Right orders and amalgamation for lattice-ordered groups

Let H _i be a sublattice subgroup of a lattice-ordered group G _i (i = 1, 2). Suppose that H ₁ and H ₂ are isomorphic as lattice-ordered groups, say by φ. In general, there is no lattice-ordered group in which G ₁ and G ₂ can be embedded (as lattice-ordered groups) so that the embeddings agree on the images of H ₁ and H ₁ ^φ . In this article we prove that the group free product of G ₁ and G ₂ amalgamating H ₁ and H ₁ ^φ is right orderable and so embeddable (as a group) in a lattice-orderable group. To obtain this, we use our necessary and sufficient conditions for the free product of right-ordered groups with amalgamated subgroup to be right orderable [BLUDOV, V. V.—GLASS, A. M. W.: Word problems, embeddings, and free products of right-ordered groups with amalgamated subgroup, Proc. London Math. Soc. (3) 99 (2009), 585–608]. We also provide new limiting examples to show that amalgamation can fail in the category of lattice-ordered groups even when the amalgamating sublattice subgroups are convex and normal (?-ideals) and solve of Problem 1.42 from [KOPYTOV, V. M.—MEDVEDEV, N. YA.: Ordered groups. In: Selected Problems in Algebra. Collection of Works Dedicated to the Memory of N. Ya. Medvedev, Altaii State University, Barnaul, 2007, pp. 15–112 (Russian)].     

A family of identities related to zero-sum and team games

We list and prove a family of binomial identities by calculating in two ways the probabilities of approximate saddlepoints occurring in random m×n matrices. The identities are easily seen to be equivalent to the evaluation of a family of Gauss ₂F₁ polynomials according to a formula of Vandermonde. We also consider some implications concerning the number of approximate pure strategy Nash equilibria we can expect in large matrix zero-sum and team games.     

A note on the efficiency and fairness of decentralized matching

This note shows by means of simulation experiments that although two-sided matching markets can always converge to a stable outcome via a decentralized process of matching and rematching, the most likely outcome need not be the median matching or a minimum-choice-count matching, whether or not the two coincide.     

On the distribution of the (un)bounded sum of random variables

We propose a general treatment of random variables aggregation accounting for the dependence among variables and bounded or unbounded support of their sum. The approach is based on the extension to the concept of convolution to dependent variables, involving copula functions. We show that some classes of copula functions (such as Marshall-Olkin and elliptical) cannot be used to represent the dependence structure of two variables whose sum is bounded, while Archimedean copulas can be applied only if the generator becomes linear beyond some point. As for the application, we study the problem of capital allocation between risks when the sum of losses is bounded.     

On the threshold dividend strategy for a generalized jump-diffusion risk model

In this paper, we generalize the Cramér-Lundberg risk model perturbed by diffusion to incorporate jumps due to surplus fluctuation and to relax the positive loading condition. Assuming that the surplus process has exponential upward and arbitrary downward jumps, we analyze the expected discounted penalty (EDP) function of Gerber and Shiu (1998) under the threshold dividend strategy. An integral equation for the EDP function is derived using the Wiener-Hopf factorization. As a result, an explicit analytical expression is obtained for the EDP function by solving the integral equation. Finally, phase-type downward jumps are considered and a matrix representation of the EDP function is presented.     

Explicit ruin formulas for models with dependence among risks

We show that a simple mixing idea allows one to establish a number of explicit formulas for ruin probabilities and related quantities in collective risk models with dependence among claim sizes and among claim inter-occurrence times. Examples include compound Poisson risk models with completely monotone marginal claim size distributions that are dependent according to Archimedean survival copulas as well as renewal risk models with dependent inter-occurrence times.     

Supply network planning in the forest supply chain with bucking decisions anticipation

We consider a two-echelon timber supply chain in which the first echelon consists of several stands to be harvested and the second echelon consists of mills to be supplied with logs of different length. This problem aims at minimizing harvesting and transportation costs for one production period, while satisfying demand expressed as a mix of volumes of specific log types. Harvesting cost, which includes felling, bucking and hauling to roadside, depends upon the number of log type to be produced and sorted. Each stand to be harvested is modeled individually with a limited number of trees of various classes of diameter and total length, which affects the productivity factors of the bucking patterns to be used. To take these characteristics into account, we propose heuristics based on columns generation to solve the supply network problem at the forest level with an anticipation of bucking operations at the stand level.     

Large deviations for random dynamical systems and applications to hidden Markov models

In this paper, we prove the large deviation principle (LDP) for the occupation measures of not necessarily irreducible random dynamical systems driven by Markov processes. The LDP for not necessarily irreducible dynamical systems driven by i.i.d. sequence is derived. As a further application we establish the LDP for extended hidden Markov models, filling a gap in the literature, and obtain large deviation estimations for the log-likelihood process and maximum likelihood estimator of hidden Markov models.     

Optimality conditions in nonconvex optimization via weak subdifferentials

In this paper we study optimality conditions for optimization problems described by a special class of directionally differentiable functions. The well-known necessary and sufficient optimality condition of nonsmooth convex optimization, given in the form of variational inequality, is generalized to the nonconvex case by using the notion of weak subdifferentials. The equivalent formulation of this condition in terms of weak subdifferentials and augmented normal cones is also presented.