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

tel-00703797, version 2 - 7 Jun 2012
Chapitre 4. Asymptotiques de la probabilité de ruine
This result is in line with Theorem 7 of Chaudry and Zubair (1994) or Theorem 3.1 of Chaudry
and Zubair (2002). It is closely related to the generalized error function
√
π
erfc(x; b) =
4 e2√
b
e 2√ √
b b
erfc x + + e
x
−2√ √
b b
erfc x − .
x
If x = 0, we get
J(a, b, x) =
√ π
2 √ a e−2√ ab = J(a, b).
If b = −1, then it is equivalent as changing the occurence of √ b by the imaginary number i.
We get
√
π
J(a, −1, x) =
4 √
e
a
2i√
a √ax i
erfc + + e
x
−2i√
a √ax i
erfc − .
x
This number is of type z1z2 + ¯z1¯z2, where z1 = e2i√a and z2 = erfc √ ax + i
x . It is easy to
check that
z1z2 + ¯z1¯z2 = 2|z1z2| cos(arg(z1) + arg(z2)) ∈ R.
Thus, we extend the notation J(a, b, x) for b = −1 by the above expression.
4.6.5 Analysis
Integration by parts
We give in this subsection the integration by part theorem for the Lebesgues and the Stieltjes
integrals from Gordon (1994)’s graduate book. Other reference books include Hildebrandt
(1971); Stroock (1994).
Lebesgues integral
Theorem (Theorem 12.5 of Gordon (1994)). Let f be Lebesgues-integrable on [a, b] and
F (x) = x
a f for each x ∈ [a, b]. If G is absolutely continuous on [a, b], then fG is Lebesguesintegrable
and we have
b
a
fG = F (b)G(b) −
b
a
F G ′ .
If the integral limits as b tends to infinity exist, one can consider the indefinite integral
version +∞
+∞
fG = lim F (b)G(b) − F G
b→+∞ ′ .
Stieltjes integral
a
Theorem (Theorem 12.14 of Gordon (1994)). Let f, g be bounded functions on a closed interval
[a, b]. If f is Stieltjes-integrable with respect to g on [a, b], then g is also Stieltjes-integrable
with respect to g on [a, b] and we have
210
b
g(t)df(t) = [g(t)f(t)]
a
b b
a −
a
a
f(t)dg(t).