introduction to l ́evy processes
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Introduction to Levy processes
Graduate lecture 22 January 2004Matthias Winkel
Departmental lecturer
(Institute of Actuaries and Aon lecturer in Statistics)
1. Random walks and continuous-time limits
2. Examples
3. Classication and construction of Levy processes
4. Examples
5. Poisson point processes and simulation
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1. Random walks and continuous-time limits
Denition 1 Let Y k
, k
1, be i.i.d. Then
S n =n
k=1Y k , n N ,
is called a random walk . 1680
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Random walks have stationary and independent increments
Y k = S k S k1 , k 1 .Stationarity means the Y
k have identical distribution.
Denition 2 A right-continuous process X t , t R + , withstationary independent increments is called Levy process .
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What are S n , n 0, and X t , t 0?
Stochastic processes ; mathematical objects, well-dened,with many nice properties that can be studied.
If you dont like this, think of a model for a stock price
evolving with time. There are also many other applications .
If you worry about negative values, think of logs of prices.
What does Denition 2 mean?
Increments X tk X tk1 , k = 1 , . . . , n , are independent andX tk X tk1 X tktk1 , k = 1 , . . . , n for all 0 = t0 < . . . < t n .Right-continuity refers to the sample paths (realisations).
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Can we obtain Levy processes from random walks?
What happens e.g. if we let the time unit tend to zero, i.e.
take a more and more remote look at our random walk ?
If we focus at a xed time, 1 say, and speed up the process
so as to make n steps per time unit , we know what happens,the answer is given by the Central Limit Theorem:
Theorem 1 (Lindeberg-Levy) If 2 = V ar ( Y 1 ) 1, then (under a weak regu-larity condition)
S nn 1 / ( n ) Z in distribution, as n .for a slowly varying . Z has a so-called stable distribution .
Think 1. The family of stable distributions has threeparameters, (0 , 2], c R + , E ( |Z | ) < < (or = 2).
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Theorem 4 If = sup { 0 : E ( |Y 1 | ) < } (0 , 2] and E ( Y 1 ) = 0 for > 1, then (under a weak regularity cond.)
X ( n )t = S [nt ]n 1 / ( n ) X t in distribution, as n .
Also, X ( n ) X where X is a so-called stable Levy process .
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One can show that ( X ct ) t0 (c1 / X t ) t0 (scaling).7
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To get to full generality, we need triangular arrays .
Theorem 5 (Khintchine) Let Y ( n )k
, k = 1 , . . . , n , be i.i.d.with distribution changing with n 1, and such that
Y ( n )1 0 , in probability, as n .
If S ( n )n Z in distribution,then Z has a so-called innitely divisible distribution .
Theorem 6 (Skorohod) In Theorem 5, k 1, n 1,X ( n )t = S
( n )[nt ] X t in distribution, as n .
Furthermore, X ( n ) X where X is a Levy process .8
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Do we really want to study Levy processes as limits?
No! Not usually. But a few observations can be made:Brownian motion is a very important Levy process.
Jumps seem to be arising naturally (in the stable case).We seem to be restricting the increment distributions to:
Denition 3 A r.v. Z has an innitely divisible distributionif Z = S ( n )n for all n 1 and suitable random walks S ( n ) .Example for Theorems 5 and 6: B ( n, p n )
P oi ( ) for
np n is a special case of Theorem 5 ( Y ( n )k B (1 , pn )),
and Theorem 6 turns out to give an approximation of the
Poisson process by Bernoulli random walks.
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2. Examples
Example 1 (Brownian motion) X t
N (0 , t ).
Example 2 (Stable process) X t stable.
Example 3 (Poisson process) X t P oi ( t/ 2)To emphasise the presence of jumps , we remove the verti-cal lines.
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3. Classication and construction of Levy proc.
Why restrict to innitely divisible increment distrib.?
You have restrictions for random walks : nstep incrementsS n must be divisible into n iid increments, for all n 2.
For a Levy process, X t , t > 0, must be divisible into n 2iid random variables
X t =
n
k=1 ( X tk/n X t ( k1) /n ) ,since these are successive increments (hence independent)
of equal length (hence identically distrib., by stationarity).
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Approximations are cumbersome. Often nicer direct argu-
ments exist in continuous time , e.g., because the class of
innitely divisible distributions can be parametrized nicely.
Theorem 7 (Levy-Khintchine) A r.v. Z is innitely divis-ible iff its characteristic function E ( eiZ ) is of the form
exp i 12 2 2 + Reix 1 ix 1{|x|1} ( dx )where R , 2 0 and is a measure on R such that
R(1
x2 ) ( dx ) 1}
and M is a martingale with jumps M t = X t 1{| X s |1}.13
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4. Examples
Z 0 E ( eZ ) = exp (0 ,
)1 ex ( dx )
Example 4 (Poisson proc.) = 0, ( dx ) = 1 ( dx ).
Example 5 (Gamma process) = 0, ( x) = f ( x) dx withf ( x) = ax 1 exp
{bx
}, x > 0. Then X t
Gamma ( at,b ).
Example 6 (stable subordinator) = 0, f ( x) = cx3 / 2
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Example 7 (Compound Poisson process) = 2 = 0.
Choose jump distribution , e.g. density g( x), intensity > 0,
( dx ) = g ( x) dx . J in Theorem 8 is compound Poisson.
Example 8 (Brownian motion) = 0, 2 > 0, 0.Example 9 (Cauchy process) = 2 = 0, ( dx ) = f ( x) dx
with f ( x) = x2 , x > 0, f ( x) = |x|2 , x < 0.
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= 5 = 0 = 218
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5. Poisson point processes and simulation
What does it mean that ( X t ) t0 is a Poisson point pro-cess with intensity measure ?
Denition 4 A stochastic process ( H t ) t0 in a measurablespace E = E {0} is called a Poisson point process withintensity measure on E if
N t ( A) = # {s t : H s A}, t 0 , A E measurablesatises
N t ( A), t 0, is a Poisson process with intensity ( A) . For A1 , . . . , A n disjoint, the processes N t ( A1 ) , . . . , N t ( An ),
t 0, are independent .19
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N t ( A), t 0, counts the number of points in A, but doesnot tell where in A they are. Their distribution on A is :
Theorem 9 (It o) For all measurable A E with M = ( A) < , denote the jump times of N t ( A) , t 0, by
T n ( A) = inf
{t
0 : N t ( A) = n
}, n
1 .
ThenZ n ( A) = H T n ( A) , n 1 ,
are independent of N t ( A) , t
0, and iid with common
distribution M 1 ( A) .
This is useful to simulate H t , t 0.20
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Simulation of ( X t ) t0Time interval [0 , 5]
> 0 small, M = (( , ) c) < N P oi (5 (( , ) c))T 1 , . . . , T N U (0 , 5) X T j M 1 ( (, ) c) 543210
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Corrections to improve approximation
First, add independent Brownian motion t + B t .If X symmetric, small means small error.If X not symmetric , we need a drift correction t
=
A
[
1 ,1]
x ( dx ) , where A = (
, ) c.
Also, one can often add a Normal variance correction C t
2 = (, ) x2 ( dx ) , C independent Brownian motion
to account for the many small jumps thrown away. Thiscan be justied by a version of Donskers theorem and
E ( X 1 ) = + [1 ,1] c x ( dx ) , V ar ( X 1 ) = 2 + Rx2 ( dx ) .
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Decomposing a general Levy process X
Start with intervals of jump sizes in sets An
with
n1An = (0 , ) , An = An , n1
An = ( , 0)
so that ( An + A
n )
1, say. We construct X ( n ) , n
Z,
independent Levy processes with jumps in An according to
( An ) (and drift correction n t ). NowX t =
nZX ( n )t , where X
(0)t = t + B t .
In practice, you may cut off the series when X ( n )t is small,
and possibly estimate the remainder by a Brownian motion.
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A spectrally negative Levy process: ((0 , )) = 02 = 0,
( dx ) = f ( x) dx ,
f ( x) = |x|5 / 2 ,x [3 , 0), = 0 .845
1 = 0,
0 .3 = 1 .651,
0 .1 = 4 .325, 0 .01 = 18,
as 0.
eps=1
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Another look at the Levy-Khintchine formula
E ( eiX 1 ) = E exp in
Z
X ( n )1 =
nZE ( eiX
( n )1 )
can be calculated explicitly, for n Z (n = 0 obvious)E ( eiX
( n )1 ) = E exp i n +
N
k=1
Z k
= exp i An[1 ,1] x ( dx ) m =0
P ( N = m ) E ( eiZ 1 )m
= exp An eix 1ix 1{|x|1} ( dx )to give for E ( eiX 1 ) the Levy-Khintchine formula
exp i 12
2 2 + Reix 1 ix 1{|x|1} ( dx )25
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SummaryWeve dened Levy processes via stationary independent
increments.
Weve seen how Brownian motion, stable processes and
Poisson processes arise as limits of random walks, indi-
cated more general results.Weve analysed the structure of general Levy processes and
given representations in terms of compound Poisson pro-
cesses and Brownian motion with drift.Weve simulated Levy processes from their marginal distri-
butions and from their Levy measure.
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