Confidence intervals of ruin probability under L'evy surplus (original) (raw)
Related papers
Risks
A statistical inference for ruin probability from a certain discrete sample of the surplus is discussed under a spectrally negative Lévy insurance risk. We consider the Laguerre series expansion of ruin probability, and provide an estimator for any of its partial sums by computing the coefficients of the expansion. We show that the proposed estimator is asymptotically normal and consistent with the optimal rate of convergence and estimable asymptotic variance. This estimator enables not only a point estimation of ruin probability but also an approximated interval estimation and testing hypothesis.
On The Expected Discounted Penalty function for Lévy Risk Processes
North American Actuarial Journal, 2006
Dufresne et al. (1991) introduced a general risk model defined as the limit of compound Poisson processes. Such a model is either a compound Poisson process itself or a process with an infinite number of small jumps. Later, in a series of now classical papers, they studied the joint distribution of the time of ruin, the surplus before ruin and the deficit at ruin [Gerber and Shiu (1997, 1998a, 1998b), Gerber and Landry (1998)]. They work with the classical and the perturbed risk models and hint that their results can be extended to gamma and inverse Gaussian risk processes. In this paper we work out this extension in the context of a more general risk model. The construction of Dufresne et al. (1991) is based on a non-negative, non-increasing function Q that governs the jumps of the process. This function, it turns out, is the tail of the Lévy measure of the process. Our aim is to extend their work to a generalized risk model driven by an increasing Lévy process. This first paper * This research was funded by a Society of Actuaries (AERF/CKER project) and the Natural Sciences and Engineering Research Council of Canada (NSERC) operating grants 368601999 and 3116602005. presents the results for the case when the aggregate claims process is a subordinator. Embedded in this wide family of risk models we find the gamma, inverse Gaussian and generalized inverse Gaussian processes.
On a Generalization from Ruin to Default in a Lévy Insurance Risk Model
Methodology and Computing in Applied Probability, 2013
In a variety of insurance risk models, ruin-related quantities in the class of expected discounted penalty function (EDPF) were known to satisfy defective renewal equations that lead to explicit solutions. Recent development in the ruin literature has shown that similar defective renewal equations exist for a more general class of quantities than that of EDPF. This paper further extends the result on the class of functions that satisfy defective renewal equations in a spectrally negative Lévy risk models. In particular, we present an operator-based approach as an alternative analytical tool in comparison with fluctuation theoretic methods used for similar quantities in the current literature.
Simulation-based inference for the finite-time ruin probability of a surplus with a long-memory
2016
We are interested in statistical inference for the finite-time ruin probability of an insurance surplus whose claim process has a long-range dependence. As an approximated model, we consider a surplus driven by a fractional Brownian motion with the Hurst parameter H > 1/2. We can compute the ruin probability via the Monte Carlo simulations if some unknown parameters in the model are decided. From discrete samples, we estimate those unknowns, by which an asymptotically normal estimator of the ruin probability is computed. An expression of the asymptotic variance is given via the Malliavin Calculus in the estimable form. As a result, we can construct a confidence interval of the finite-time ruin probability. Since the ruin is usually rare event, an importance sampling technique is sometimes usuful in computation in practice.
Journal of Applied Probability, 2012
Consider a general bivariate Lévy-driven risk model. The surplus process Y , starting with Y 0 = x > 0, evolves according to dY t = Y t− dR t − dP t for t > 0, where P and R are two independent Lévy processes representing, respectively, a loss process in a world without economic factors and a process describing return on investments in real terms. Motivated by a conjecture of Paulsen, we study the finite-time and infinite-time ruin probabilities for the case in which the loss process P has a Lévy measure of extended regular variation and the stochastic exponential of R fulfills a moment condition. We obtain a simple and unified asymptotic formula as x → ∞, which confirms Paulsen's conjecture.
Asymptotic Ruin Probabilities of the Lévy Insurance Model under Periodic Taxation
ASTIN Bulletin, 2009
Recently, Albrecher and his coauthors have published a series of papers on the ruin probability of the Lévy insurance model under the so-called loss-carry-forward taxation, meaning that taxes are paid at a certain fixed rate immediately when the surplus of the company is at a running maximum. In this paper we assume periodic taxation under which the company pays tax at a fixed rate on its net income during each period. We devote ourselves to deriving explicit asymptotic relations for the ruin probability in the most general Lévy insurance model in which the Lévy measure has a subexponential tail, a convolution-equivalent tail, or an exponential-like tail.
Consider an insurance risk model proposed by Paulsen in a series of papers. In this model, the surplus process is described as a general bivariate Lévy-driven risk process in which one Lévy process, representing a loss process in a world without economic factors, is compounded by another Lévy process, describing return on investments. Motivated by a conjecture of Paulsen, we study the …nite-time and in…nite-time ruin probabilities for the case in which the …rst Lévy process has a heavy-tailed Lévy measure and the second one ful…lls a moment condition. We obtain a simple uni…ed asymptotic formula, which con…rms Paulsen's conjecture.
Discounted penalty function at Parisian ruin for Lévy insurance risk process
Insurance: Mathematics and Economics
In the setting of a Lévy insurance risk process, we present some results regarding the Parisian ruin problem which concerns the occurrence of an excursion below zero of duration bigger than a given threshold r. First, we give the joint Laplace transform of ruin-time and ruin-position (possibly killed at the first-passage time above a fixed level b), which generalises known results concerning Parisian ruin. This identity can be used to compute the expected discounted penalty function via Laplace inversion. Second, we obtain the q-potential measure of the process killed at Parisian ruin. The results have semi-explicit expressions in terms of the q-scale function and the distribution of the Lévy process.