Processus de Markov, de Levy, Files d'attente, Actuariat et Fiabilité ...

(ρ is sometimes called ”geometric compounding parameter” (due to its role in the Benes-
Pollaczek-Khinchin decomposition of ruin probabilities as geometric sums–see (?)) A00 and
In conclusion
ψ ∗ (s) =
∫ ∞
0
⇐⇒ ψ ∗ (s) =
ψ(0) = P 0 [Ȳ = 0] = 1 − ρ = κ′ (0)
.
c
e −su ψ(u)du = λm 1 − λ ¯F ∗ (s)
= 1 κ(s) s − κ′ (0)
κ(s) = 1 s − p
κ(s) ,
∫ ∞
0
e −su ψ(u)du = κ′ (0)
κ(s) =
1 − ρ
s(1 − ρb ∗ e (s)) (4.8) PK
Note also that ψ ∗ (s) = ∫ ∞
e −su P[Ȳ ≤ u]du = s ∫ ∞
e −su 1−ρ
f
0 0 Ȳ (u)du =
s(1−ρb ∗ e (s)).
Finally,
we obtain the famous Pollaczek-Khinchine formula for the Laplace transform of the density
of the supremum of a spectrally positive Levy process :
φ(s) =
∫ ∞
0
e −su f Ȳ (u)du =
1 − ρ
(4.9)
1 − ρb ∗ PKf
e (s),
where φ(s) is defined by ψ ∗ (s) = φ(s) ⇐⇒ ψ ∗ (s) = 1−φ(s) § .
s s
In conclusion, a careful treatment of the singularity at 0 finds simultaneously the Laplace
transform and the ”auxiliary boundary unknown” ψ(0).
4.6 Exponential claims
Exemple 4.6.1 With exponential claim sizes of intensity µ, the ultimate ruin probability
is :
ψ(x) = ρe −γx . (4.10) expruin
Here −γ = −µ+λ/c > 0 is the unique negative root of the Cramér Lundberg equation :
κ(s) = 0, (4.11) CL
with κ(s) = log ( E 0 e sX(1)) = cs−λs/(µ+s) being the ”cumulant generating function/Laplace
exponent/symbol” of the process X(t) –see below, and
ρ = − κ′ (0)
κ ′ (−γ) = λ < 1. (4.12) CLct
µc
It turns out that whenever a negative solution −γ of (4.24) exists, the expression
− κ′ (0)
κ ′ (−γ) e−γx is asymptotic to ψ(x)! This is the famous Cramér Lundberg approximation.
The ruin probability killed at rate q is also exponential :
ψ q (x) = (1 + s −
µ )exs −
= λ/c
µ + s +
e xs −
,
the exponent being the unique negative root of the Cramér Lundberg equation :
κ(s) = q, (4.13) CLq
which reduces here to the quadratic equation cs 2 + s(cµ − λ − q) − qµ = 0.
§ in fact, φ(s) has a probabilistic interpretation, being one of the two factors of the celebrated Wiener-Hopf
decomposition –see (?), Ber and φ(s) = E[e −sȲ ] = P[Ȳ = 0] + ∫ ∞
0
e −su f Ȳ (u)du = sψ ∗ (s) = κ′ (0)s
κ(s)