Le XIImecongrès international d'actuaires

Abstract

We study the following inverse thinning problem for renewal processes: for which completely monotone functions f is f/(p+qf), 0?p?1, q=1-p, completely monotone? A characterisation of such f's is given. We also study the case when f comes from a gamma distribution, and present some ideas for more general results.

The intention of this note is to add some information to a paper by Yannaros (1985), in which thinned renewal processes are considered. Let X_n , n?1, be i.i.d. non-negative random variables, distributed according to a probability measure µ, and let S_n = X ₁+...+X_n (with S ₀=0) be the corresponding renewal process. Replacing µ by the probability measure ν=∑_n?1 pq^n-1 µ^n* (µ^n* =µ* ... µ*, n times) we get a new renewal process, obtained from the original one by independently at each stage preserving the process with probability p. Here and below q= 1-p, and to avoid trivialities we assume that 0 Let µ^(s) = ∫[0,∞) exp (-sx)µ(dx) , s?0, denote the Laplace transform of µ. Then ν^=pµ/(1-µ^). We will study the inverse problem: given a completely monotone function ψ, when does ψ(p+qψ) define a completely monotone function. A complete characterisation, and some of its consequences, is given in §§ 1–3 below. In §§ 4–5 we study the gamma distribution. It is proved that the inverse problem has a negative solution when the parameter a > 1, i.e. 1/(p + q(1 + s) ^a ) is not completely monotone then. In Yannaros (1985) this was proved for a=2, 3, ... with entirely different methods. (That 1/(p+q(1+s)^a is completely monotone for 0?a?1 is easily seen; cf. Yannaros (1985). Finally, in § 6 we give some suggestions to more general results related to thinning. Perhaps the most interesting problem is to find sufficiently general conditions for an absolutely monotone function to have a Bernstein function as its inverse.     

A generalized penalty function for a class of discrete renewal processes

Analysis of a generalized Gerber–Shiu function is considered in a discrete-time (ordinary) Sparre Andersen renewal risk process with time-dependent claim sizes. The results are then applied to obtain ruin-related quantities under some renewal risk processes assuming specific interclaim distributions such as a discrete K _n distribution and a truncated geometric distribution (i.e. compound binomial process). Furthermore, the discrete delayed renewal risk process is considered and results related to the ordinary process are derived as well.     

Potential measures and expected present value of operating costs until ruin in renewal risk models with general interclaim times

In this paper, a Sparre Andersen risk process with arbitrary interclaim time distribution is considered. We analyze various ruin-related quantities in relation to the expected present value of total operating costs until ruin, which was first proposed by Cai et al. [(2009a). On the expectation of total discounted operating costs up to default and its applications. Advances in Applied Probability 41(2), 495–522] in the piecewise-deterministic compound Poisson risk model. The analysis in this paper is applicable to a wide range of quantities including (i) the insurer's expected total discounted utility until ruin; and (ii) the expected discounted aggregate claim amounts until ruin. On one hand, when claims belong to the class of combinations of exponentials, explicit results are obtained using the ruin theoretic approach of conditioning on the first drop via discounted densities (e.g. Willmot [(2007). On the discounted penalty function in the renewal risk model with general interclaim times. Insurance: Mathematics and Economics 41(1), 17–31]). On the other hand, without any distributional assumption on the claims, we also show that the expected present value of total operating costs until ruin can be expressed in terms of some potential measures, which are common tools in the literature of Lévy processes (e.g. Kyprianou [(2014). Fluctuations of L'evy processes with applications: introductory lectures, 2nd ed. Berlin Heidelberg: Springer-Verlag]). These potential measures are identified in terms of the discounted distributions of ascending and descending ladder heights. We shall demonstrate how the formulas resulting from the two seemingly different methods can be reconciled. The cases of (i) stationary renewal risk model and (ii) surplus-dependent premium are briefly discussed as well. Some interesting invariance properties in the former model are shown to hold true, extending a well-known ruin probability result in the literature. Numerical illustrations concerning the expected total discounted utility until ruin are also provided.     

On stationary point processes and Markov chains

Abstract

Introductory. In the theory of random processes we may distinguish between ordinary processes and point processes. The former are concerned with a quantity, say x (t), which varies with time t, the latter with events, incidences, which may be represented as points along the time axis. For both categories, the stationary process is of great importance, i. e., the special case in which the probability structure is independent of absolute time. Several examples of stationary processes of the ordinary type have been examined in detail (see e. g. H. Wold 1). The literature on stationary point processes, on the other hand, has exclusively been concerned with the two simplest cases, viz. the Poisson process and the slightly more general process arising in renewal theory (see e. g. J. Doob 3).     

Optimal capital and risk allocations for law- and cash-invariant convex functions

In this paper we provide a complete solution to the existence and characterization problem of optimal capital and risk allocations for not necessarily monotone, law-invariant convex risk measures on the model space L ^p for any p∈[1,∞]. Our main result says that the capital and risk allocation problem always admits a solution via contracts whose payoffs are defined as increasing Lipschitz-continuous functions of the aggregate risk. Filipović is supported by WWTF (Vienna Science and Technology Fund). Svindland gratefully acknowledges financial support from Munich Re Grant for doctoral students and hospitality of the Research Unit of Financial and Actuarial Mathematics, Vienna University of Technology. We thank Beatrice Acciaio and Walter Schachermayer for fruitful discussions and an anonymous referee for helpful remarks.     

Sequential Quasi-Credibility for Scale Dispersion Models

The sequential approach to credibility, developed by Landsman and Makov [(1999a) On stochastic approximation and credibility. Scand. Actuarial J. 1, 15-31; (1999b) Sequential credibility evaluation for symmetric location claim distributions. Insurance: Math. Econ. 24, 291-300] is extended to the scale dispersion family, which contains distributions often used in actuarial science: log-normal, Weibull, Half normal, Stable, Pareto, to mention only a few. For members of this family a sequential quasi-credibility formula is devised, which can also be used for heavy tailed claims. The results are illustrated by a study of log-normal claims.     

Nils Ekholm

Abstract

The problem of χ² tests of a linear hypothesis H₀ for ‘matched samples’ in attribute data has been discussed earlier by the author (Bennett, 1967, 1968). This note presents corresponding results for the hypothesis that the multinomial probabilities p satisfy (c ?1) functional restrictions: F ₁(p) = 0, ... , F _C?1(p) = 0. An explicit relationship between the usual ‘goodness-of-fit’ χ² and the modified minimum χ² (=χ*²) of Jeffreys (1938) and Neyman (1949) is demonstrated for this situation. An example of the test for the 2 × 2 × 2 contingency table is given and compared with the solution of Bartlett (1935).     

A Stochastic Model for Pensionable Service

Abstract

In this paper, a stochastic model is developed for the prediction of the employee's cumulative pensionable service over his working life, as a function of his unemployment periods and his completed lengths of service, under a service requirement constraint. Distributional and first passagetime problems are considered for the total pensionable service and a generalized renewal equation is formally solved. Further useful results are obtained and numerical applications are given.     

Optimal risk sharing with non-monotone monetary functionals

We consider the problem of sharing pooled risks among n economic agents endowed with non-necessarily monotone monetary functionals. In this framework, results of characterization and existence of optimal solutions are easily obtained as extensions from the convex risk measures setting. Moreover, the introduction of the best monotone approximation of non-monotone functionals allows us to compare the original problem with the one which involves only ad hoc monotone criteria. The explicit calculation of optimal risk sharing rules is provided for particular cases, when agents are endowed with well-known preference relations.      

Private Investment, Public Aid and Endogenous Divergence in the Evolution of Urban Neighborhoods

This paper offers a novel explanation for urban blight and endogenous divergence in the overall quality and wealth of neighborhoods and simultaneously derives the salient features of actual urban renewal and other aid programs from optimizing government behavior based on collective public preferences. These features appear when the objective of such public aid programs is to restore the ex ante distribution of wealth or property values within a blighted neighborhood, while equilibria exhibiting deficient levels of private investment and blight itself can arise when residents accurately anticipate the potential provision of public aid to an affected neighborhood and ex ante investment in private insurance diminishes neighborhood eligibility for such aid. Examples of antipodal equilibria in which urban renewal programs entirely crowd out local private investment or in which neighborhood residents invest in efficient levels of private mitigation illustrate these results, which stand in direct contrast to both traditional explanations of urban blight and to the new “social-interaction” models of neighborhood divergence.     

On a Class of Renewal Risk Processes

Abstract

In this paper I show how methods that have been applied to derive results for the classical risk process can be adapted to derive results for a class of risk processes in which claims occur as a renewal process. In particular, claims occur as an Erlang process. I consider the problem of finding the survival probability for such risk processes and then derive expressions for the probability and severity of ruin and for the probability of absorption by an upper barrier. Finally, I apply these results to consider the problem of finding the distribution of the maximum deficit during the period from ruin to recovery to surplus level 0.     

Ruin probabilities expressed in terms of ladder height distributions

Abstract

During the latest few years much attention has been given to the study of the ruin problem of a risk business when the epochs of claims form a renewal process. The study of this problem was initiated by E. S. Andersen (1957). Thorin has then in a series of papers (Thorin, 1970, 1971a, 1971b) shown that the Wiener-Hopf technique, originally developed by Cramer (1955) in the case of a Poisson process, can be used in this more general case, and Takacs (1970) has derived results similar to those of Thorin by an entirely new technique.     

Drawdown analysis for the renewal insurance risk process

In this paper, we study some drawdown-related quantities in the context of the renewal insurance risk process with general interarrival times and phase-type distributed jump sizes. We make use of some recent results on the two-sided exit problem for the spectrally negative Markov additive process and a fluid flow analogy between certain queues and risk processes to solve for the two-sided exit problem of the renewal insurance risk process. The two-sided exit quantities are later shown to be central to the analysis of drawdown quantities including the drawdown time, the drawdown size, the running maximum (minimum) at the drawdown time, the last running maximum time prior to drawdown, the number of jumps before drawdown and the number of excursions from running maximum before drawdown. Finally, we consider another application of our methodology for the study of the expected discounted dividend payments until ruin.     

Remark on a theorem by Gurland

Abstract

Let X ₁,X ₂,...,X _n be a random sample of size from a distribution with probability density function p(x|θ), where the unknown parameter θ belongs to a non-degenerate interval I. The unknown true value of θ will be denoted by θ₀.     

Testing for Volatility Jumps in the Stochastic Volatility Process

This paper considers the stochastic volatility process with contemporaneous and correlated jumps in returns and volatility, which was proposed by Eraker, B., Johannes, M. and Poison, N. G. (Journal of Finance 53, 2003, 1269--1300) and proposes the Lagrange multiplier test for the presence of jumps in volatility. The test statistic is derived by regarding the degenerate density of volatility jumps with zero variance under the null as Dirac's delta function. The correlation parameter between jumps, which is a nuisance parameter unidentified under the null, is cancelled out in this test statistic and hence the test is free from the Davies problem (Davies, R. B., Biometrika 64, 1977, 247–254).     

Solvable local and stochastic volatility models: supersymmetric methods in option pricing

In this paper we provide an extensive classification of one- and two-dimensional diffusion processes which admit an exact solution to the Kolmogorov (and hence Black–Scholes) equation (in terms of hypergeometric functions). By identifying the one-dimensional solvable processes with the class of integrable superpotentials introduced recently in supersymmetric quantum mechanics, we obtain new analytical solutions. In particular, by applying supersymmetric transformations on a known solvable diffusion process (such as the Natanzon process for which the solution is given by a hypergeometric function), we obtain a hierarchy of new solutions. These solutions are given by a sum of hypergeometric functions, generalizing the results obtained in a paper by Albanese et al. (Albanese, C., Campolieti, G., Carr, P. and Lipton, A., Black–Scholes goes hypergeometric. Risk Mag., 2001, 14, 99–103). For two-dimensional processes, more precisely stochastic volatility models, the classification is achieved for a specific class called gauge-free models including the Heston model, the 3?/?2-model and the geometric Brownian model. We then present a new exact stochastic volatility model belonging to this class.     

Stationarity and the start of a renewal process

Abstract

In his short article Professor Seal considers inter alia the suitability of the renewal approach to the occurrence scheme in risk theory. He gives a catalog of four facts which according to him make the renewal approach less reasonable. About the two first facts he himself says that they may be removed as more actual experience becomes available so I will not discuss them but consider the two last only     

Numerical evaluation of the compound Poisson distribution: Recursion or Fast Fourier Transform?

1. The problem

The finite vector p=(p ₁,p ₂, ...,p_s ) defines a probability distribution on the integers 1,2, ...,s.