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

4.7 Rational symbols
FILES D’ATTENTE, FIABILITÉ, ACTUARIAT
The final result of the analytic approach is to yield Laplace transforms; for example, the
Laplace transform of the perpetual survival probability is given by the famous Pollaczek-
Khinchin formula :
ψ ∗ (s) =
∫ ∞
0
e −su ψ(u)du =
∫ ∞
0
e −su P[Ȳ ≤ u]du = κ′ (0)
κ(s)
(4.15) PK1
where κ(s) is the ”symbol” of the process.
From the Pollaczek-Khinchine formula(4.15), it may be seen that if the claims have a
rational Laplace transform
¯B ∗ (s) =
∑ K−1
k=0 a ks k
s K + ∑ K−1
k=0 b (4.16) Lappar
ksk, then the same will be true of the Laplace transform of Ȳ and of the ultimate ruin probability.
In this case, the Laplace transform may be inverted explicitly, either by a) the
matrix-exponential formalism, or by b) splitting into partial fractions, yielding mixtures of
exponentials involving the roots of the Cramer-Lundberg equation § .
Exemple 4.7.1 The matrix exponential formalism. For light tailed claims with rational
Laplace transform (4.16), the distribution may also be expressed in ”matrix exponential” form
¯B(x) = βe Bx 1, ⇐⇒ ¯B ∗ (s) = β(sI − B) −1 1 (4.17) matexp
with B a matrix of order K –see for example (?).
AsmBla
This representation renders Laplace inversion unnecessary, and the ruin probability is
”explicit” (see (?)) A00 :
ψ(u) = ηe Qu 1, where Q = B + (−B)1η, η = β(−B) −1 . (4.18) phaseruin
Note though that the matrix exponential representation is overparameterized (K 2 parameters
instead of the 2K parameters of the Laplace transform). Since the claims distribution is never
certain, and the statistical estimation of matrix-exponential or phase-type representations is
a notoriously difficult problem, the formula (4.18) is not necessarily the best way to approach
the problem.
Exemple 4.7.2 Suppose the claims are mixtures of distinct exponentials with positive or
negative weights
¯B(x) =
I∑
β i e −bix ,
i=1
∑
β i = 1, b(x) ≥ 0, x ∈ R +
i
(with mean m 1 = ∑ I
i=1 β ib −1
i ).
Let us apply the Pollaczek-Khinchin formula, exploiting the fact that for Cramér Lundberg
processes, the symbol κ(s) = log ( E 0 e −sX(1)) may be written as κ(s) = s(c − λ ¯B ∗ (s)).
§ For non-rational symbols, the task of Laplace inversion is more challenging; for example, the case of
lognormal claims is not so straightforward.