Etude des marchés d'assurance non-vie à l'aide d'équilibres de ...

tel-00703797, version 2 - 7 Jun 2012
4.6. Appendix
A large part of this paper focuses on the discrete time framework. We have seen that
discrete time ruin probability asymptotics can be obtained in a similar way as in the continuous
case. However, deriving asymptotics for the claim distribution is far more difficult: complex
analysis is necessary to tackle the binomial alternating sum issue. Furthermore, the nonuniqueness
of discrete copulas is also studied. We quantify the maximal difference between
the continuous and the discrete settings. Despite the issues encountered with discrete copula,
the latent variable approach is considered in many other articles, e.g., Joe (1997); Frees and
Wang (2006); Channouf and L’Ecuyer (2009); Braeken et al. (2007); Leon and Wu (2010).
As mentioned in Albrecher et al. (2011), the approach proposed in this paper to derive
new explicit formula can be used for more general risk models. It could be interesting to
test whether the A + B/u still applies for the ruin probability, say, with phase-type claim
distributions. Beyond the study of ruin probability, the mixing approach and the asymptotic
rule might probably be used for finite-time ruin probabilities and the Gerber-Shiu function.
4.6 Appendix
4.6.1 Usual special functions
List of common special functions
Let us recall the definition of the so-called special functions, see Olver et al. (2010) for a
comprehensive and updated list. In most cases, the definition does not limit to z ∈ R, but
can be extended to z ∈ C.
– the gamma function Γ(α) = ∞
0 xα−1 e −x dx,
– the lower incomplete gamma function γ(α, z) = z
0 xα−1 e −x dx,
– the upper incomplete gamma function Γ(α, z) = ∞
z xα−1 e −x dx,
– the beta function β(a, b) = 1
0 xa−1 (1 − x) b−1 dx,
– the incomplete beta function β(a, b, z) = z
0 xa−1 (1 − x) b−1 dx,
– the complementary incomplete beta function ¯ β(a, b, z) = 1
z xa−1 (1−x) b−1 dx = β(b, a, 1−
z),
– the error function erf(z) = 2
z
√
π 0 e−t2dt,
– the complementary error function erfc(z) = 2
– the binomial coefficient n
k =
n!
k!(n−k)! .
Known asymptotics of usual special functions
√ π
+∞
z e−t2dt, In this subsection, we list asymptotics of the main special functions, see Olver et al. (2010).
Gamma function Known as the Stirling formula, the asymptotic expansion for Γ(z) is
Γ(z) ∼ +∞ e −z z z
2π
z
1 + 1
12z
gk
+ · · · + + . . .
zk for z ∈ C and | arg(z)| < π with g0 = 1, g1 = 1/12, g2 = 1/288, g3 = −139/51480. See page
141 of Olver et al. (2010).
We recall that the incomplete gamma functions are defined as
γ(a, z) =
,
z
x
0
α−1 e −x ∞
dx, and Γ(a, z) = x
z
α−1 e −x dx.
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