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tel-00703797, version 2 - 7 Jun 2012
4.3. Asymptotics – the A + B/u rule
The expressions derived when Θ is Lévy distributed, are much more complex than in the continuous
time framework. In Subsection 4.3.3, we study asymptotics for the survival function.
The ruin probability can be computed using I3(−1, u + 1, α 2 /4, θ0).
Proposition 4.2.3. Let us consider the discrete time framework of Subsection 4.2.2 with a
latent variable Θ Lévy distributed Le(α).
α
ψ(u) = erfc
2 √
+
θ0
1 − q
4qu+1 u+1
u + 1
(−1)
j
j=0
j
e α√
j−1 α
erfc
2 √ +
θ0
j − 1
θ0
+e −α√
j−1 α
erfc
2 √ −
θ0
j − 1
θ0 ,
with the convention ∗ √ −1 = i, θ0 = − log(1 − q) and for u ≥ 0.
4.3 Asymptotics – the A + B/u rule
This section is the core of the paper, where we establish the A + B/u asymptotic rule for
the ultimate ruin probability for both continuous and discrete time models. We also obtain
an expansion of the ruin probability as a power series of 1/u. Finally, we investigate the
asymptotic behavior of the resulting claim distribution, which requires a special treatment
with complex analysis.
4.3.1 Notation
We recall basics and notation of the asymptotic analysis; see e.g. Jones (1997), Olver
et al. (2010). We introduce the standard Landau notation O(), o() and ∼. One says that f is
asymptotically bounded by g as x → x0, denoted by
f(x) = O(g(x)) ,
x0
if there exists K, δ > 0, such that for all 0 < |x − x0| < δ, we have |f(x)| ≤ K|g(x)|. In other
words, in a neighborhood of x0 excluding x0, |f(x)/g(x)| is bounded.
Then, f is said to be asymptotically smaller than g as x → x0, denoted by
f(x) = o(g(x)) ,
x0
if for all ɛ > 0, there exists δ > 0, such that for all 0 < |x − x0| < δ, we have |f(x)| ≤ ɛ|g(x)|.
That is to say, in a neighborhood of x0 excluding x0, |f(x)/g(x)| tends to 0.
And finally, f is asymptotically equivalent to g around x0, if the ratio of f over g tends to
1, i.e.,
f(x) ∼ g(x) ⇔
x0
f(x)
g(x) −→ 1.
x→x0
This is equivalent to f(x) = g(x) + o(g(x)).
The asymptotic idea aims to approximate a complicated function f at x0 by a sum of
known and tractable terms g(x), controlling the error by o(g(x)). Note that x0 can be +∞.
∗. One can check that the term j = 0 is still a real number.
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