Page 333 - A First Course In Stochastic Models

328 ADVANCED RENEWAL THEORY

This implies that
 
 k

P {D n+1 > x} = P U j > x for some 1 ≤ k ≤ n , x ≥ 0.
j=1
 
Since lim n→∞ P {E n } = P {lim n→∞ E n } for any monotone sequence {E n } of
events, it follows that lim n→∞ P {D n > x} exists for all x ≥ 0. Moreover,
 
 k k

lim P {D n > x} = P X j − σ τ j > σx for some k ≥ 1 , x ≥ 0.
n→∞  
j=1 j=1
Together this relation and (8.4.1) prove the result (8.4.2).

A renewal equation for the ruin probability
We now turn to the determination of the ruin probability Q(x). For that purpose,
we derive first an integro-differential equation for Q(x). For ease of presentation
we assume that the probability distribution function B(x) of the claim sizes has a
probability density b(x). Fix x > 0. To compute Q(x − x) with x small, we
condition on what may happen in the first t = x/σ time units. In the absence
of claims, the company’s capital grows from x − x to x. However, since claims
arrive according to a Poisson process with rate λ, a claim occurs in the first x/σ
time units with probability λ x/σ + o( x), in which case the company’s capital
becomes x − S if S is the size of that claim. A ruin occurs if S > x. Thus, by
conditioning, we get for fixed x > 0,

λ x λ x
∞
Q(x − x) = 1 − Q(x) + b(y) dy
σ σ x
λ x x
+ Q(x − y)b(y) dy + o( x).
σ 0
Subtracting Q(x) from both sides of this equation, dividing by h = − x and
letting x → 0, we obtain the integro-differential equation
λ λ λ x
′
Q (x) = − {1 − B(x)} + Q(x) − Q(x − y)b(y) dy, x > 0. (8.4.3)
σ σ σ 0
Equation (8.4.3) can be converted into an integral equation of the renewal type. To
do so, note that
d x
Q(x − y){1 − B(y)} dy
dx 0
x
′
= Q(0){1 − B(x)} + Q (x − y){1 − B(y)} dy
0