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

fig1
X(t): Risk process: skipfree upward Y(t): workload process, skip
free downward, reflected at 0
Birth−death process:skfree
in both directions.
x
u
T
−y
τ(q)
1 2 1 2 1 0 1 2 1 Busy period
Fig. 4.1 – Skip-free (spectrally one sided) processes
Remarque 4.1.2 A different perspective is obtained studying instead the ”aggregate loss
random variable”
Y (t) = u − X(t) = C(t) − c t
representing the excess of claims over the profit. Note that the process Y (−t) obtained by
reversing also the time has the same finite dimensional distributions as X(t).
The spectrally positive process Y (t) is also of interest in in queueing theory, where it is
used to model the M/G/1 queue workload § .
Remarque 4.1.3 The processes X(t), Y (t) are particular examples of spectrally negative
and spectrally positive Levy processes. Many results about the Cramér Lundberg process,
when expressed in terms of the ”Laplace exponent/symbol”
κ(s) = log E 0 e sX(1) ,
continue to be valid under this greater generality.
Remarque 4.1.4 A natural discrete space analogue is to assume u ∈ N, and to replace the
premium and claims with a combined random walk with an ”upwards skip-free” distribution
p k , k ∈ {1, 0, −1, −2, ...}.
4.2 The ruin problem
Let T be the first passage time of a stochastic process X below 0 :
T := inf{t ≥ 0 : X(t) < 0} = inf{t ≥ 0 : Y (t) > u} = τ Y (u).
The objects of interest in ruin theory are the ”finite-time” and ”ultimate” ruin probabilities
Ψ(t, u) = P u [T ≤ t] = P[τ Y (u) ≤ t],
ψ(u) = P u [T < ∞],
and the related ”survival” and ”late ruin” probabilities
Ψ(t, u) = P u [T > t] = 1 − Ψ(t, u)
˜Ψ(t, u) = P u [t ≤ T < ∞] = ψ(u) − Ψ(t, u).
§ In this case, one is interested in the reflected process Y (t) − inf s≤t Y (s) describing the buffer content.
The stationary distribution of the M/G/1 workload coincides with the eventual ruin probability.