Unit-linked life insurance policies: Optimal hedging in partially observable market models

In this paper we investigate the hedging problem of a unit-linked life insurance contract via the local risk-minimization approach, when the insurer has a restricted information on the market. In particular, we consider an endowment insurance contract, that is a combination of a term insurance policy and a pure endowment, whose final value depends on the trend of a stock market where the premia the policyholder pays are invested. To allow for mutual dependence between the financial and the insurance markets, we use the progressive enlargement of filtration approach. We assume that the stock price process dynamics depends on an exogenous unobservable stochastic factor that also influences the mortality rate of the policyholder. We characterize the optimal hedging strategy in terms of the integrand in the Galtchouk–Kunita–Watanabe decomposition of the insurance claim with respect to the minimal martingale measure and the available information flow. We provide an explicit formula by means of predictable projection of the corresponding hedging strategy under full information with respect to the natural filtration of the risky asset price and the minimal martingale measure. Finally, we discuss applications in a Markovian setting via filtering.     

Geometric approach to global asymptotic stability for the SEIRS models in epidemiology

In this paper, we present a more general criterion for the global asymptotic stability of equilibria for nonlinear autonomous differential equations based on the geometric criterion developed by Li and Muldowney. By applying this criterion, we obtain some results for the global asymptotic stability of SEIRS models with constant recruitment and varying total population size. Based on these results, we give a complete affirmative answer to Liu–Hethcote–Levin conjecture. Furthermore, an affirmative answer to Li–Graef–Wang–Karsai’s problem for SEIR model with permanent immunity and varying total population size is given.     

Proximal-gradient algorithms for fractional programming

In this paper, we propose two proximal-gradient algorithms for fractional programming problems in real Hilbert spaces, where the numerator is a proper, convex and lower semicontinuous function and the denominator is a smooth function, either concave or convex. In the iterative schemes, we perform a proximal step with respect to the nonsmooth numerator and a gradient step with respect to the smooth denominator. The algorithm in case of a concave denominator has the particularity that it generates sequences which approach both the (global) optimal solutions set and the optimal objective value of the underlying fractional programming problem. In case of a convex denominator the numerical scheme approaches the set of critical points of the objective function, provided the latter satisfies the Kurdyka-?ojasiewicz property.     

Approximate solutions and scalarization in set-valued optimization

Abstract

Certain notions of approximate weak efficient solutions are considered for a set-valued optimization problem based on vector and set criteria approaches. For approximate solutions based on the vector approach, a characterization is provided in terms of an extended Gerstewitz’s function. For the set approach case, two notions of approximate weak efficient solutions are introduced using a lower and an upper quasi order relations for sets and further compactness and stability aspects are discussed for these approximate solutions. Existence and scalarization using a generalized Gerstewitz’s function are also established for approximate solutions, based on the lower set order relation.     

Two optimization problems in linear regression with interval data

We consider a linear regression model where neither regressors nor the dependent variable is observable; only intervals are available which are assumed to cover the unobservable data points. Our task is to compute tight bounds for the residual errors of minimum-norm estimators of regression parameters with various norms (corresponding to least absolute deviations (LAD), ordinary least squares (OLS), generalized least squares (GLS) and Chebyshev approximation). The computation of the error bounds can be formulated as a pair of max–min and min–min box-constrained optimization problems. We give a detailed complexity-theoretic analysis of them. First, we prove that they are NP-hard in general. Then, further analysis explains the sources of NP-hardness. We investigate three restrictions when the problem is solvable in polynomial time: the case when the parameter space is known apriori to be restricted into a particular orthant, the case when the regression model has a fixed number of regression parameters, and the case when only the dependent variable is observed with errors. We propose a method, called orthant decomposition of the parameter space, which is the main tool for obtaining polynomial-time computability results.     

An adaptive nonmonotone global Barzilai–Borwein gradient method for unconstrained optimization

The Barzilai–Borwein (BB) gradient method has received many studies due to its simplicity and numerical efficiency. By incorporating a nonmonotone line search, Raydan (SIAM J Optim. 1997;7:26–33) has successfully extended the BB gradient method for solving general unconstrained optimization problems so that it is competitive with conjugate gradient methods. However, the numerical results reported by Raydan are poor for very ill-conditioned problems because the effect of the degree of nonmonotonicity may be noticeable. In this paper, we focus more on the nonmonotone line search technique used in the global Barzilai–Borwein (GBB) gradient method. We improve the performance of the GBB gradient method by proposing an adaptive nonmonotone line search based on the morphology of the objective function. We also prove the global convergence and the R-linear convergence rate of the proposed method under reasonable assumptions. Finally, we give some numerical experiments made on a set of unconstrained optimization test problems of the CUTEr collection. The results show the efficiency of the proposed method in the sense of the performance profile introduced (Math Program. 2002;91:201–213) by Dolan and Moré.