Approximation of the ruin probability using the scaled Laplace transform inversion

Abstract

The problem of recovering the ruin probability in the classical risk model based on the scaled Laplace transform inversion is studied. It is shown how to overcome the problem of evaluating the ruin probability at large values of an initial surplus process. Comparisons of proposed approximations with the ones based on the Laplace transform inversions using a fixed Talbot algorithm as well as on the ones using the Trefethen–Weideman–Schmelzer and maximum entropy methods are presented via a simulation study.

Introduction

Recovering a function from its Laplace transform represents a very severe ill-posed inverse problem (Tikhonov and Arsenin [1]). That is why the regularization is very helpful in situations using the Laplace transform inversion. In Mnatsakanov et al. [2], the regularized inversion of the Laplace transform (Chauveau et al. [3]) has been used for approximation as well as for estimation of the ruin probability. Under the conditions on ruin probability, the upper bound for integrated squared error was derived, and rate of convergence of order 1/log n was obtained in the classical risk model. See also Shimizu [4] for application of this approach in estimating the expected discount penalty function in the L$\acute{e}$vy risk model. Our simulation study shows that the approximation rate derived in Mnatsakanov et al. [2] is not optimal. This motivated our interest to improve the rate using the scaled Laplace transform inversion suggested by Mnatsakanov and Sarkisian [5].

There are many approaches which deal with approximating the ruin probability ψ. See, for example, Gzyl et al. [6], Avram et al. [7], and Zhang et al. [8] among others. A very interesting approximation based on the Trefethen–Weideman–Schmelzer (TWS) method (see [9]) is constructed in Albrecher et al. [10]. In the latter work the authors assume that the claim size distribution represents a completely monotone function.

Note that there are several difficulties associated with inverting the Laplace transform. For example, when applying the Pad$\acute{\mathrm{e}}$ approximation one cannot guarantee the positivity of the obtained approximation of ψ, and its rate of convergence. Gzyl et al. [6] applied the maximum entropy (ME) method to approximate ψ. In order to reduce the ill-conditioning of ME approximation, the authors used the fractional moments of the exponential transform and derived an accurate approximation of ψ by reducing the ruin problem to the Hausdorff moment problem on [0, 1].

It is worth mentioning that the moment-recovered (MR) constructions proposed in Section 2 also enable us to reduce the Stieltjes moment problem to the Hausdorff one (cf. with Corollary 1 (iii), as well as Corollary 4 (ii)–(iii) in Mnatsakanov [11] and [12], respectively).

Note also that the Laplace transform inversions proposed in [5], [6], [7], [8], [9], [10], [11], [12] do not require the claim size distribution F to be a completely monotone function. See, for example, Gzyl et al. [6], where the ME method and MR-approach proposed in Mnatsakanov [12] are compared. In particular, the cases with gamma (a, β)) (with a > 1) and uniform on [0, 1] claim size distributions are considered. To conduct the comparison the authors used the formula from [12] (see also (A.4) in the Appendix) with the number of integer moments $\alpha =60$. In the case of the gamma (2, 1) model we show that the construction (17) has a better performance in terms of sup-norm when compared to the ME counterpart from [6] when α ≥ 60 and the optimal scaling parameter 1 < b ≤ e. As a result we obtained an accurate approximation of the ruin probability ψ (see Fig. 2(a) in Section 4). Calculations for large values of the parameter α ≥ 60 have been performed using a new programming language called SmartXML being developed by Artak Hakobyan (see his web page: www.oroptimizer.com). The calculations performed using SmartXML avoid many numerical problems, and perform very well for models related to the Hausdorff, Stieltjes, and Hamburger moment problems.

The main aim of present article is to derive the upper bound for the rate of MR-approximation (4) of a function f in $\mathit{sup}$-norm and demonstrate its performance in the ruin problem via a simulation study. We show that the MR-construction of ruin probability, see (17) below, with the large value of α and appropriately chosen parameter b, performs better than its ME counterpart (cf. with [6]) and is comparable with the ones using TWS and a fixed Talbot (FT) algorithms (cf. with [10]).

The remainder of this article is organized as follows. In Section 2 we introduce two MR-approximations of a function and its derivative, see (3) and (4), respectively, and provide the upper bound for MR-approximation (4); the upper bound for (3) has been already established in [5] (see also Theorem 1A in the Appendix). In Section 3, we propose two approximations (see (17) and (20)) of the ruin probability ψ, which are based on the finite number of values of Laplace (${\mathcal{L}}_{\psi }$) and Laplace–Stieltjes (L_G) transforms of ψ and $G=1-\psi ,$ respectively:

$$\begin{array}{c}\hfill {\mathcal{L}}_{\psi }\left(s\right)={\int }_0^{\infty }{e}^{-s\tau }\psi \left(\tau \right)d\tau \end{array}$$

$$\begin{array}{c}\hfill L_G\left(s\right)={\int }_0^{\infty }{e}^{-s\tau }dG\left(\tau \right),s\in [0,\infty ):={\mathbb{R}}_{+}.\end{array}$$

In Section 4 we consider two models (gamma and log-normal) in order to make a comparison with the approaches developed in [6] and [10]. In Examples 1 and 2 the performance of the proposed approximations are demonstrated graphically via Figs. 1 and 2 and Tables 1–4. In the case of the gamma (2, 1) model, for several values of α and the scaling parameter b, the maximum deviations between approximants and the true ruin probability are recorded in Table 1. When the claim size distribution is gamma, we compared the MR-approximations (17) and (20) with the approximant ψ_FT that is based on FT algorithm.

In the case when the claim size distribution is log-normal (Example 3), several values of MR-approximation ψ_α,b defined in (17) are compared with those of ψ_TWS (see Table 5 below and Table 9 in [10]).

Finally, in the Appendix, we recall two results derived in [5] and [12], where the rates of MR-approximations of a cumulative distribution function (cdf) $F:{\mathbb{R}}_{+}\to [0,1]$ as well as the Laplace transform inversions in bivariate and univariate cases are presented.