One-year Value-at-Risk for longevity and mortality

Upcoming new regulation on regulatory required solvency capital for insurers will be predominantly based on a one-year Value-at-Risk measure. This measure aims at covering the risk of the variation in the projection year as well as the risk of changes in the best estimate projection for future years. This paper addresses the issue how to determine this Value-at-Risk for longevity and mortality risk. Naturally, this requires stochastic mortality rates. In the past decennium, a vast literature on stochastic mortality models has been developed. However, very few of them are suitable for determining the one-year Value-at-Risk. This requires a model for mortality trends instead of mortality rates. Therefore, we will introduce a stochastic mortality trend model that fits this purpose. The model is transparent, easy to interpret and based on well known concepts in stochastic mortality modeling. Additionally, we introduce an approximation method based on duration and convexity concepts to apply the stochastic mortality rates to specific insurance portfolios.     

Accounting for regime and parameter uncertainty in regime-switching models

As investment guarantees become increasingly complex, realistic simulation of the price becomes more critical. Currently, regime-switching models are commonly used to simulate asset returns. Under a regime switching model, simulating random asset streams involves three steps: (i) estimate the model parameters given the number of regimes using maximum likelihood, (ii) choose the number of regimes using a model selection criteria, and (iii) simulate the streams using the optimal number of regimes and parameter values. This method, however, does not properly incorporate regime or parameter uncertainty into the generated asset streams and therefore into the price of the guarantee. To remedy this, this article adopts a Bayesian approach to properly account for those two sources of uncertainty and improve pricing.     

Motivic zeta functions of abelian varieties, and the monodromy conjecture

We prove for abelian varieties a global form of Denef and Loeser?s motivic monodromy conjecture, in arbitrary characteristic. More precisely, we prove that for every tamely ramified abelian variety A over a complete discretely valued field with algebraically closed residue field, its motivic zeta function has a unique pole at Chai?s base change conductor c(A) of A, and that the order of this pole equals one plus the potential toric rank of A. Moreover, we show that for every embedding of Q? in C, the value exp(2πic(A)) is an ?-adic tame monodromy eigenvalue of A. The main tool in the paper is Edixhoven?s filtration on the special fiber of the Néron model of A, which measures the behavior of the Néron model under tame base change.     

Curves with a prescribed number of rational points

We show that for any finite field Fq, any N?0 and all sufficiently large integers g there exist curves over Fq of genus g having exactly N rational points.     

Failure of the Hasse principle for Enriques surfaces

We construct an Enriques surface X over Q with empty étale-Brauer set (and hence no rational points) for which there is no algebraic Brauer–Manin obstruction to the Hasse principle. In addition, if there is a transcendental obstruction on X, then we obtain a K3 surface that has a transcendental obstruction to the Hasse principle.     

The p-rank strata of the moduli space of hyperelliptic curves

We prove results about the intersection of the p-rank strata and the boundary of the moduli space of hyperelliptic curves in characteristic p?3. This yields a strong technique that allows us to analyze the stratum of hyperelliptic curves of genus g and p-rank f. Using this, we prove that the endomorphism ring of the Jacobian of a generic hyperelliptic curve of genus g and p-rank f is isomorphic to Z if g?4. Furthermore, we prove that the Z/?-monodromy of every irreducible component of is the symplectic group Sp2g(Z/?) if g?3, and ?≠p is an odd prime (with mild hypotheses on ? when f=0). These results yield numerous applications about the generic behavior of hyperelliptic curves of given genus and p-rank over finite fields, including applications about Newton polygons, absolutely simple Jacobians, class groups and zeta functions.     

Block companion Singer cycles, primitive recursive vector sequences, and coprime polynomial pairs over finite fields

We discuss a conjecture concerning the enumeration of nonsingular matrices over a finite field that are block companion and whose order is the maximum possible in the corresponding general linear group. A special case is proved using some recent results on the probability that a pair of polynomials with coefficients in a finite field is coprime. Connection with an older problem of Niederreiter about the number of splitting subspaces of a given dimension are outlined and an asymptotic version of the conjectural formula is established. Some applications to the enumeration of nonsingular Toeplitz matrices of a given size over a finite field are also discussed.     

The Conway–Sloane tetralattice pairs are non-isometric

Conway and Sloane constructed a 4-parameter family of pairs of isospectral lattices of rank four. They conjectured that all pairs in their family are non-isometric, whenever the parameters are pairwise different, and verified this for classical integral lattices of determinant up to 10⁴. In this paper, we use our theory of lattice invariants to prove this conjecture.     

?-adic properties of smallest parts functions

We prove explicit congruences modulo powers of arbitrary primes for three smallest parts functions: one for partitions, one for overpartitions, and one for partitions without repeated odd parts. The proofs depend on ?-adic properties of certain modular forms and mock modular forms of weight 3/2 with respect to the Hecke operators T(?2m).     

On the action of permutations on distances between values of rational functions mod p

For the finite field Fp one may consider the distance between r₁(n) and r₂(n), where r₁, r₂ are rational functions in Fp(x). We study the effect to such distances by applying all possible permutations to the elements.     

Generalization of a theorem of Carlitz

We generalize Carlitz? result on the number of self-reciprocal monic irreducible polynomials over finite fields by showing that similar explicit formula holds for the number of irreducible polynomials obtained by a fixed quadratic transformation. Our main tools are a combinatorial argument and Hurwitz genus formula.     

A Hecke correspondence theorem for automorphic integrals with symmetric rational period functions on the Hecke groups

Marvin Knopp showed that entire automorphic integrals with rational period functions satisfy a Hecke correspondence theorem, provided the rational period functions have poles only at 0 or ∞. For other automorphic integrals the corresponding Dirichlet series has a functional equation with a remainder term that arises from the nonzero poles of the rational period function. In this paper we prove a Hecke correspondence theorem for a class of automorphic integrals with rational period functions on the Hecke groups. We restrict our attention to automorphic integrals of weight that is twice an odd integer and to rational period functions that satisfy a symmetry property we call “Hecke-symmetry.” Each remainder term satisfies two relations (the second of which is new in this paper) corresponding to the two relations for the rational period function.     

A risk-based model for the valuation of pension insurance

In the US, defined benefit plans are insured by the Pension Benefit Guaranty Corporation (PBGC). Taking account of the fact that the PBGC covers only the residual deficits of the pension fund the sponsoring company is unable to cover and that the plans can be prematurely terminated, we consider a model that accounts for the joint dynamics of the pension fund’s and sponsoring firm’s assets in order to effectively determine the risk-based pension premium for the insurance provided by the PBGC. We obtain a closed-form pricing formula for this risk-based premium. Its magnitude depends highly on the investment portfolio of the pension fund and of the sponsoring company as well as the correlation between these two portfolios.     

Topological flatness of local models for ramified unitary groups. I. The odd dimensional case

Local models are certain schemes, defined in terms of linear-algebraic moduli problems, which give étale-local neighborhoods of integral models of certain p-adic PEL Shimura varieties defined by Rapoport and Zink. When the group defining the Shimura variety ramifies at p, the local models (and hence the Shimura models) as originally defined can fail to be flat, and it becomes desirable to modify their definition so as to obtain a flat scheme. In the case of unitary similitude groups whose localizations at Qp are ramified, quasi-split GUn, Pappas and Rapoport have added new conditions, the so-called wedge and spin conditions, to the moduli problem defining the original local models and conjectured that their new local models are flat. We prove a preliminary form of their conjecture, namely that their new models are topologically flat, in the case n is odd.     

On the irrationality of factorial series III

In this paper we derive some irrationality and linear independence results for series of the form where is either a non-negative integer sequence with υ_n = o(log n/log log n) or a non-decreasing integer sequence with .     

Homology of -types and Hopf type formulas

The tripleability of the category of crossed n-cubes is studied. The leading cotriple homology of these homotopy (n+1)-types is investigated, describing it as Hopf type formulas.     

Construction of rational surfaces yielding good codes

In the present article, we consider Algebraic Geometry codes on some rational surfaces. The estimate of the minimum distance is translated into a point counting problem on plane curves. This problem is solved by applying the upper bound à la Weil of Aubry and Perret together with the bound of Homma and Kim for plane curves. The parameters of several codes from rational surfaces are computed. Among them, the codes defined by the evaluation of forms of degree 3 on an elliptic quadric are studied. As far as we know, such codes have never been treated before. Two other rational surfaces are studied and very good codes are found on them. In particular, a [57,12,34] code over F₇ and a [91,18,53] code over F₉ are discovered, these codes beat the best known codes up to now.     

Compactness of Schrödinger semigroups with unbounded below potentials

By using the super Poincaré inequality of a Markov generator L₀ on L²(μ) over a σ-finite measure space (E,F,μ), the Schrödinger semigroup generated by L₀−V for a class of (unbounded below) potentials V is proved to be L²(μ)-compact provided μ(V?N)<∞ for all N>0. This condition is sharp at least in the context of countable Markov chains, and considerably improves known ones on, e.g., Rd under the condition that V(x)→∞ as |x|→∞. Concrete examples are provided to illustrate the main result.     

New methods for generating permutation polynomials over finite fields

We present two methods for generating linearized permutation polynomials over an extension of a finite field Fq. These polynomials are parameterized by an element of the extension field and are permutation polynomials for all nonzero values of the element. For the case of the extension degree being odd and the size of the ground field satisfying , these parameterized linearized permutation polynomials can be used to derive non-parameterized nonlinear permutation polynomials via a recent result of Ding et al.     

Optimal inverse Littlewood–Offord theorems

Let ηi, i=1,…,n, be iid Bernoulli random variables, taking values ±1 with probability . Given a multiset V of n integers v₁,…,vn, we define the concentration probability as A classical result of Littlewood–Offord and Erd?s from the 1940s asserts that, if the vi are non-zero, then ρ(V) is O(n−1/2). Since then, many researchers have obtained improved bounds by assuming various extra restrictions on V.About 5 years ago, motivated by problems concerning random matrices, Tao and Vu introduced the inverse Littlewood–Offord problem. In the inverse problem, one would like to characterize the set V, given that ρ(V) is relatively large.In this paper, we introduce a new method to attack the inverse problem. As an application, we strengthen the previous result of Tao and Vu, obtaining an optimal characterization for V. This immediately implies several classical theorems, such as those of Sárközy and Szemerédi and Halász.The method also applies to the continuous setting and leads to a simple proof for the β-net theorem of Tao and Vu, which plays a key role in their recent studies of random matrices.All results extend to the general case when V is a subset of an abelian torsion-free group, and ηi are independent variables satisfying some weak conditions.