Introduction to L ́evy processes

8/13/2019 Introduction to L evy processes

1/26

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

1


2/26

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

4

0

-4

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 .

2


3/26

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).

3


4/26

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).

6


7/26

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 .

1680

4

0

-464320

64

0

-642561280

1024

0

-1024

One can show that ( X ct ) t0 (c1 / X t ) t0 (scaling).7


8/26

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


9/26

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.

9


10/26

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.

2561280

16

0

-162561280

1024

0

-10241680

8

4

0

10


11/26

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).

11


12/26

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


14/26

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

14


15/26

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.

15


16/26

16


17/26

17


18/26

= 5 = 0 = 218


19/26

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


20/26

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


21/26

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

40

20

0

-20

-40

-60

543210

1

0

-1

zoom

543210

40

20

0

-20

-40

-60

21


22/26

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 ) .

22


23/26

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.

23


24/26

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

151050

8

4

0

-4

eps=0.01

151050

4

0

-4

-8

eps=0.3

151050

4

0

-4

-8

eps=0.1

151050

4

0

-4

-8

24


25/26

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


26/26

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.

26

Documents

Introduction to L ́evy processes