Exact and asymptotic results for insurance risk models with surplus-dependent premiums

In classical insurance risk theory, the collective renewal risk model describes the amount of surplus U (t) of an insurance portfolio at time t by

where c represents a constant rate of premium inflow, N (t) is a renewal process that counts the number of claims incurred during the time interval (0, t] and (X k ) k≥0 is a sequence of independent and identically distributed (i.i.d.) claim sizes with distribution function F X and density f X (also independent of the claim arrival process N (t)). Let (τ k ) k≥0 be the i.i.d. sequence of interclaim times. One of the crucial quantities to investigate in this context is the probability that at some point in time the surplus in the portfolio will not be sufficient to cover the claims, which is called the probability of ruin

where U (0) = u ≥ 0 is the initial capital in the portfolio and

A related, more general quantity is the expected discounted penalty function, which penalizes the ruin event for both the deficit at ruin and the surplus before ruin,

where δ ≥ 0 is a discount rate and the penalty w(x, y) is a bivariate function. (Φ(u) is often referred to as the Gerber-Shiu function, see [8]).

The classical collective risk model is based on the assumption of a constant premium rate c. However, it is clear that it will often be more realistic to let premium amounts depend on the current surplus level. In this case, the risk process (1.1) is replaced by

Hence, in between jumps (claims) the risk process moves deterministically along the curve ϕ(u, t), which satisfies the partial differential equation ∂ϕ ∂t = p(u) ∂ϕ ∂u ; ϕ(u, 0) = u.

There are only a few situations for which exact expressions for ψ(u) are known for surplus-dependent premiums. One such case is the Cramér-Lundberg risk model (where N (t) is a homogeneous Poisson process with intensity λ) and the linear premium function p(u) = c + εu, which has the interpretation of an interest rate ε on the available surplus. In the case of exponential claims, it was already shown by [21] that the probability of ruin then has the form

where Γ(η, x) = ∞ x t η-1 e -t dt is the incomplete gamma function (for extensions to finite-time ruin probabilities, see [11,12] and [3]). In fact, for the Cramér-Lundberg risk model with exponential claims and general monotone premium function p(u), one has the explicit expression

exp {λq(x) -µx} dx,

where 1/γ 0 ≡ 1 + λ ∞ 0 p(x) -1 exp {λq(x) -µx} dx and q(x) ≡

x 0 1 p(y) dy is assumed finite for x > 0 (see [22]). Since for surplus-dependent premiums the probabilistic approach based on random equations does not work, and also the usual analytic methods lead to difficulties because the equations become too complex, it is a challenge to derive explicit solutions beyond the one given above.

In this paper we will employ a method based on boundary problems and Green’s operators to derive closed-form solutions and asymptotic properties of ψ(u) and Φ(u) under more general model assumptions.

For that purpose we will employ the algebraic operator approach developed in [2]. However, since that approach was restricted to linear ordinary differential equations (LODEs) with constant coefficients, we will have to extend the theory to tackle the variable-coefficients equations that occur in the present context.

In Section 2 we derive the boundary problem for the Gerber-Shiu function Φ(u) in a renewal risk model with claim and interclaim distributions having rational Laplace transform. For solving it, we employ a new symbolic method, described in Section 3. This allows to construct integral representations for the solution of inhomogeneous LODEs with variable coefficients, for given initial values, under a stability condition. In Section 4 we derive a general asymptotic expansion for the discounted penalty function in the renewal model framework. Subsequently, Section 5 is dedicated to the more specific case of compound Poisson risk models with exponential claims, for which we have second-order LODEs. More specifically, in 5.1 we derive exact solutions for a generic premium function p(u). Further, in 5.2, we consider some interesting particular cases of p(u). In 5.3 we identify the necessary conditions a premium function should satisfy such that the asymptotic analysis is possible and the assumptions necessary for the asymptotic results in Section 4 are validated. We will end by giving concrete examples of such premium functions and their asymptotics.

Throughout the paper we will assume that U (t) → ∞ a.s. This assumption is satisfied for example when p(u) > EX/Eτ + ς for some ς > 0 and sufficiently large u; see e.g. [4].

2. Deri