Appl Math Comput

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.CC999999/Intramural CDC HHS/United States2016-10-01T00:00:00Z26752796PMC470470

Fourier-cosine method for ruin probabilities

In theory, ruin probabilities in classical insurance risk models can be expressed in terms of an infinite sum of convolutions, but its inherent complexity makes efficient computation almost impossible. In contrast, Fourier transforms of convolutions could be evaluated in a far simpler manner. This feature aligns with the heuristic of the recently popular work by Fang and Oosterlee, in particular, they developed a numerical method based on Fourier transform for option pricing. We here promote their philosophy to ruin theory. In this paper, we not only introduce the Fourier-cosine method to ruin theory, which has O(n)O(n) computational complexity, but we also enhance the error bound for our case that are not immediate from Fang and Oosterlee (2009). We also suggest a robust method on estimation of ruin probabilities with respect to perturbation of the moments of both claim size and claim arrival distributions. Rearrangement inequality will also be adopted to amplify the Fourier-cosine method, resulting in an effective global estimation.postprin

Markov chain approximations to scale functions of L\'evy processes

We introduce a general algorithm for the computation of the scale functions of a spectrally negative L\'evy process $X$, based on a natural weak approximation of $X$ via upwards skip-free continuous-time Markov chains with stationary independent increments. The algorithm consists of evaluating a finite linear recursion with its (nonnegative) coefficients given explicitly in terms of the L\'evy triplet of $X$. Thus it is easy to implement and numerically stable. Our main result establishes sharp rates of convergence of this algorithm providing an explicit link between the semimartingale characteristics of $X$ and its scale functions, not unlike the one-dimensional It\^o diffusion setting, where scale functions are expressed in terms of certain integrals of the coefficients of the governing SDE.Comment: 46 pages, 4 figure

Applications of the Scaled Laplace Transform in some Financial and Risk Models

In this work, we propose several approximations for the evaluation of some risk measures and option prices based on the inversion of the scaled version of the Laplace transform which was suggested by Mnatsakanov and Sarkisian (2013). The classical risk model is considered for the evaluation of probability of ultimate ruin. Approximations of the inverse function of the ruin probability is proposed and its natural extension to the computation of Value at Risk, a benchmark risk measure for insurance and finance sectors, is proposed. The recovery of the distributions of bivariate models and bivariate aggregate claims amount on insurance policies is suggested. The proposed method is also applied to the Black-Scholes model for the estimation of option prices. Simulation studies and results are presented to demonstrate the performance of the proposed method