Approximating the Density of the Time to Ruin via Fourier-Cosine Series Expansion

Abstract

In this paper, the density of the time to ruin is studied in the context of the classical compound Poisson risk model. Both one-dimensional and two-dimensional Fourier-cosine series expansions are used to approximate the density of the time to ruin, and the approximation errors are also obtained. Some numerical examples are also presented to show that the proposed method is very efficient.

Estimating the Expected Discounted Penalty Function in a Compound Poisson Insurance Risk Model with Mixed Premium Income

“…For the bias Φ − Φ K,a , by similar arguments to those of Zhang [65], we obtain the following result.…”

“…For example, Chau et al [63,64] used the COS method to compute the ruin probability and Gerber-Shiu function in the Lévy risk models. Zhang [65] applied the COS method to compute the density of the time to ruin in the classical risk model. Yang et al [66] used a two-dimensional COS method to estimate the discounted density function of the deficit at ruin in the classical risk model with stochastic income where the premiums are only described by a compound Poisson process and the premium sizes are exponential distributed.…”

Valuing Guaranteed Minimum Death Benefits by Cosine Series Expansion

“…Except for option pricing, this method has been adopted in insurance ruin theory. For example, Chau et al [25,26] used the 1D COS method to compute the ruin probability and the expected discounted penalty function; Zhang [27] approximated the density function of the time to ruin by both 1D and 2D COS methods; Yang et al [28] proposed a nonparametric estimator for the deficit at ruin by the 2D COS method; Wang et al [29] and Huang et al [30] used the 1D COS method to estimate the expected discounted penalty function under some risk models with stochastic premium income. The COS method has also been used by some authors to value variable annuities.…”

Estimating the Gerber-Shiu Expected Discounted Penalty Function for Lévy Risk Model

“…For the general Gerber-Shiu function, Shimizu [36] estimated it by Laplace inversion in the compound Poisson insurance risk model. Zhang [37,38] proposed an estimator by Fourier-sinc and Fourier-cosine series expansion. Zhang and Su [39,40] and Su et al [41] proposed a more efficient estimator by Laguerre series expansion method.…”