof a LévyProcess
Some SamplePathPropertiesRecurrence andtransience
Stock Modelwith JumpsJump Diffusion
Some Sample Path Properties
Some Sample Path Properties
43/49 ,
SomeElementson Lévy
Processes
OutlineLiterature
IntroductionBasic DefinitionsPoisson Processetc.
Basic Aspectson LévyProcessesFamousProcessesMain PropertiesExamples
Structure ofLévyProcessesJump ProcessDecompositionof a LévyProcess
Some SamplePathPropertiesRecurrence andtransience
Stock Modelwith JumpsJump Diffusion
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
)44/49 ,
SomeElementson Lévy
Processes
OutlineLiterature
IntroductionBasic DefinitionsPoisson Processetc.
Basic Aspectson LévyProcessesFamousProcessesMain PropertiesExamples
Structure ofLévyProcessesJump ProcessDecompositionof a LévyProcess
Some SamplePathPropertiesRecurrence andtransience
Stock Modelwith JumpsJump Diffusion
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.
45/49 ,
SomeElementson Lévy
Processes
OutlineLiterature
IntroductionBasic DefinitionsPoisson Processetc.
Basic Aspectson LévyProcessesFamousProcessesMain PropertiesExamples
Structure ofLévyProcessesJump ProcessDecompositionof a LévyProcess
Some SamplePathPropertiesRecurrence andtransience
Stock Modelwith JumpsJump Diffusion
Stock Model with Jumps
Stock Model with Jumps
46/49 ,
SomeElementson Lévy
Processes
OutlineLiterature
IntroductionBasic DefinitionsPoisson Processetc.
Basic Aspectson LévyProcessesFamousProcessesMain PropertiesExamples
Structure ofLévyProcessesJump ProcessDecompositionof a LévyProcess
Some SamplePathPropertiesRecurrence andtransience
Stock Modelwith JumpsJump Diffusion
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.
47/49 ,
SomeElementson Lévy
Processes
OutlineLiterature
IntroductionBasic DefinitionsPoisson Processetc.
Basic Aspectson LévyProcessesFamousProcessesMain PropertiesExamples
Structure ofLévyProcessesJump ProcessDecompositionof a LévyProcess
Some SamplePathPropertiesRecurrence andtransience
Stock Modelwith JumpsJump Diffusion
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”
48/49 ,
SomeElementson Lévy
Processes
OutlineLiterature
IntroductionBasic DefinitionsPoisson Processetc.
Basic Aspectson LévyProcessesFamousProcessesMain PropertiesExamples
Structure ofLévyProcessesJump ProcessDecompositionof a LévyProcess
Some SamplePathPropertiesRecurrence andtransience
Stock Modelwith JumpsJump Diffusion
49/49 ,