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

FILES D’ATTENTE, FIABILITÉ, ACTUARIAT
Solving the IDE (??) in this case may be achieved either by either
– Inverting the Laplace transform (4.15) by partial fractions :
ψ ∗ (s) =
=
∫ ∞
0
s+µ
e −su ψ(u)du = cψ(0) − λ 1
s(c −
λ ) s+µ
=
sψ(0) + ψ(0)µ − λ/c
s(s + µ − λ/c)
A
s + µ − λ/c + B s ⇐⇒ ψ(x) = Ae−x(µ−λ/c) + B
Note that the net profit condition µ − λ/c > 0 and ψ(∞) = 0 imply B = 0. Thus, s
must simplify in the fraction above and therefore the relation
ψ(0) = λµ−1
c
= λEC 1
c
must hold. This relation turns out to be true for the Cramer Lundberg model with
arbitrary claims distribution having finite mean!
Letting l = c − λEC 1 denote the ”loading per time unit”, we conclude that ψ(0) and
ψ(0) are proportional respectively to the expected payments and loading per time
unit.
– Solving the Sturm-Liouville IDE equation (4.7) by reducing it to a ODE with constant
coefficients and then looking for combinations of exponentials (or ”looking directly”)
– by probabilistic approaches like the general ”ladder decomposition” for the Cramer
Lundberg process due to Dubordieu(1952)-Benes(1957)-Kendall(1957) (4.21) or martingale
stopping -see below.
Notes : 1) Replacing the initial risk process (4.1) by one with exponential claims and
identical first three moments, and applying the simple exponential claims formula (4.10)
yields the ”simple DeVylder approximation” in terms of the first three moments of the claims
distribution. This was slightly extended by Ramsay (?) Ram92 using mixed exponential claims of
order two; the result however is not that simple anymore, due to the complicated moment
matching and of the dependence of the Cramer-Lundberg roots r i on the moments.
2) For Gamma claims with shape parameter α < 1,, the ultimate and killed ruin probabilities
are also explicit (due to Thorin), though rather complicated – see Grandell & Segerdahl
GS
(?).
Exemple 4.6.2 Let us estimate now the eventual ruin probabilities by a Bayesian approach,
assuming an exponential distribution with uncertain parameter µ, which has a prior Gamma
distribution Γ α,β (dµ).
After n observations ⃗ C = (C 1 , ..., C n ), the posterior density of µ, f(µ|⃗c) is
Γ α+n,β+Sn (dµ),
where S n = ∑ n
i=1 c i, and the Bayes estimates of µ and m = µ −1 are :
∫
ˆµ n = µf(µ| C)dµ ⃗ = n + α
S n + β ,
∫
ˆm n = µ −1 f(µ| C)dµ ⃗ = S n + β
α + n − 1 .