Page 331 - A First Course In Stochastic Models

326 ADVANCED RENEWAL THEORY

The exact values of D x are obtained by computer simulation. The length of the
simulation run has been taken long enough to ensure that the half-width of the
95% confidence interval for the simulated probability is no more than 0.001. The
lifetime L of a unit has a Weibull distribution with mean E(L) = 1 and the repair
time R of a unit has a gamma distribution with mean E(R) = 0.125. The squared
2
coefficients of variation of the lifetime and the repair time are c = 0.5, 1 and
L
2
c = 0, 0.5, 1, 4. For the length of the interval we have taken t 0 = 1.
R
8.4 RUIN PROBABILITIES
In many applied probability problems asymptotic expansions provide a simple alter-
native to computationally intractable solutions. A nice example is the ruin proba-
bility in risk theory. Suppose claims arrive at an insurance company according to
a Poisson process {N(t)} with rate λ. The successive claim amounts X 1 , X 2 , . . .
are positive, independent random variables having a common probability distri-
bution function B(x) with finite mean µ. The claim amounts are independent of
the arrival process. In the absence of claims, the company’s reserve increases at a
constant rate of σ > 0 per time unit. It is assumed that σ > λµ, i.e. the average
premium received per time unit is larger than the average claim rate. Denote by
the compound Poisson variable
N(t)

X(t) = X k
k=1
the total amount claimed up to time t. If the company’s initial reserve is x > 0,
then the company’s total reserve at time t is x + σt − X(t). We say that a ruin
occurs at time t if x + σt − X(t) < 0 and x + σu − X(u) ≥ 0 for u < t. Let

Q(x) = P {X(t) > x + σt for some t ≥ 0}.

Then Q(x) is the probability that a ruin will ever occur when the initial capital is
x. Since a ruin can occur only at the claim epochs, we can equivalently write
 
 k

Q(x) = P X j − σT k > x for some k ≥ 1 , (8.4.1)
j=1
 
where T k is the epoch at which the kth claim occurs for k = 1, 2, . . . . We are
interested in the asymptotic behaviour of Q(x) for large x.
The ruin probability Q(x) arises in a variety of contexts. As another example
consider a production/inventory situation in which demands for a given product
arrive according to a Poisson process. The successive demands are independent
and identically distributed random variables. On the other hand, inventory replen-
ishments of the product occur at a constant rate of σ > 0 per time unit. In this
context, the ruin probability Q(x) represents the probability that a shortage will
ever occur when the initial inventory is x.