240 S. Cheng et al. Insurance: Mathematics and Economics 26 2000 239–250
In Section 3 we consider the discounted power martingale. For a given discount factor, the base must satisfy Lundberg’s fundamental equation, which has always one solution between zero and one, and a larger, second
solution under some regularity conditions for the claim amount distribution. The main goal of the paper is to study the function fx, y; u, which is the “discounted” probability of ruin, given an initial surplus of u, such that the
surplus just before ruin is x, and the deficit at ruin is y. With this function, the expected present value of a penalty due at ruin can be calculated. In Section 4, an explicit formula for fx, y; 0 is obtained. In Section 5, we show that
f
x, y; u can be expressed in a transparent fashion through a function hu, which does not depend on x or y, and is the solution of a certain recursive equation. In Section 6, an asymptotic formula for hu is derived and used to
obtain an asymptotic formula for fx, y; u. Applications include an explicit expression for the expected discounted value of a payment of 1 at the time of ruin, a discrete-time version of Dickson’s formula, formulas for the expected
time of ruin, as well as some asymptotic results.
2. The compound binomial model
We consider a discrete-time model, in which the number of insurance claims is governed by a binomial process N
t, t=0, 1, 2,. . . In each time period, the probability of a claim is q, 0q1, and the probability of no claim is 1−q. The claim occurrences in different time periods are independent events. The individual claim amounts X
1
, X
2
, X
3
,. . . are mutually independent, identically distributed, positive and integer-valued random variables; they are independent of the binomial process
{Nt}. Put X=X
1
and let px =
PrX = x, x =
1, 2, 3, . . . 2.1
be the common probability function of the individual claim amounts. The value of px is zero if x is not a positive integer. This is called a compound binomial model, and has been considered by Gerber 1988, Shiu 1989, Willmot
1993, Dickson 1994, DeVylder 1996 Chapter 10, DeVylder and Marceau 1996 Section 2, and Cheng and Wu 1998a,b.
The compound binomial model can also be used to model the case where there can be more than one claim in each time period. Then we assume that the total claims in each time period are mutually independent, identically
distributed and integer-valued random variables. Let Y
j
denote the sum of the claims in period j. We consider PrY
j
= 0 = 1 − q,
PrY
j
= y =
qpy, y =
1, 2, 3, . . . To avoid confusion, we shall not refer to this interpretation in this paper.
We also assume that the premium received in each time period is one. We do not necessarily make the assumption that the premiums contain a positive security loading, i.e.
1 − qEX 0, 2.2
may not hold. Let the initial surplus be u, which is a nonnegative integer. For t=0, 1, 2,. . . , the surplus at time t is
U t = u + t − [X
1
+ X
2
+ · · · + X
N t
]. 2.3
“Ruin” is the event that Ut≤0 for some t≥1. We suppose that p11 so that ruin is possible. Let T =
inf{t ≥ 1 : U t ≤ 0} 2.4
denote the time of ruin. We are interested in the joint probability distribution of the time of ruin, T, the surplus just before ruin, UT−1, and the surplus at ruin, UT. For x, y=0, 1, 2,. . . , t=1, 2, 3,. . . , define
f x, y, t ; u = Pr[U T − 1 = x, U T = −y, T = t|U 0 = u].
2.5
S. Cheng et al. Insurance: Mathematics and Economics 26 2000 239–250 241
Note that f0, y, t; u can be different from 0 only if t=1 and u=0; obviously f
0, y, 1; 0 = qpy + 1. 2.6
Let v be a discount factor 0v1. Our primary goal is to explore the “discounted” probability f x, y; u =
∞
X
t = 1
v
t
f x, y, t ; u. 2.7
Using the function fx, y; u, we can calculate the expected discounted value of a “penalty” which is due at ruin and may depend on the surpluses just before and at ruin. We may also view v as a parameter; then 2.7 is the formula
of a probability generating function.
3. Lundberg’s fundamental equation
