**2. Model Description**

Consider the Sparre Andersen risk model for an insurance surplus process defined as

$$R(t) = \mu + \varepsilon t - \sum\_{i=1}^{N(t)} X\_{i\prime} \quad t \ge 0,\tag{1}$$

where *u* ≥ 0 is the initial capital, *c* > 0 is the constant premium rate and the i.i.d. positive random variables {*Xi*}*i*≥1 with distribution function *FX* represent the claim sizes. The counting process {*N*(*t*), *t* ≥ 0} denotes the number of claims within [0, *t*] and is defined as *N*(*t*) = max{*n* ∈ N : *W*1 + *W*2 + ... *Wn* ≤ *<sup>t</sup>*}, where the interclaim times *Wi* are assumed to be i.i.d. with common distribution function *K*, independent of the claim sizes; see, e.g., Asmussen and Albrecher (2010). We also assume *c*E *W* > E *X*, providing a positive safety loading condition.

Now, let *T* = inf{*t* ≥ 0 : *R*(*t*) < 0} be the time of ultimate ruin. Then, the ruin probability is defined as

$$\psi(\mathfrak{u}) = \mathbb{P}(T < \infty \mid \mathcal{R}(0) = \mathfrak{u}). \tag{2}$$

The ruin probability satisfies the defective renewal equation

$$
\psi(u) = \phi \int\_0^u \psi(u - x) dH(x) + \phi \overline{H}(u), \qquad u \ge 0,\tag{3}
$$

where *φ* = *ψ*(0), *<sup>H</sup>*(*u*) is the distribution of the ascending ladder height associated with the surplus process *S*(*t*) := *u* − *R*(*t*) and *<sup>H</sup>*(*u*) = 1 − *<sup>H</sup>*(*u*), for *u* ≥ 0; see, e.g., Willmot et al. (2001). The solution to Equation (3) is the Pollaczek–Khintchine-type formula

$$\psi(u) = \sum\_{n=1}^{\infty} (1 - \phi) \phi^n \overline{H}^{\*n}(u),\tag{4}$$

i.e., *ψ*(*u*) is a geometric compound tail with geometric parameter *φ*; see Section 1.2.3 in Willmot and Woo (2017) for details.

Although Equation (4) provides a closed-form formula for the ruin probability, it is impractical, because the ladder height distribution *<sup>H</sup>*(*u*) is not available in most cases of interest. However, when the distribution *K* of the interclaim times has a rational Laplace transform, *<sup>H</sup>*(*u*) has an explicit form (Li and Garrido 2005), which we recall in the next subsection. In the sequel, we will then use this as a starting point for developing highly accurate approximations for *ψ*(*u*), which is of particular interest for heavy-tailed claim sizes.

### *The Ladder Height Distribution with Interclaim Times of Rational Laplace Transform*

We assume now that the Laplace transform of the interclaim times is a rational function of the form

$$\bar{k}(s) = \frac{\mu^\* + s\beta(s)}{\prod\_{n=1}^N (s + \mu\_n)},\tag{5}$$

where *μn* > 0, ∀*n* = 1, ... , *N*, *μ*∗ = ∏*Nn*=<sup>1</sup> *μn* and *β*(*s*) is a polynomial of degree *N* −2 or less. Obviously, E*W* = − ˜ *k*(0) = ∑*Nn*=<sup>1</sup> 1*μn* − *β*(0) *μ*∗ . If ˜*fX*(*s*) = +∞ 0 *e*<sup>−</sup>*sxdFX*(*x*) is the Laplace–Stieltjes transform (LST) of the claim sizes, it is shown in Li and Garrido (2005) that the generalised Lundberg equation

$$\frac{\prod\_{n=1}^{N} (\mu\_n - \alpha s)}{\mu^\* - \csc \beta (-\csc)} = \tilde{f}\_X(s), \qquad s \in \mathbb{C}, \tag{6}$$

has exactly *N* roots *ρ*1, *ρ*2,. . . , *ρN*, with *ρN* = 0 and (*ρn*) > 0, *n* = 1, 2, ... , *N* − 1. These roots play an important role in the evaluation of the ladder height distribution and the geometric parameter *φ*. Denote with *FX*(*x*) the complementary cumulative distribution function (ccdf) of the claim sizes, and consider the Dickson–Hipp operator

$$\mathcal{T}\_r f(\mathbf{x}) := \int\_{\mathbf{x}}^{\infty} e^{-r(y-\mathbf{x})} f(y) dy = \int\_0^{\infty} e^{-ry} f(y+\mathbf{x}) dy,\tag{7}$$

for a function *f*(*x*) (see Dickson and Hipp 2001). Moreover, let *ρ*∗ = ∏*<sup>N</sup>*−<sup>1</sup> *<sup>n</sup>*=1 *ρ<sup>n</sup>*. Then, as shown in Li and Garrido (2005), the ccdf of the ascending ladder heights is calculated via the formula

$$\overline{H}(u) = \frac{1}{\phi c^N} \sum\_{n=1}^N \frac{\mu^\* - c\rho\_n \beta (-c\rho\_n)}{\prod\_{\substack{k=1\\k \neq n}}^N (\rho\_k - \rho\_n)} \mathcal{T}\_{\overline{\rho\_n}} \overline{\mathcal{F}}\_X(u), \tag{8}$$

where

$$\phi = 1 - \frac{\mu^\*(c \mathbb{E} \mathcal{W} - \mathbb{E}X)}{\rho^\*} < 1. \tag{9}$$

Although the ladder height distribution in this model has an explicit formula, it is difficult to evaluate *ψ*(*u*) either via Equation (4) or by taking Laplace transforms (an equivalent formula to the Pollaczek–Khinchine in the Cramér–Lundberg model). In particular, this is the case when the claim sizes follow a heavy-tailed distribution, as already mentioned in Section 1. As a result, in such cases, opting for approximations seems a natural solution.

In the next section, we will study error bounds for *ψ*(*u*) when the ladder height distribution is approximated by a phase-type distribution. In particular, we will provide an efficient algorithm to construct approximations for *ψ*(*u*) when approximating *<sup>H</sup>*(*u*) by the subclass of hyperexponential distributions.

### **3. Spectral Approximation for the Ruin Probability**

The starting point for the approximation of *ψ*(*u*) is its geometric compound tail representation in Equation (4). Note that this representation is similar to the Pollaczek–Khintchine formula for *ψ*(*u*) in the Cramér–Lundberg model where *φ* is replaced by the average amount of claim per unit time *ρ* < 1 and the ladder height distribution is equal to the stationary excess claim size distribution. Therefore, following the reasoning in Vatamidou et al. (2014), we will approximate the ladder height distribution by a hyperexponential distribution (which has a rational Laplace transform), to construct approximations for the ruin probability.

### *3.1. Error Bound for the Ruin Probability*

Let *H* ˆ (*u*) be an approximation of the ladder height distribution *<sup>H</sup>*(*u*) and *ψ*<sup>ˆ</sup>(*u*) be the exact result we obtain from (4) when we use *H* ˆ (*u*). From Equation (4) and the triangle inequality, the error between the ruin probability and its approximation then is

$$\left|\psi(u) - \hat{\psi}(u)\right| \le \sum\_{n=1}^{\infty} (1 - \phi) \phi^n \left|\overrightarrow{\mathcal{H}}^{\*n}(u) - \overleftarrow{\mathcal{H}}^{\*n}(u)\right|.\tag{10}$$

If we define the sup norm distance between two distribution functions *F*1 and *F*2 as <sup>D</sup>(*<sup>F</sup>*1, *<sup>F</sup>*2) := sup*x* |*<sup>F</sup>*1(*x*) − *<sup>F</sup>*2(*x*)|, *x* ≥ 0 (also referred to as Kolmogorov metric), the following result holds.

**Theorem 1.** *A bound for the approximation error of the ruin probability is*

$$\left|\psi(\mu) - \psi(\mu)\right| \le \frac{\mathcal{D}(H, \hat{H})(1 - \phi)\phi}{\left(1 - \phi H(\mu)\right)\left(1 - \phi \hat{H}(\mu)\right)}, \qquad \forall \mu > 0.1$$

**Proof.** The result is a direct application of Theorem 4.1 of Peralta et al. (2018) by (i) choosing the functions *F* ˆ 1 and *F* ˆ 2 to be *H* and *H* ˆ , respectively; (ii) taking *ρ* = *φ*; and (iii) recognising that sup*y*<*u* +11*H*(*y*) − *H*ˆ (*y*)11, ≤ <sup>D</sup>(*<sup>H</sup>*, *H*ˆ ).

**Remark 1.** *As* lim*u*→+∞ *<sup>H</sup>*(*u*) = lim*u*→+∞ *H*ˆ (*u*) = 1*, it is immediately obvious that the bound converges to* <sup>D</sup>(*<sup>H</sup>*, *H*ˆ )*φ*/(1 − *φ*)*, which means that the bound is asymptotically uniform in u.*

To sum up, when the ladder height distribution is approximated with some desired accuracy, a bound for the ruin probability is guaranteed by Theorem 1. Although this result holds for any approximation *H* ˆ of *H*, we will in the sequel focus on hyperexponential approximations, as these lead to very tractable expressions and at the same time are sufficiently accurate for the purpose. Consequently, our next goal is to construct an algorithm to approximate the ladder height distribution by a hyperexponential distribution.
