Levy process

Levy process

Let

(

Ω

,

Ψ

,

P

,

(

ℱ

)

0

≤

t

<

∞

)

be a

filtered
probability space

. A Lèvy process on that space is an

stochastic
process

L

:

[

0

,

∞

)

×

Ω

→

ℜ

n

that has the
following

properties

:

1.

L

has increments

independent

of the past: for any

t

≥

0

and for all

s

≥

0

,

L

t

+

s

-

L

t

in independent of

ℱ

t

2.

L

has

stationary increments

: if

t

≥

s

≥

0

then

L

t

-

L

s

and

L

t

-

s

have the same

distribution

. This particulary implies that

L

t

+

s

-

L

t

and

L

s

have the same distribution.

3.

L

is continous in

probability

: for any

t

,

s

∈

[

0

,

∞

)

,

lim

t

→

s

=

X

s

, the limit taken in probability.

Some important properties of any Lèvy processes

L

are:

1.

There exist a

modification

of

L

that has càdlàg paths a.s. (càdlàg paths means that the paths are

continuous

from the right and that the left limits exist for any

t

≥

0

).

2.

L

t

is an infinite divisible

random variable

for all

t

∈

[

0

,

∞

)

3.

Lèvy -Itô decomposition

:

L

can be written as the sum of a diffusion, a continuous

Martingale

and a pure jump process; i.e:

L

t

=

α

⁢

t

+

σ

⁢

B

t

+

∫

|

x

|

<

1

x

⁢

𝑑

N

~

t

⁢

(

⋅

,

d

⁢

x

)

+

∫

|

x

|

≥

1

x

⁢

𝑑

N

t

⁢

(

⋅

,

d

⁢

x

)

for all

t

≥

0

where

α

∈

ℜ

,

B

t

is a standard

brownian motion

.

N

is defined to be the Poisson random measure of the Lèvy process (the process that counts the jumps): for any Borel

A

in

ℜ

n

such that

0

∉

c

⁢

l

⁢

(

A

)

then

N

t

⁢

(

⋅

,

A

)

:=

∑

0

<

s

≤

t

1

A

⁢

(

Δ

⁢

L

s

)

, where

Δ

⁢

L

s

:=

L

s

-

L

s

-

; and

N

~

t

⁢

(

⋅

,

A

)

=

N

t

⁢

(

⋅

,

A

)

-

t

⁢

E

⁢

[

N

1

⁢

(

⋅

,

A

)

]

is the compensated jump process, which is a martingale.

4.

Lèvy -Khintchine formula

: from the previous property it can be shown that for any

t

≥

0

one has that

E

⁢

[

e

i

⁢

u

⁢

L

t

]

=

e

-

t

⁢

ψ

⁢

(

u

)

where

ψ

⁢

(

u

)

=

-

i

⁢

α

⁢

u

+

σ

2

2

⁢

u

2

+

∫

|

x

|

≥

1

(

1

-

e

i

⁢

u

⁢

x

)

⁢

𝑑

ν

⁢

(

x

)

+

∫

|

x

|

<

1

(

1

-

e

i

⁢

u

⁢

x

+

i

⁢

u

⁢

x

)

⁢

𝑑

ν

⁢

(

x

)

with

α

∈

ℜ

,

σ

∈

[

0

,

∞

)

and

ν

is a

positive

, borel,

σ

-finite

measure

called

Lèvy measure

. (Actually

ν

⁢

(

⋅

)

=

E

⁢

[

N

1

⁢

(

⋅

,

A

)

]

). The second formula is usually called the Lèvy exponent or Lèvy symbol of the process.

5.

L

is a

semimartingale

(in the classical sense of being a sum of a

finite variation process

and a

local martingale

), so it is a

good

integrator, in the

stochastic

sense.

Some important examples of Lèvy processes include: the

Poisson
Process

, the Compound Poisson process, Brownian Motion, Stable Processes,
Subordinators, etc.

Bibliography

•

Protter, Phillip (1992).

Stochastic Integration

and

Differential Equations

. A New Approach. Springer-Verlag, Berlin, Germany.

•

Applebaum David (2004). Lèvy Procesess and

Stochastic Calculus

. Cambridge University Press, Cambrigde, UK.

Title

Levy process

Canonical

name

LevyProcess1

Date of creation

2013-03-22 17:58:09

Last modified on

2013-03-22 17:58:09

Owner

juansba (18789)

Last modified by

juansba (18789)

Numerical id

11

Author

juansba (18789)

Entry type

Definition

Classification

msc 60G20

Generated on Sat Feb 10 11:58:59 2018 by

LaTeXML