**6. Conclusions**

In this paper, we considered the ruin probability of the Sparre Andersen model with heavy-tailed claim sizes and interclaim times with rational Laplace transform. Using the geometric random sum representation, we developed an explicit bound and also constructed a spectral approximation by approximating the c.m. ladder height distribution with a hyperexponential one. Our spectral approximation algorithm advances on the algorithm established in Vatamidou et al. (2014) in various aspects. We provide below a summary of our conclusions both for the spectral approximation and the bound.


To sum up, the spectral approximation is highly accurate for all values of *u* as opposed to the heavy-tail approximation, which fails to provide a good fit for small values. Moreover, it is accompanied by a rather tight bound.

Finally, note that the results of this paper are also valid for the risk model with two-sided jumps, i.e.,

$$\mathcal{R}(t) = \mu + ct + \sum\_{j=1}^{N\_+(t)} Y\_j - \sum\_{i=1}^{N\_-(t)} X\_{i\prime} \quad t \ge 0,\tag{13}$$

where *u*, *c* and *Xi* are defined as before, whereas *N*+(*t*) and *<sup>N</sup>*−(*t*) are independent Poisson processes with intensities *λ*+ and *λ*−, respectively; see, e.g., Albrecher et al. (2010). In addition, the sequence {*Yj*}*i*≥1 of i.i.d. r.v.'s, independent of {*Xi*}*j*≥1, *N*+(*t*) and *<sup>N</sup>*−(*t*), and having the common d.f. *GY* that belongs to the class of distributions with rational Laplace transform, are the sizes of premium payments. The positive security loading condition in this model becomes *c* + *λ*+E*Y* > *λ*−E*X*.

Let *τn* be the time when the *n*th claim occurs with *τ*0 = 0. As ruin occurs only at the epochs when claims occur, we define the discrete time process *R* ˇ = {*R*ˇ *n* : *n* = 0, 1, 2, ... }, where *R*ˇ 0 = 0 and *R* ˇ *n* = *R* <sup>ˇ</sup>(*<sup>τ</sup>n*), which denotes the surplus immediately after the *n*th claim, i.e.,

$$\check{R}\_{\text{ll}} = \boldsymbol{\mu} + c\boldsymbol{\tau}\_{\text{n}} + \sum\_{j=1}^{N\_+\left(\tau\_{\text{n}}\right)} Y\_j - \sum\_{i=1}^{n} X\_i = \boldsymbol{\mu} + c\boldsymbol{\tau}\_{\text{n}} - \sum\_{i=1}^{n} X\_{i\prime} \qquad \boldsymbol{n} = 0, 1, 2, \dots, \tag{14}$$

where *τ*ˇ*n* = *τn* + ∑*<sup>N</sup>*+(*<sup>τ</sup>n*) *j*=1 *Yj*/*c* with *τ*ˇ0 = 0. Equation (14) corresponds to the discrete-time embedded process of the Sparre Andersen risk model (1), and the counting process *N*(*t*) denotes the number of claims up to time *t* with the modified interclaim times *Wi* = *τ*ˇ*j* − *<sup>τ</sup>*<sup>ˇ</sup>*j*−1. Clearly,

$$\tilde{k}(s) = \frac{\lambda\_-}{\lambda\_- + s + \lambda\_+ \left(1 - \tilde{g}\_Y(s/c)\right)},\tag{15}$$

where *g*˜*Y*(*s*) = +∞ 0 *e*<sup>−</sup>*sxdGY*(*x*) is the Laplace transform of the premium payments; see Dong and Liu (2013). Let now *T* ˇ = inf{*t* ≥ 0 | *R*<sup>ˇ</sup>(*t*) < 0} and *ψ*<sup>ˇ</sup>(*u*) = P(*T*<sup>ˇ</sup> < ∞ | *R*<sup>ˇ</sup>(0) = *<sup>u</sup>*). Obviously, *ψ* <sup>ˇ</sup>(*u*) = *ψ*(*u*).

**Author Contributions:** Both authors contributed equally.

**Funding:** Financial support from the Swiss National Science Foundation Project 200021\_168993 is gratefully acknowledged.

**Conflicts of Interest:** The authors declare no conflict of interest.
