Page 336 - A First Course In Stochastic Models

RUIN PROBABILITIES 331

where the load factor ρ is defined by ρ = λµ/σ. Applying the key renewal theorem
from Section 8.2 to the renewal equation (8.4.8), we find

lim Q (x) = γ,
∗
x→∞
where the constant γ is given by

−1

(1 − ρ) λ ∞ δy
γ = ye {1 − B(y)} dy .
δ σ 0
This yields the asymptotic expansion
−δx
Q(x) ∼ γ e as x → ∞, (8.4.9)
where f (x) ∼ g(x) as x → ∞ means that lim x→∞ f (x)/g(x) = 1. This is an
extremely important result. The asymptotic expansion is very useful for practical
purposes in view of the remarkable finding that already for relatively small values
of x the asymptotic estimate predicts quite well the exact value of Q(x) when the
load factor ρ is not very small. To illustrate this, Table 8.4.1 gives the numerical
values of Q(x) and the asymptotic estimate Q asy (x) = γ e −δx for several examples.
2
We take µ = 1 and σ = 1. The squared coefficient of variation c of the claim size
X
2
2
2
X is c = 0 (deterministic distribution), c = 0.5 (E 2 distribution) and c = 1.5
X X X
(H 2 distribution with balanced means). The load factor ρ is 0.2, 0.5 and 0.8. It
turns out that the closer ρ is to 1, the earlier the asymptotic expansion applies.
Table 8.4.1 Exact and asymptotic values for Q(x)
c 2 = 0 c 2 = 0.5 c 2 = 1.5
X X X
x Q(x) Q asy (x) Q(x) Q asy (x) Q(x) Q asy (x)
ρ = 0.2 0.5 0.11586 0.07755 0.12462 0.14478 0.13667 0.09737
1 0.02288 0.03007 0.07146 0.07712 0.09669 0.07630
2 0.00196 0.00210 0.02144 0.02188 0.05234 0.04685
3 0.00015 0.00015 0.00617 0.00621 0.03025 0.02877
5 7.20E-7 7.20E-7 0.00050 0.00050 0.01095 0.01085
ρ = 0.5 0.5 0.35799 0.30673 0.37285 0.38608 0.39390 0.34055
1 0.17564 0.18817 0.26617 0.26947 0.31629 0.28632
2 0.05304 0.05356 0.13106 0.13126 0.21186 0.20239
5 0.00124 0.00124 0.01517 0.01517 0.07179 0.07149
10 2.31E-6 2.31E-6 0.00042 0.00042 0.01262 0.01262
ρ = 0.8 0.5 0.70164 0.67119 0.71197 0.71709 0.72705 0.70204
1 0.55489 0.56312 0.62430 0.62549 0.66522 0.65040
2 0.36548 0.36601 0.47582 0.47589 0.56345 0.55825
5 0.10050 0.10050 0.20959 0.20959 0.35322 0.35299
10 0.01166 0.01166 0.05343 0.05343 0.16444 0.16444