Page 334 - A First Course In Stochastic Models

RUIN PROBABILITIES 329

x
x

= Q(0){1 − B(x)} − Q(x − y)}{1 − B(y)} − Q(x − y)b(y) dy
0 0
x
= Q(x) − Q(x − y)b(y) dy.
0
Hence (8.4.3) can be rewritten as
λ λ d x
′
Q (x) = − {1 − B(x)} + Q(x − y){1 − B(y)} dy (8.4.4)
σ σ dx 0
for x > 0. Integrating both sides of this equation gives

λ x λ x
Q(x) = Q(0) − {1 − B(y)} dy + Q(x − y){1 − B(y)} dy (8.4.5)
σ 0 σ 0
for all x ≥ 0. The unknown constant Q(0) is easily determined by taking the
Laplace transforms of both sides of (8.4.5). Using the relations (E.5), (E.6) and
(E.7) in Appendix E and noting lim x→∞ Q(x) = 0, it is readily verified that

Q(0) = λµ/σ,

where µ = E(X) is the mean claim size. The details are left to the reader. Hence
the integro-differential equation (8.4.3) is equivalent to
x

Q(x) = a(x) + Q(x − y)h(y) dy, x ≥ 0, (8.4.6)
0
where the functions a(x) and h(x) are given by
λ x λ
a(x) = Q(0) − {1 − B(y)} dy and h(x) = {1 − B(x)}, x ≥ 0.
σ 0 σ
The equation (8.4.6) has the form of a standard renewal equation except that the
function h(x), x ≥ 0, is not a proper probability density. It is true that the function
h is non-negative, but

∞ λ ∞ λµ
h(x) dx = {1 − B(x)} dx = < 1.
0 σ 0 σ
Thus h is the density of a distribution whose total mass is less than 1 with a defect
of 1−λµ/σ. Equation (8.4.6) is called a defective renewal equation.

Asymptotic expansion for the ruin probability
A very useful asymptotic expansion of Q(x) can be given when it is assumed that
the probability density of the claim size (service time) is not heavy-tailed. To be
more precise, the following assumption is made.