Lvy Copula Habib Esmaeili

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Lévy Copula Habib Esmaeili

• 20 July 2006
• Munich University of Technology

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Overview 1- Lévy Process -definition,
infinite divisibility, Lévy-Khinchin
representation,
decomposition of Lévy process, Lévy measure 2-
Ordinary Copula -F-volume, d-increasing and
grounded function, marginal function,
copula, Sklars theorem, 3- Lévy
copula - Tail integral, margins, Lévy
copula, extension of Sklars theorem,
some example, Archimedean Lévy copula,
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• 1-Lévy Process
• A cad lag (right continuous and left limits)
  stochastic process on
  with values in such that
  is called a Lévy Process if it possesses the
  following properties
• 1-Independent increments
• 2-Stationary increments
• 3-Stochastic continuity

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Infinite Divisibility A probability distribution
F on is said to be infinitely divisible if
for any integer there exists
random variables such that
has distribution
F. Proposition Let be a Lévy
Process. Then for every t, 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.
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Lévy-Khinchin representation Let be
a Lévy process on . Its characteristic
function can be represented as a positive
definite matrix (covariance matrix of Brownian
Motion) A positive measure (Lévy measure
which describes the jumps of ) a vector
(the drift parameter of BM)
means that X has a finite
number of jumps.
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The triplet is called characteristic
triplet. One can decompose a Lévy process in a
unique way into a sum of two independent
components - a continuous Gaussian process
with the drift and covariance matrix of
Brownian motion denoted by . - a Lévy motion
part without a Gaussian component. The most
important properties of the second part are
determined by the Lévy measure . If the Lévy
measure is finite, then the levy process has only
finitely many jumps in any interval of a finite
length. (Compound Poisson Process)
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Lévy measure Let be a Lévy process on
. The measure for a set
is the expected number, per unit time, of jumps
whose size belongs to .
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2- Ordinary Copula F-volume Example
(two-dimensional case)
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D-increasing function Grounded
function (one dimensional) Marginal function
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Copula An d-dimensional copula is a function C
with domain Such that - C is grounded
and d-increasing. - C has margins
, which satisfy for all u in
0,1. Sklars Theorem Let F be an
d-dimensional distribution function with margins
. Then there exists an
d-dimensional Copula C such that for all
,
Conversely,.
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The role of Copula is twofold 1- gives a
complete characterization of a possible
dependence structure of a random vector given
the margins 2- construct a
multidimensional distribution
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3- Lévy Copula Question Can Copulas define the
dependence structure in dynamic
multidimensional models? Many authors who
successfully used one dimensional Lévy process
with jumps, continued applications of
multidimensional Lévy process that dominated by
BM. Also a few people attempted to parameterize
the dependence structure of multidimensional Lévy
process by the Copula . This method has some
drawbacks 1- The copula may depend on t,
and usually cannot be
computed from . 2- For infinitely
divisible margins, it is unclear which
copula yield a 2-dimensinal infinitely divisible
law.
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3- It would be inconvenient to model dependence
using the Copula of probability
distribution, whereas the laws of
components are usually specified via their
Lévy measures. Example Let
be a Lévy process with we can construct
from the jumps of as From the dynamic
point of view and are completely
dependent, but the copula of and is
not that of a complete dependence because is
not a deterministic function of .
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Key Idea 1- For Lévy process, dependence of
jumps should be studied using the Lévy measure
(and no probability measure), because the
knowledge of jumps and jump dependence is in the
Lévy measure. 2- The Lévy measure for Lévy
process plays the same role as the probability
measure for random Variables. 3- To model the
dependence, we must construct Copulas for Lévy
measures. Principal difference - Lévy
measures are not necessarily finite and they may
have a non-singularity at zero.
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Tail Integral A d-dimensional tail integral is
a function
, such that 1- is a
d-increasing function. 2- U is equal to zero if
at least one of its arguments is equal
to . 3- U is finite everywhere except at
zero and Margins of a tail
integral
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A tail integral for a Lévy measure is defined
as Lévy Copula for process with positive jumps
(S-copula) A d-dimensional Lévy Copula is a
d-increasing and grounded function
with margins
, which satisfy for all u
in . Extension of Sklars Theorem Let
U be a d-dimensional tail integral with margins
. There exists a Lévy Copula F
such that If the margins are continuous on
, this Lévy Copula
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is unique. Otherwise, its values are uniquely
determined on .
Conversely, if F is a Lévy Copula and
are one dimensional tail integrals,
then U defines a d-dimensional tail integral. -
Tail integral of a Lévy process is
left-continuous in each variable. It integrates
near 0 Reconstruction Lévy measure from
their tail integral Corollary Let U be a
d-dimensional tail integral, left continuous in
each variable and satisfying the above
properties. Then there exists a unique Lévy
measure v
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on such that U is the tail integral of
v. Marginal Lévy measure The result of the
theorem 1- all types of the dependence of Lévy
process (with only positive jumps), including
complete and incomplete dependence can be
represented by Lévy copulas. 2- one can
construct multivariate Lévy process models by
specifying separately jump dependence structure
and one-dimensional laws.
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Example of positive Lévy Copula 1-
Independence Let be a
d-dimensinal Lévy process (with only
positive jumps), its components
are independent if and only if its Lévy
measure is The tail
integral of this Lévy measure is its
Lévy Copula has the form
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2- Complete dependence Increasing set A
subset is called increasing if for
every two vectors and
either or . A
d-dimensional Lévy process is called completely
dependence or comonotonic if there exists an
increasing subset S of such that every
jump is in S. - Let is a
Lévy process with increasing jumps, whose Lévy
measure is supported by a non-decreasing set S.
Then Lévy Copula of is the complete
dependence Lévy Copula given by
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Construction of Lévy Copula (Archimedean Lévy
Copula) Let be strictly decreasing continuous
function from to such that
, and has
derivatives up to the order d on with
. Then defines a d-dimensional
positive Lévy Copula. Example Clayton- Lévy
Copula
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The family of Clayton- Lévy Copula is too
important, because it concludes as limits for
the complete dependence and for
the independence Lévy Copula. A paper on the
application of Clayton- Lévy Copula has been
written by Bregman Kluppelberg for using to
estimate ruin probability in multivariate models.

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Jan Kallsen Peter Tankov have recently
extended positive Lévy Copula to general Lévy
Copula. This function can defined dependence
structure for general Lévy processes. Definition
A function is called Lévy
Copula if
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integral for general Lévy processes Tail
(one-dimensional Case) Let v be a Lévy measure
on R. The tail integral of v is a function
defined by
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References 1- Tankov, Peter (2003) Dependence
structure of spectrally positive multidimensional
Levy processes. Available at www.cmap.polytechniq
ue.fr 2- Cont, Rama and Tankov, Peter (2004)
Financial Modelling with Jump processes Chapman
Hall. 3- Kyprianou, Andreas (2005) Introductory
Lectures on Fluctuations of Lévy processes with
application , Springer. 4- Kallsen, Jan
Tankov, Peter (2006) Characterization of
dependence of multidimensional Lévy processes
Using Lévy Copulas, To appear in Journal of
Multivariate Analysis. Available at
www.mathfinance.ma.tum.de/personen/kallsen.php
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5- Bregman, Y. Kluppelberg, C. (2005) Ruin
estimation in Multivariate models with Clayton
dependence structure. Available at
www-m4.ma.tum.de/Papers/