*3.2. Completely Monotone Claim Sizes*

We are mostly interested in evaluating ruin probabilities when the claim sizes follow a heavytailed distribution, such as Pareto or Weibull. These two distributions belong to the class of completely monotone distributions.

**Definition 1.** *A pdf f is said to be completely monotone (c.m.) if all derivatives of f exist and if*

$$(-1)^n f^{(n)}(u) \ge 0 \text{ for all } u > 0 \text{ and } n \ge 1.$$

Completely monotone distributions can be approximated arbitrarily closely by hyperexponentials; see, e.g., Feldmann and Whitt (1998). Here, we provide a method to approximate a completely monotone ladder height distribution with a hyperexponential one to achieve any desired accuracy for the ruin probability. The following result is standard; see Feller (1971).

**Theorem 2.** *A ccdf F is completely monotone if and only if it is the Laplace–Stieltjes transform of some probability distribution S defined on the positive half-line, i.e.,*

$$\mathcal{F}(u) = \int\_0^\infty e^{-yu} dS(y). \tag{11}$$

*We call S the spectral cdf.* **Remark 2.** *With a slight abuse of terminology, we will say that a function S is the spectral cdf of a distribution if it is the spectral cdf of its ccdf.*

Note that Theorem 2 also extends to the case where *<sup>S</sup>*(*y*) is not a distribution but simply a finite measure on the positive half-line, i.e., a function *f* is completely monotone if and only if it can be expressed as the Laplace–Stieltjes integral of such a finite measure *<sup>S</sup>*(*y*). We will show that under the assumption that the claim size distribution is c.m. and the ladder height distribution is c.m. too. We first need the following intermediate result.

**Lemma 1.** *If the ccdf FX*(*u*) *is c.m., then* T*ρn FX*(*u*) *is a c.m. function,* ∀*n* = 1, . . . , *N.*

**Proof.** Assume that the claim sizes are completely monotone, i.e., *FX*(*u*) = ∞0 *<sup>e</sup>*<sup>−</sup>*uydS*(*y*), for some spectral cdf *<sup>S</sup>*(*y*). In this case, it holds that

$$\begin{aligned} \langle \mathcal{T}\_{\rho\_n} \mathsf{F}\_X(u) \rangle &= \int\_{t=0}^{\infty} e^{-\rho\_n t} \mathsf{F}\_X(t+u) dt = \int\_{t=0}^{\infty} e^{-\rho\_n t} \int\_{y=0}^{\infty} e^{-(t+u)y} dS(y) dt \\ &= \int\_{y=0}^{\infty} e^{-\mu y} dS(y) \int\_{t=0}^{\infty} e^{-(y+\rho\_n)t} dt = \int\_{0}^{\infty} e^{-\mu y} \frac{dS(y)}{y+\rho\_n} = \int\_{0}^{\infty} e^{-\mu y} dS \,\mathcal{T}\_{\rho\_n}(y), \end{aligned}$$

where *dS*T*ρn* (*y*) = *dS*(*y*) *y*+*ρn* , *n* = 1, ... , *N*, is a finite measure on the positive half-line with *<sup>S</sup>*T*ρn* (+∞)=(<sup>1</sup> − ˜ *fX*(*ρn*))/*ρ<sup>n</sup>*, *n* = 1, . . . , *N* − 1, and *<sup>S</sup>*T0 (+∞) = E*X*.

We can now state the following result.

**Proposition 1.** *If the ccdf FX*(*u*) *is c.m., i.e., FX*(*u*) = ∞0 *<sup>e</sup>*<sup>−</sup>*uydS*(*y*)*, for some spectral cdf <sup>S</sup>*(*y*)*, then the ladder height distribution is c.m. too, i.e., <sup>H</sup>*(*u*) = ∞0 *<sup>e</sup>*<sup>−</sup>*uydSH*(*y*)*, where SH*(*y*) *is a spectral cdf such that*

$$dS\_H(y) = \frac{1}{\Phi c^N} \sum\_{n=1}^N \frac{\mu^\* - c\rho\_n \beta(-c\rho\_n)}{(y + \rho\_n) \prod\_{\substack{k=1\\k \neq n}}^N (\rho\_k - \rho\_n)} dS(y).$$

**Proof.** It was proven in Chiu and Yin (2014) that the ascending ladder height distribution in the Sparre Andersen model is c.m. if the claim size distribution is c.m, meaning that *<sup>H</sup>*(*u*) can be represented as the Laplace–Stieltjes transform of some spectal cdf *SH*(*y*). Due to the uniqueness of Laplace transforms, it, therefore, suffices to find the formula of the spectral cdf *SH*(*y*) by applying Lemma 1 to (8).

We show in the next section how to utilise the above results to construct approximations for the ruin probability *ψ*(*u*) that have a guaranteed error bound given by Theorem 1.
