Analytical best upper bounds on stop-loss premiums

We provide a list of best upper bounds on the stop-loss premium E(X?t)₊ corresponding to the risk X and the retention limit t. Various information (moments, unimodality, symmetry,…) on the distribution F of X is taken into account.     

Further improved recursions for a class of compound Poisson distributions

In the present paper we develop more efficient recursive formulae for the evaluation of the t-order cumulative function Γ^th(x) and the t-order tail probability Λ^th(x) of the class of compound Poisson distributions in the case where the derivative of the probability generating function of the claim amounts can be written as a ratio of two polynomials. These efficient recursions can be applied for the exact evaluation of the probability function (given by De Pril [De Pril, N., 1986a. Improved recursions for some compound Poisson distributions. Insurance Math. Econom. 5, 129-132]), distribution function, tail probability, stop-loss premiums and t-order moments of stop-loss transforms of compound Poisson distributions. Also, efficient recursive algorithms are given for the evaluation of higher-order moments and r-order factorial moments about any point for this class of compound Poisson distributions. Finally, several examples of discrete claim size distributions belonging to this class are also given.     

Bounds on compound distributions and stop-loss premiums

In the actuarial literature a lot of attention is given to the approximation of aggregate claims distributions by compound Poisson and Polya distributions and their subsequent numerical evaluation. The present contribution derives bounds for the tail of compound distributions and stop-loss premiums. The bounds are obtained in an elementary manner based on a version of the Chebyshev inequality. The main point of this contribution consists in deriving bounds with explicit dependence on the distribution function itself as well as on some partial moments of it.     

Restricted one way analysis of variance using the empirical likelihood ratio test

Empirical likelihood (EL) ratio tests are developed for testing for or against the hypothesis that k-population means μ₁,μ₂,…,μ_k are isotonic with respect to some quasi-order ? on {1,2,…,k}. The null asymptotic distributions are derived and are shown to be of chi-bar squared type. The asymptotic power of the proposed test for testing for equality of these means against the order restriction is derived under contiguous alternatives and a simulation study is carried out to investigate the finite sample behaviors of this test. In addition, an adjusted EL test is used to improve the small size performance of our test and an example is also discussed to illustrate the theoretical results.     

How retention levels influence the variability of the total risk under reinsurance

Recently, Escudero and Ortega (Insur. Math. Econ. 43:255–262, 2008) have considered an extension of the largest claims reinsurance with arbitrary random retention levels. They have analyzed the effect of some dependencies on the Laplace transform of the retained total claim amount. In this note, we study how dependencies influence the variability of the retained and the reinsured total claim amount, under excess-loss and stop-loss reinsurance policies, with stochastic retention levels. Stochastic directional convexity properties, variability orderings, and bounds for the retained and the reinsured total risk are given. Some examples on the calculation of bounds for stop-loss premiums (i.e., the expected value of the reinsured total risk under this treaty) and for net premiums for the cedent company under excess-loss, and complementary results on convex comparisons of discounted values of benefits for the insurer from a portfolio with risks having random policy limits (deductibles) are derived.      

Convex bounds on multiplicative processes, with applications to pricing in incomplete markets

Extremal distributions have been extensively used in the actuarial literature in order to derive bounds on functionals of the underlying risks, such as stop-loss premiums or ruin probabilities, for instance. In this paper, the idea is extended to a dynamic setting. Specifically, convex bounds on multiplicative processes are derived. Despite their relative simplicity, the extremal processes are shown to produce reasonably accurate bounds on option prices in the classical trinomial model for incomplete markets.     

The influence of sequential extremal processes on the partial sum process

In this paper we derive general upper bounds for the total variation distance between the distributions of a partial sum process in row-wise independent, non-negative triangular arrays and the sum of a fixed number of corresponding extremal processes. As a special case we receive bounds for the supremum distance between the distribution functions of a partial sum and the sum of corresponding upper extremes which improve upon existing results. The outcome may be interpreted as the influence of large insurance claims on the total loss. Moreover, under an additional infinitesimal condition we also prove explicit bounds for limits of the above quantities. Thereby we give a didactic and elementary proof of the Ferguson–Klass representation of Lévy processes on ?_?≥?0 which reflects the influence of extremal processes in insurance.     

Best bounds on the stop loss premium in case of known range,expectation, variance and mode of the risk

The following problems are solved analytically in this note. Find the maximum and the minimum of the stop-loss premium E(R ? t)₊ corresponding to the risk R and retention limit t when R has a unimodal distribution with known range, mode, expectation and variance. Find the distribution leading to this maximum and minimum.     

Weak Convergence Results for Multiple Generations of a Branching Process

We establish limit theorems involving weak convergence of multiple generations of critical and supercritical branching processes. These results arise naturally when dealing with the joint asymptotic behavior of functionals defined in terms of several generations of such processes. Applications of our main result include a functional central limit theorem (CLT), a Darling–Erdös result, and an extremal process result. The limiting process for our functional CLT is an infinite dimensional Brownian motion with sample paths in the infinite product space (C ₀[0,1])^∞, with the product topology, or in Banach subspaces of (C ₀[0,1])^∞ determined by norms related to the distribution of the population size of the branching process. As an application of this CLT we obtain a central limit theorem for ratios of weighted sums of generations of a branching processes, and also to various maximums of these generations. The Darling–Erdös result and the application to extremal distributions also include infinite-dimensional limit laws. Some branching process examples where the CLT fails are also included.     

Maximization of the variance of a stop-loss reinsured risk

We solve the following problems: Find the maximum variance of a stop-loss reinsured risk (R?t)₊ when only the range of the risk R and its two first moments are known. Find the distribution of R leading to that maximum variance.The solution is complete for all practical values of the involved parameters.     

Maximization,under equality constraints,of a functional of a probability distribution

Let $\text{F}$ be a family of probability distributions. Let O, C₁…C_n be real functions on $\text{F}$. Let z₁…z_n be real numbers. Then we consider the problem of maximization of the object function O(F)(F?$\text{F}$) under the equality constraints C₁(F)=z₁(i=1,…,n) . The theory is developed in order to solve problems of the following kind: Find the maximal variance of a stop-loss reinsured risk under partial information on the risk such as its range and two first moments.     

On the Hong–Krahn–Szego inequality for the p-Laplace operator

Given an open set Ω, we consider the problem of providing sharp lower bounds for λ ₂(Ω), i.e. its second Dirichlet eigenvalue of the p-Laplace operator. After presenting the nonlinear analogue of the Hong–Krahn–Szego inequality, asserting that the disjoint unions of two equal balls minimize λ ₂ among open sets of given measure, we improve this spectral inequality by means of a quantitative stability estimate. The extremal cases p = 1 and p = ∞ are considered as well.     

Bounds on solutions to the boundary-free incompressible Navier-Stokes initial value problem

The integral formulation of the Navier-Stokes initial value problem for boundary-free, incompressible fluid flow is used to establish Volterra integral inequalities on the physical velocity field u_μ(x, t). Standard comparison theorems for Volterra equations are then applied to obtain weakly singular, nonlinear Volterra equations of the second kind for upper bounds on $\text{¦ u}{}_{\mathrm{\mu }}\text{(}\text{x}\text{, t)¦}$. With local existence and uniqueness guaranteed and smoothness of solutions characterized by results on Volterra equations, solutions for bounds may be obtained by standard analytical and numerical methods. A specific example is considered. Existence and uniqueness of local solutions u_μ(x, t) to the Navier-Stokes problem is then guaranteed within the radius of convergence of bounds on $\text{¦ u}{}_{\mathrm{\mu }}\text{(}\text{x}\text{, t)¦}$, thereby precluding local breakdown phenomena. In addition, bounds on $\text{¦ u}{}_{\mathrm{\mu }}\text{(}\text{x}\text{, t)¦}$ are used to obtain improved duration times for convergence of local iteration solutions for u_μ(x, t). Finally, a new technique for establishing sufficient conditions for global existence, based on successive application of Volterra comparison theorems, is indicated.     

Properties of solutions of Riccati equation

The paper studies some properties of solutions of the Riccati equation

$y'(t) + a(t)y^2 (t) + b(t)y(t) + c(t) = 0$

on a semiaxis [t ₀, +∞) for different types of initial value sets. Two types of solutions are singled out: normal, that are in a sense stable, and extremal, that are non-stable in the Lyapunov sense. Relations expressing the extremal solutions by means of a given normal solution in quadratures and elementary functions are obtained and some relations between solutions the extendable to [t ₀, +∞) are derived.

     

Inequalities for characteristic functions of multidimensional distributions

We obtain lower and upper bounds for the absolute values of characteristic functions of multivariate distributions F and also derive a lower bound on the norm of the zeroes of a characteristic function in terms of moments of the norm of the random vector with distribution F. Similar results are obtained for characteristic functions of probability measures on a separable Hilbert space.     

Expectiles,Omega Ratios and Stochastic Ordering

In this paper we introduce the expectile order, defined by X ≤ _e Y if e _α (X) ≤e _α (Y) for each α ∈ (0, 1), where e _α denotes the α-expectile. We show that the expectile order is equivalent to the pointwise ordering of the Omega ratios, and we derive several necessary and sufficient conditions. In the case of equal means, the expectile order can be easily characterized by means of the stop-loss transform; in the more general case of different means we provide some sufficient conditions. In contrast with the more common stochastic orders such as ≤ _{s t} and ≤ _{c x} , the expectile order is not generated by a class of utility functions and is not closed with respect to convolutions. As an illustration, we compare the ≤ _{s t} , ≤ _{i c x} and ≤ _e orders in the family of Lomax distributions and compare Lomax distributions fitted to real world data of natural disasters in the U.S. caused by different sources of weather risk like storms or floods.     

On the normal approximation of a binomial random sum

We present upper bounds of the L _s norms of the normal approximation for random sums of independent identically distributed random variables X ₁ , X ₂ , . . . with finite absolute moments of order 2 + δ, 0 < δ ≤ 1, where the number of summands N is a binomial random variable independent of the summands X ₁ , X ₂ , . . . . The upper bounds obtained are of order (E N) ^?δ/2 for all 1 ≤ s ≤ ∞.     

A classification of invariant distributions and convergence of imprecise Markov chains

We analyse the structure of imprecise Markov chains and study their convergence by means of accessibility relations. We first identify the sets of states, so-called minimal permanent classes, that are the minimal sets capable of containing and preserving the whole probability mass of the chain. These classes generalise the essential classes known from the classical theory. We then define a class of extremal imprecise invariant distributions and show that they are uniquely determined by the values of the upper probability on minimal permanent classes. Moreover, we give conditions for unique convergence to these extremal invariant distributions.     

Sharp distribution free lower bounds for spread options and the corresponding optimal subreplicating portfolios

We derive in closed form distribution free lower bounds and optimal subreplicating strategies for spread options in a one-period static arbitrage setting. In the case of a continuum of strikes, we complement the optimal lower bound for spread options obtained in [Rapuch, G., Roncalli, T., 2002. Pricing multiasset options and credit derivatives with copula, Credit Lyonnais, Working Papers] by describing its corresponding subreplicating strategy. This result is explored numerically in a Black-Scholes and in a CEV setting. In the case of discrete strikes, we solve in closed form the optimization problem in which, for each asset S₁ and S₂, forward prices and the price of one option are used as constraints on the marginal distributions of each asset. We provide a partial solution in the case where the marginal distributions are constrained by two strikes per asset. Numerical results on real NYMEX (New York Mercantile Exchange) crack spread option data show that the one discrete lower bound can be far and also very close to the traded price. In addition, the one strike closed form solution is very close to the two strike.