Lvy Processes

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Lévy Processes

• David Suda

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Lévy Processes an introduction

• processes with independent stationary increments
• they are the analogues of random walks in
  continuous time
• they form special subclasses of both
  semimartingales and Markov processes
• they are the simplest examples of processes whose
  sample paths are càdlag functions
• typical examples of Lévy processes are the Wiener
  process, Poisson process, compound Poisson process

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Lévy Processes an introduction

• called Lévy processes in honour of the French
  mathematician Paul Lévy (1886-1971)
• Paul Lévy major work related to Lévy processes is
  his 1934 paper Sur les integrales dont les
  elements sont des variables aléatoires
  independentes

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Lévy Processes an introduction

• major contributions to theory on Lévy processes
  by Alexander Khintchine (Russia) and Kiyosi Ito
  (Japan)
• Khintchine, A. (1937) A new derivation of a
  formula by Paul Lévy
• Ito, K. (1942) On stochastic processes
  (Infinitely divisible laws of probability)
• the first time Lévy processes entered financial
  econometrics was in 1963 when Mandelbrot proposed
  them as models for cotton prices
• they have also found their applicability in
  quantum field theory

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Lévy Processes a definition

• A càdlàg stochastic process on
  with values in such that
  is called a Lévy process if it
  possesses the following properties
• for every increasing sequence of times
  , the random variables
  are independent
  (independent increments)
• the law of does not depend on
  (stationary increments)
• for all ,
  (stochastic
  continuity)

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Lévy Processes infinite divisibility

• A probability distribution F on is said to
  be infinitely divisible if for any integer
  , there exists independent identically
  distributed random variables such
  that has distribution F.
• Examples of infinitely divisible distributions
  Gaussian, gamma, -stable distributions,
  lognormal, Pareto, Student distributions

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Lévy Processes infinite divisibility

• Proposition (Infinite divisibility and Lévy
  processes) Given a Lévy process , then
  for every , has an infinitely divisible
  distribution. Conversely, if F is an infinitely
  divisible distribution, then there exists a Lévy
  process such that the distribution of
  is given by F.
• Examples
• Gaussian distribution ? Wiener Process
• Poisson distribution ? Poisson Process

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Lévy Processes characteristic exponent

• Let be a random variable on
  taking values in and governed by
  probability measure
• . Its characteristic function
  is defined by

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Lévy Processes characteristic exponent

• Proposition (Characteristic function of a Lévy
  process) Let be a Lévy process on
  . There exists a continuous function
  called the characteristic exponent of
  such that

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Lévy Processes characteristic exponent

• Examples
• standard Brownian motion
• Gaussian process (Brownian motion with drift)
• Poisson process

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Examples of Lévy Processes Brownian Motion
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Examples of Lévy Processes Poisson Process
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Examples of Lévy Processes Compound Poisson
Process

• A compound Poisson process with intensity
  and jump size distribution f is a stochastic
  process defined as where
  jumps sizes are independent
  identically distributed with distribution f and
  is a Poisson process with intensity
  , independent from .
• The Poisson process is a special case of the
  compound Poisson process, with a fixed jump size
  1.

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Examples of Lévy Processes Compound Poisson
Process
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Examples of Lévy Processes Compound Poisson
Process

• Properties of the Compound Poisson Process
• is a compound Poisson process if and
  only if it is a Lévy process and its sample paths
  are piecewise constant functions
• if is a compound Poisson process on
  then its characteristic exponent is
  given by
• for all , where denotes the
  jump intensity and f is the jump distribution.

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Jump Processes

• The jump process , defined by
  
  
  
• 
  for each .
• Examples
• the jump process of a Poisson process takes
  values in 0,1
• the jump process of a compound Poisson process
  can take any value in
• in the case of Brownian motion, the jump process
  for all

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Jump Processes

• Proposition If is a Lévy process then,
  for fixed ,
  
• almost surely.
• Let be a Lévy process on . Then
  the measure on defined by
• is called the Lévy measure of .
  Hence, is the expected number per unit
  time of jumps whose size belongs to
• .
• jump measure we denote by
  the number of jump times of
  between t1 and t2

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Jump Processes

• If is a compound Poisson process
  with intensity
• and jump size distribution f, then is
  a Poisson random
• measure on with intensity
  measure
• that is (amongst other conditions)

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A General Representation for Lévy Processes

• Can every Lévy process be represented in the
  form
• where is a Poisson random measure?
• is finite for any compact set that
  does not
• contain zero, otherwise the càdlàg property
  would be
• contradicated, but in general it is not
  necessarily a
• finite measure and this restriction allows it
  to blow
• up at zero. Hence the reason for the following
  
• decomposition.

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Lévy-Itô Decomposition

• Theorem (Lévy-Itô decomposition) Let
  be a Lévy process on and its Lévy
  measure then
• is a Radon measure on and
  verifies
• the jump measure is a Poisson random
  measure on
• with intensity measure
• there exists a vector and a d-dimensional Wiener
  process with covariance matrix
  such that

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Lévy-Itô Decomposition
The terms in the above decomposition are
independent and the convergence in the last term
is almost sure and uniform in on .
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Lévy-Itô Decomposition

• The first two terms of this decomposition are the
  continuous part of the Lévy process, consisting
  of a linear drift and a Wiener process.
• The last two terms are discontinuous processes,
  incorporating the jumps of the Lévy process.
  These last two terms are, respectively, a
  compound poisson process and a square integrable
  pure jump martingale with an almost surely finite
  number of jumps on each finite time interval of
  magnitude less than 1 .
• The triplet is called the
  characteristic triplet of the Lévy process

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Lévy-Itô Decomposition

• There is nothing special about the threshold
  .
• The reason why we integrate by a compensated
• Poisson process in the last term is that
  may
• have a singularity at zero and there can be
  infinitely
• many small jumps, meaning that the compound
• Poisson component would not converge.
• This last term does not affect the case when
  
• does not have a singularity at 0.

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Lévy-Khintchine Representation

• Let be a Lévy process on
  with characteristic triplet . Then

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Special Types of Lévy Processes

• Stable Lévy Processes
• is said to be a stable random variable if
  there exists real valued sequences (cn)n?? and
  (dn)n?? with each cngt0 such that
• where are independent
  copies of
• . In particular is said to be
  strictly stable if dn0 for all n.

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Special Types of Lévy Processes

• Stable Lévy Processes
• A stable Lévy process is a Lévy process
  such that each is a
  stable random variable.
• Since in stable Lévy Processes, is a
  stable random variable then the distribution of
  is the same as the distribution of its
  increments.

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Special Types of Lévy Processes

• Stable Lévy Processes
• Examples of stable distributions
• Gaussian distribution (?2)
• Cauchy distribution (?1)
• Lévy distribution (?0.5)

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Special Types of Lévy Processes

• Stable Lévy Processes
• Stable Lévy processes have a Lévy measure of the
  form
• The constant?? is called the index of stability,
  and stable distributions with index ? are also
  referred to as ?-stable distributions.

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Special Types of Lévy Processes

• Subordinators
• A Lévy process is said to be a
  subordinator if it is one-dimensional and
  non-decreasing almost surely.
• If is a subordinator then its Lévy
  symbol takes the form
• where ?0 and the Lévy measure ? satisfy the
  additional requirements ?(-?,0)0 and

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Special Types of Lévy Processes

• Examples of subordinators
• Poisson processes
• compound Poisson processes with ? valued jumps
  (e.g. compound Poisson process with exponential
  jumps)

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Pathwise properties of Lévy Processes

• Compound Poisson processes have piecewise
  constant trajectories.
• A Lévy process has paths of finite variation, and
  is therefore a finite variation Lévy process, if
  and
• .
• If the Lévy process is a subordinator (an
  increasing Lévy process) then its sample paths
  are almost surely non-decreasing.
• In general, paths of Lévy processes are càdlàg
  functions.

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Modification of a Lévy Process

• Let and be stochastic
  processes defined on the same probability space.
  Then is said to be a modification of
  if for each
• If is a modification of a Lévy process
  then is a Lévy process with the
  same characteristics as .

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Modification of a Lévy Process

• Every Lévy process has a càdlag modification that
  is itself a Lévy process.
• If a Lévy process has càdlàg paths, then it has
  an augmented natural filtration which is right
  continuous.

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Distributional Properties of Lévy Processes

• The distribution associated with a Lévy process
  is always infinitely divisible and has a
  characteristic function of the previously
  described form, but does not always have a
  continuous density (e.g. if it is a compound
  Poisson process as there is an atom at zero for
  all t)
• The tail behaviour of a Lévy process and its
  moments are determined by the Lévy measure.

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Markov Property of Lévy Processes

• Lévy processes have the strong Markov property
  and are hence, themselves, Markov processes (more
  specifically Feller processes).
• Lévy processes are Feller processes with
  translation invariant semigroups, that is
• where for
  .

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Markov Property of Lévy Processes

• Let be a Lévy process with Lévy
  characteristic exponent and
  characteristics
• . Then the associated Feller semigroup
• is given by

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Markov Property of Lévy Processes

• If is a Lévy process on with
  characteristic triplet . Then the
  infinitesimal generator of is defined
  for any by

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Lévy Processes and Semimartingale Theory

• By the Lévy-Itô decomposition, all Lévy processes
  are semimartingales because a Lévy process can be
  split into a sum of a square integrable
  martingale and a finite variation process.
• If is a Lévy process and, for some
  ,
  for all t, then the process
  
• is a martingale process.

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Lévy Processes and Semimartingale Theory

• If is a Lévy process with Lévy
  symbol ?? then, for each u??d, then
• is a complex-valued martingale.

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Stochastic Calculus based on Lévy Processes

• Examples
• Brownian stochastic integrals
• Poisson stochastic integrals

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Stochastic Calculus based on Lévy Processes

• A Lévy-type stochastic integral can be written in
  the form
• Weiner-Lévy integral

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Stochastic Calculus based on Lévy Processes

• Itôs formula for Lévy-type stochastic integral

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Stochastic Calculus based on Lévy Processes

• A generalization of time-homogenous Brownian
  motion driven stochastic differential equations
  are stochastic differential equations which have
  an additional independent Poisson random measure.
• See Applebaum (2004) for further details.

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References

• Applebaum, D. (2004) Lévy Processes and
  Stochastic Calculus
• Bertoin, J. (1996), Lévy Processes
• Cont, R. and Tankov, P. (2004) Financial
  Modelling With Jump Processes