1. BACKGROUND AND MOTIVATION

Ross [2] considered the following classical insurance risk problem. Claims are made against an insurance company according to a homogeneous Poisson process with rate λ > 0. The successive claim amounts are independent and identically distributed (i.i.d.) nonnegative random variables Y₁,Y₂,… which are, in turn, independent of the claim process. Claim amounts have the common cumulative distribution function F and finite mean μ, 0 < μ < ∞. The insurance company has initial capital x and receives premiums and other income at a constant rate c > 0 per unit time. The ruin probability is given by

where v(t) is the number of claims made up to time t. We assume ρ = λμ/c < 1 to avoid the trivial case in which p ≡ 1.

The ruin probability has been characterized (see [3] and [4]) by

where T has geometric distribution

and T,X₁,X₂,… are mutually independent, with each X_i having the equilibrium distribution associated with the renewal process generated by Y₁,Y₂,…; that is,

Note that from Eq. (2) and (3), it follows immediately that p(0) = ρ (any sum involving

will be defined to be 0).

Let {N(t), t ≥ 0} be the renewal process generated by X₁,X₂,…, and Z₁,Z₂,…,Z_r be mutually independent exponential random variables each having mean x/r and independent of {N(t), t ≥ 0}. Ross [2] exploited the relation

and the fact that S_r = Z₁ + Z₂ +···+ Z_r converges in probability to x to approximate Eq. (5) by ρm(r) for large r, where

and developed a recursive relation for computing m(r) given by

with

for j = 0,1,…,r. A natural question is: What is the rate at which |p(x) − ρm(r)| → 0 as r → ∞?

Before addressing this question, consider the general problem of approximating p(x) in Eq. (2) when X_i ≥ 0 has an arbitrary continuous cumulative distribution function G(x) = P{X₁ ≤ x} and ρ in Eq. (3) is an arbitrary fixed number in (0,1). This is the general problem of approximating a tail probability for a geometrically compounded continuous life distribution. The above method of Ross [2] for approximating p(x) by ρm(r) still has merit, with the coefficients in the recursions (7) now given by

for j = 0,1,…,r. In the next section, the convergence to zero for |p(x) − ρm(r)| is first established in this setting for general continuous life distribution G, and then some convergence rates are given under various additional restrictions on G. These are then related back to the original insurance risk problem.

2. CONVERGENCE

First, consider the general problem with G continuous. We have p(0) = ρ, so we need only to consider x > 0. Fix an x > 0. Conditioning on S_r, we have ρm(r) = Ep(S_r), and, consequently, in view of the relation

it is sufficient, by the bounded convergence theorem and the fact that S_r converges in probability to x as r → ∞, to show that p is continuous at x. Denoting by G⁽ⁿ⁾ the n-fold convolution of G with itself, as h → 0,

by the bounded convergence theorem, establishing the continuity of p at x. This establishes ρm(r) → p(x) as r → ∞ and confirms the utility of the Ross [2] method for this problem.

To establish rates of convergence, additional assumptions are needed on G. Suppose first that G has a density g. Then, p is differentiable at x > 0. To see this, note that for x > 0, p(0) − p(x) is given by

by Fubini's theorem, since the integrand is nonnegative. It follows that the derivative of p is given by

Now, if g(x) ≤ M < ∞ for x ≥ 0, it follows easily that |p′(x)| ≤ Mρ for x > 0. From the first-order Taylor expansion (with θ a number between 0 and 1),

it follows then that

the last equality following from the fact that S_r has the same distribution as

where ξ₁,ξ₂,… are i.i.d. exponential random variables with mean 1, and the calculations

An application of Stirling's approximation then easily gives the rate

as r → ∞. Note that this case applies to the original insurance risk problem, since there g(x) = (1/μ)(1 − F(x)) for x ≥ 0, and so g(x) ≤ (1/μ) for all x ≥ 0.

Finally, if G has a density g which itself has a bounded derivative g′ for x > 0, then p(x) will have a bounded second derivative for x > 0, given by

Assuming that this is true for the moment, taking sup_x>0|g′(x)| = 2Q < ∞ and then taking expectations in a second-order Taylor expansion gives

where

so that

as r → ∞. Finally, to see that p will have a bounded second derivative for x > 0 if g′ is bounded for x > 0, note that since G has a density, Eq. (10) gives

Now, since g′ is bounded, then by the mean value theorem, where x − u > 0 and x + h − u > 0,

is bounded by a constant, so by the bounded convergence theorem, Eq. (11) follows, which is bounded for all x > 0.

For the original insurance risk problem, if F has a bounded derivative, then g(x) = (1/μ)(1 − F(x)) will have a bounded derivative and we get a O(1/r) convergence rate for ρm(r) to p(x).