ensaecours1
ENSAE, 2004
1
Plan
Processus de Poisson
Processus mixtes
Processus de L´evy
2
Processus de Poisson
1. Counting processes and stochastic integral
2. Standard Poisson process
3. Inhomogeneous Poisson Processes
4. Stochastic intensity processes
3
1 Counting processes and stochastic inte-
gral
Nt=nif t∈[Tn,T
n+1[
+∞otherwise
or, equivalently
Nt=
n≥1
11{Tn≤t}=
n≥1
n11{Tn≤t<Tn+1}.
We denote by Nt−the left-limit of Nswhen s→t, s < t
and by ∆Ns=Ns−Ns−the jump process of N.
4
The stochastic integral
t
0
CsdNs
is defined as
(CN)t=t
0
CsdNs=]0,t]
CsdNs=
∞
n=1
CTn11{Tn≤t}.
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