Lévy Models - Computational Methods for Quantitative Finance - page 124

={

X t :

≥ }

Note that for an adapted, càdlàg stochastic process X t 0 starting in

X 0 = 0on (Ω, F , P , F ) (i) and (ii) imply (iii). We can associate to X ={ X t : t ∈

[ 0 ,T ]} a random measure J X on [ 0 ,T ]×R ,

X t = 0

J X (ω, · ) = 1 (t,X t ) ,

t ∈[ 0 ,T ]

which is called the jump measure . For any measurable subset B B)

counts then the number of jumps of X occurring between 0 and t whose amplitude
belongs to B . The intensity of J X is given by the Lévy measure.

⊂ R , J X ( [ 0 ,t ]×

Definition 10.1.2 Let X be a Lévy process. The measure ν on R defined by

),
is called the Lévy measure of X . ν(B) is the expected number, per unit time, of
jumps whose size belongs to B .
The Lévy measure satisfies
R

ν(B) = E[ # { t ∈[ 0 , 1 ]: X t = 0 ,X t ∈ B }] ,B ∈ B ( R

∧ z 2 ν( d z) < ∞

. Using the Lévy-Itô decomposi-
tion, we see that every Lévy process is uniquely defined by the drift γ , the variance
σ 2 and the Lévy measure ν . The triplet (σ 2 ,ν,γ) is the characteristic triplet of the
process X .

1

Theorem 10.1.3 (Lévy-Itô decomposition) Let X be a Lévy process and ν its Lévy
measure . Then , there exist a γ , σ and a standard Brownian motion W such that

t

X t = γt + σW t + xJ X ( d s, d x)

0 | x |≥ 1

t

+ lim

ε

x(J X ( d s, d x) − ν( d x) d s)

↓ 0

0 ε ≤| x |≤ 1

= γt + σW t + X s 1

{| X s |≥ 1 }

0 ≤ s ≤ t

t

x J X ( d s, d x), + lim

ε ↓

(10.1)

0

ε ≤| x |≤

0 1

where J X is the jump measure of X .

Proof See [143, Chap. 4].

The characteristic triplet could also be derived from the Lévy-Khinchine repre-
sentation.

Theorem 10.1.4 (Lévy-Khinchine representation) Let X be a Lévy process with
characteristic triplet (σ 2 ,ν,γ) . Then , for t ≥ 0,

e iξX t e − tψ(ξ) ,ξ

E[ ]= ∈ R ,

1 ν( d z). (10.2)

1
2 σ 2 ξ 2

e iξz

with ψ(ξ) =− iγξ + + − + iξz 1

{| z |≤ 1 }

R