Quantile hedging for equity-linked contracts

We consider an equity-linked contract whose payoff depends on the lifetime of the policy holder and the stock price. We provide the best strategy for an insurance company assuming limited capital for the hedging. The main idea of the proof consists in reducing the construction of such strategies for a given claim to a problem of superhedging for a modified claim, which is the solution to a static optimization problem of the Neyman-Pearson type. This modified claim is given via some sets constructed in an iterative way. Some numerical examples are also given.     

On the mode-change problem for random measures

The classical change-point problem in modern terms, i.e., the mode-change problem, is stated for sufficiently general set-indexed random processes, namely for random measures. A method is shown for solving this problem both in the general form and for the intensity of compound Poisson random measures. The results obtained are novel for the change-point problem, too.     

Arranging apples in an array

Departing from a real-world problem we consider a greedy algorithm which either finds a 0–1 matrix with prescribed row and column sums or terminates with the conclusion that no such matrix exists.The problem has been pending in the literature for about 30 years. Our proof of the validity of the algorithm is established along with an account of earlier findings and appears to be considerably simpler than any other proof hitherto proposed. Various open questions conclude the paper.Dedicated to Peter Naur on the occasion of his 60th birthday     

Estimation of a distribution under generalized censoring

This paper focuses on the problem of the estimation of a distribution on an arbitrary complete separable metric space when the data points are subject to censoring by a general class of random sets. If the censoring mechanism is either totally observable or totally ordered, a reverse probability estimator may be defined in this very general framework. Functional central limit theorems are proven for the estimator when the underlying space is Euclidean. Applications are discussed, and the validity of bootstrap methods is established in each case.     

K-sample subsampling in general spaces: The case of independent time series

The problem of subsampling in two-sample and K-sample settings is addressed where both the data and the statistics of interest take values in general spaces. We focus on the case where each sample is a stationary time series, and construct subsampling confidence intervals and hypothesis tests with asymptotic validity. Some examples are also given, and the problem of optimal block size choice is discussed.     

On qualitative robustness of support vector machines

Support vector machines (SVMs) have attracted much attention in theoretical and in applied statistics. The main topics of recent interest are consistency, learning rates and robustness. We address the open problem whether SVMs are qualitatively robust. Our results show that SVMs are qualitatively robust for any fixed regularization parameter λ. However, under extremely mild conditions on the SVM, it turns out that SVMs are not qualitatively robust any more for any null sequence λ_n, which are the classical sequences needed to obtain universal consistency. This lack of qualitative robustness is of a rather theoretical nature because we show that, in any case, SVMs fulfill a finite sample qualitative robustness property.For a fixed regularization parameter, SVMs can be represented by a functional on the set of all probability measures. Qualitative robustness is proven by showing that this functional is continuous with respect to the topology generated by weak convergence of probability measures. Combined with the existence and uniqueness of SVMs, our results show that SVMs are the solutions of a well-posed mathematical problem in Hadamard’s sense.     

Does stock return predictability affect ESO fair value?

Executive Stock Options (ESOs) are modified American options that cannot be valued using standard methods. With a few exceptions, the literature has discussed the ESO fair value by assuming unpredictable stock returns which are not supported by the available empirical evidence. In this paper we obtain the fair value of American ESOs when stock returns are predictable and, specifically, driven by the trending Ornstein–Uhlenbeck process of Lo and Wang (1995). We solve the executive’s portfolio allocation problem for a simple buy-and-hold strategy when his wealth can be distributed between a risk-free asset and a market portfolio. This problem is jointly solved with the executive’s optimal exercise policy. We find that executives tend to wait longer the higher the predictability, independently of the composition of executive’s asset menu. We have also analyzed the implications under the FAS123R proposals for the ESO fair value and found that, even for low autocorrelations, there is a meaningful mispricing when unpredictable returns are erroneously assumed.     

Finite homogeneous deformations of symmetrically loaded compressible membranes

The equilibrium problem of nonlinear, isotropic and hyperelastic square membranes, stretched by a double symmetric system of dead loads, is investigated. Depending on the form of the stored energy function, the problem considered may admit asymmetric solutions in addition to the expected symmetric solutions. For compressible materials, the mathematical condition allowing the computation of these asymmetric solutions is given. Moreover, explicit expressions for evaluating critical loads and bifurcation points are derived. Results and basic relations obtained for general isotropic materials are then specialized for a compressible Mooney–Rivlin material and a broad numerical analysis is performed. The qualitatively more interesting branches of asymmetric equilibrium are shown and the influence of the material parameters is discussed. Finally, using the energy criterion, some stability considerations are made.     

Finding squares and rectangles in sets of points

A number of problems concerning sets of points in the plane are studied, e.g. establishing whether it contains a subset of size 4, which are the vertices of a square or rectangle. Both the problems of finding axis-parallel squares and rectangles, and arbitrarily oriented squares and rectangles are studied. Efficient algorithms are obtained for all of them. Furthermore, we investigate the generalizations tod-dimensional space, where the problem is to find hyperrectangles and hypercubes. Also, upper and lower bounds are given for combinatorial problems on the maxium number of subsets of size 4, of which the points are the vertices of a square or rectangle. Then we state an equivalence between the problem of finding rectangles, and the problem of findingK _{2, 2} subgraphs in bipartite graphs. Thus we immediately have an efficient algorithm for this graph problem.This work was partially supported by the ESPRIT Basic Research Action No. 3075 (project ALCOM). Work of the second author was also supported by the Dutch Organisation for Scientific Research (N.W.O.).     

Homogeneity tests based on several progressively Type-II censored samples

In this paper, we discuss the problem of testing the homogeneity of several populations when the available data are progressively Type-II censored. Defining for each sample a univariate counting process, we can modify all the methods that were developed during the last two decades (see e.g. [P.K. Andersen, Ø. Borgan, R. Gill, N. Keiding, Statistical Models Based on Counting Processes, Springer, New York, 1993]) for use to this problem. An important aspect of these tests is that they are based on either linear or non-linear functionals of a discrepancy process (DP) based on the comparison of the cumulative hazard rate (chr) estimated from each sample with the chr estimated from the whole sample (viz., the aggregation of all the samples), leading to either linear tests or non-linear tests. Both these kinds of tests suffer from some serious drawbacks. For example, it is difficult to extend non-linear tests to the K-sample situation when K?3. For this reason, we propose here a new class of non-linear tests, based on a chi-square type functional of the DP, that can be applied to the K-sample problem for any K?2.     

Lévy risk model with two-sided jumps and a barrier dividend strategy

In this paper, we consider a general Lévy risk model with two-sided jumps and a constant dividend barrier. We connect the ruin problem of the ex-dividend risk process with the first passage problem of the Lévy process reflected at its running maximum. We prove that if the positive jumps of the risk model form a compound Poisson process and the remaining part is a spectrally negative Lévy process with unbounded variation, the Laplace transform (as a function of the initial surplus) of the upward entrance time of the reflected (at the running infimum) Lévy process exhibits the smooth pasting property at the reflecting barrier. When the surplus process is described by a double exponential jump diffusion in the absence of dividend payment, we derive some explicit expressions for the Laplace transform of the ruin time, the distribution of the deficit at ruin, and the total expected discounted dividends. Numerical experiments concerning the optimal barrier strategy are performed and new empirical findings are presented.     

Superefficient estimation of the marginals by exploiting knowledge on the copula

We consider the problem of estimating the marginals in the case where there is knowledge on the copula. If the copula is smooth, it is known that it is possible to improve on the empirical distribution functions: optimal estimators still have a rate of convergence n^−1/2, but a smaller asymptotic variance. In this paper we show that for non-smooth copulas it is sometimes possible to construct superefficient estimators of the marginals: we construct both a copula and, exploiting the information our copula provides, estimators of the marginals with the rate of convergence logn/n.     

Second-order accuracy of depth-based bootstrap confidence regions

We consider the problem of setting bootstrap confidence regions for multivariate parameters based on data depth functions. We prove, under mild regularity conditions, that depth-based bootstrap confidence regions are second-order accurate in the sense that their coverage error is of order n⁻¹, given a random sample of size n. The results hold in general for depth functions of types A and D, which cover as special cases the Tukey depth, the majority depth, and the simplicial depth. A simulation study is also provided to investigate empirically the bootstrap confidence regions constructed using these three depth functions.     

Variance component testing in semiparametric mixed models

It is of considerable interest to test for heteroscedasticity in statistical studies. In this paper, we investigate such a problem under the framework of a semiparametric mixed model. A score test is proposed for the hypothesis that all the variance components are zero. We establish the asymptotic property of the test, and examine its performance in a simulation study. The test is illustrated with the analysis of a longitudinal study of measurements of serum creatinine.     

Efficient reduction for path problems on circular-arc graphs

An efficient reduction process for path problems on circular-arc graphs is introduced. For the parity path problem, this reduction gives anO(n+m) algorithm for proper circular-arc graphs, and anO(n+m) algorithm for general circular-arc graphs. This reduction also gives anO(n+m) algorithm for the two path problem on circular-arc graphs.     

Fractional-order regularization and wavelet approximation to the inverse estimation problem for random fields

The least-squares linear inverse estimation problem for random fields is studied in a fractional generalized framework. First, the second-order regularity properties of the random fields involved in this problem are analysed in terms of the fractional Sobolev norms. Second, the incorporation of prior information in the form of a fractional stochastic model, with covariance operator bicontinuous with respect to a certain fractional Sobolev norm, leads to a regularization of this problem. Third, a multiresolution approximation to the class of linear inverse problems considered is obtained from a wavelet-based orthogonal expansion of the input and output random models. The least-squares linear estimate of the input random field is then computed using these orthogonal wavelet decompositions. The results are applied to solving two important cases of linear inverse problems defined in terms of fractional integral operators.     

Valuation and hedging of participating life-insurance policies under management discretion

The valuation and hedging of participating life insurance policies, also known as with-profits policies, is considered. Such policies can be seen as European path-dependent contingent claims whose underlying security is the investment portfolio of the insurance company that sold the policy. The fair valuation of these policies is studied under the assumption that the insurance company has the right to modify the investment strategy of the underlying portfolio at any time. Furthermore, it is assumed that the issuer of the policy does not setup a separate portfolio to hedge the risk associated with the policy. Instead, the issuer will use its discretion about the investment strategy of the underlying portfolio to hedge shortfall risks. In that sense, the insurer’s investment portfolio serves simultaneously as the underlying security and as the hedge portfolio. This means that the hedging problem can not be separated from the valuation problem. We investigate the relationship between risk-neutral valuation and hedging of these policies in complete and incomplete financial markets.     

Canonical solution of the SU(3) ↓ SO(3) reduction problem from the SU(3) pattern calculus

Because of its numerous applications to physics, there have been many solutions published on the problem of reducing a given irreducible representation (irrep) of the unitary unimodular group SU(3) into irreps of the proper orthogonal subgroup SO(3). Such solutions are generally based on an arbitrary construction of a nonorthogonal basis of the highest weight space for an irrep of SO(3), followed by an equally arbitrary orthonormalization procedure. This paper presents a unique solution of this problem based on the intrinsic structure of the multiplicity function, which is a function M _L(p, q) giving the number of times irrep[L] of SO(3) is contained in irrep[pq0] of SU(3). This structure is implemented uniquely into the reduction problem through the use of the SU(3) pattern calculus.Work performed under the auspices of the U.S. Department of Energy.     

An optimal time algorithm for finding a maximum weight independent set in a tree

The maximum weight independent set problem for a general graph is NP-hard. But for some special classes of graphs, polynomial time algorithms do exist for solving it. Based on the divide-and-conquer strategy, Pawagi has presented anO(|V|log|V|) time algorithm for solving this problem on a tree. In this paper, we propose anO(|V|) time algorithm to improve Pawagi's result. The proposed algorithm is based on the dynamic programming strategy and is time optimal within a constant factor.     

Detection and correction of outliers in the bivariate chain-ladder method

The expected profit or loss of a non-life insurance company is determined for the whole of its multiple business lines. This implies the study of the claims reserving problem for a portfolio consisting of several correlated run-off triangles. A popular technique to deal with such a portfolio is the multivariate chain-ladder method of Merz and Wüthrich (2008). However, it is well known that the chain-ladder method is very sensitive to outlying data. For the univariate case, we have already developed a robust version of the chain-ladder method. In this article we propose two techniques to detect and correct outlying values in a bivariate situation. The methodologies are illustrated and compared on real examples from practice.