Filter‐based portfolio strategies in an HMM setting with varying correlation parametrizations

We consider portfolio optimization in a regime‐switching market. The assets of the portfolio are modeled through a hidden Markov model (HMM) in discrete time, where drift and volatility of the single assets are allowed to switch between different states. We consider different parametrizations of the involved asset covariances: statewise uncorrelated assets (though linked through the common Markov chain), assets correlated in a state‐independent way, and assets where the correlation varies from state to state. As a benchmark, we also consider a model without regime switches. We utilize a filter‐based expectation‐maximization (EM) algorithm to obtain optimal parameter estimates within this multivariate HMM and present parameter estimators in all three HMM settings. We discuss the impact of these different models on the performance of several portfolio strategies. Our findings show that for simulated returns, our strategies in many settings outperform naïve investment strategies, like the equal weights strategy. Information criteria can be used to detect the best model for estimation as well as for portfolio optimization. A second study using real data confirms these findings.     

A new smoothing scheme for mathematical programs with complementarity constraints

In this paper, we consider a mathematical program with complementarity constraints (MPCC). We present a new smoothing scheme for this problem, which makes the primal structure of the complementarity part unchanged mostly. For the new smoothing problem, we show that the linear independence constraint qualification (LICQ) holds under some conditions. We also analyze the convergence behavior of the smoothing problem, and get some sufficient conditions such that an accumulation point of stationary points of the smoothing problems is C (M, B)-stationarity respectively. Based on the smoothing problem, we establish an algorithm to solve the primal MPCC problem. Some numerical experiments are given in the paper.     

Semiparametric estimation in regression with missing covariates using single-index models

We investigate semiparametric estimation of regression coefficients through generalized estimating equations with single-index models when some covariates are missing at random. Existing popular semiparametric estimators may run into difficulties when some selection probabilities are small or the dimension of the covariates is not low. We propose a new simple parameter estimator using a kernel-assisted estimator for the augmentation by a single-index model without using the inverse of selection probabilities. We show that under certain conditions the proposed estimator is as efficient as the existing methods based on standard kernel smoothing, which are often practically infeasible in the case of multiple covariates. A simulation study and a real data example are presented to illustrate the proposed method. The numerical results show that the proposed estimator avoids some numerical issues caused by estimated small selection probabilities that are needed in other estimators.

     

On Random Sampling Without Replacement from a Finite Population

We consider the three progressively more general sampling schemes without replacement from a finite population: simple random sampling without replacement, Midzuno sampling and successive sampling. We (i) obtain a lower bound on the expected sample coverage of a successive sample, (ii) show that the vector of first order inclusion probabilities divided by the sample size is majorized by the vector of selection probabilities of a successive sample, and (iii) partially order the vectors of first order inclusion probabilities for the three sampling schemes by majorization. We also show that the probability of an ordered successive sample enjoys the arrangement increasing property and for sample size two the expected sample coverage of a successive sample is Schur convex in its selection probabilities. We also study the spacings of a simple random sample from a linearly ordered finite population and characterize in several ways a simple random sample.     

A Smoothing Method for a Mathematical Program with P-Matrix Linear Complementarity Constraints

We consider a mathematical program whose constraints involve a parametric P-matrix linear complementarity problem with the design (upper level) variables as parameters. Solutions of this complementarity problem define a piecewise linear function of the parameters. We study a smoothing function of this function for solving the mathematical program. We investigate the limiting behaviour of optimal solutions, KKT points and B-stationary points of the smoothing problem. We show that a class of mathematical programs with P-matrix linear complementarity constraints can be reformulated as a piecewise convex program and solved through a sequence of continuously differentiable convex programs. Preliminary numerical results indicate that the method and convex reformulation are promising.     

Ultra-analytic effect of Cauchy problem for a class of kinetic equations

The smoothing effect of the Cauchy problem for a class of kinetic equations is studied. We firstly consider the spatially homogeneous nonlinear Landau equation with Maxwellian molecules and inhomogeneous linear Fokker-Planck equation to show the ultra-analytic effects of the Cauchy problem. Those smoothing effect results are optimal and similar to heat equation. In the second part, we study a model of spatially inhomogeneous linear Landau equation with Maxwellian molecules, and show the analytic effect of the Cauchy problem.     

Method of resurgent analysis in atomic collision theory

We consider the atomic collision problem in the adiabatic approximation. We show that the transition probabilities can be evaluated in this approximation using the tools of resurgent analysis. We suggest a computational algorithm for the transition probabilities and give the mathematical foundation of this algorithm. The analysis is carried out using the example of the two-level Landau-Zener model. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 112, No. 2, pp. 308–323, August, 1997.     

A Multinomial Approximation Approach for the Finite Time Survival Probability Under the Markov-modulated Risk Model

In this paper, we consider the problem of computing different types of finite time survival probabilities for a Markov-Modulated risk model and a Markov-Modulated risk model with reinsurance, both with varying premium rates. We use the multinomial approximation scheme to derive an efficient recursive algorithm to compute finite time survival probabilities and finite time draw-down survival probabilities. Numerical results show that by comparing with MCMC approximation, discretize approximation and diffusion approximation methods, the proposed scheme performs accurate results in all the considered cases and with better computation efficiency.

     

Determining lower and upper bounds on probabilities of atomic propositions in sets of logical formulas represented by digraphs

In this paper we consider the problem of determining lower and upper bounds on probabilities of atomic propositions in sets of logical formulas represented by digraphs. We establish a sharp upper bound, as well as a lower bound that is not in general sharp. We show further that under a certain condition the lower bound is sharp. In that case, we obtain a closed form solution for the possible probabilities of the atomic propositions.The second author is partially supported by ONR grant N00014-92-J-1028 and AFOSR grant 91-0287.     

Improving demand forecasting accuracy using nonlinear programming software

We address the problem of forecasting real time series with a proportion of zero values and a great variability among the nonzero values. In order to calculate forecasts for a time series, the model coefficients must be estimated. The appropriate choice of values for the smoothing parameters in exponential smoothing methods relies on the minimization of the fitting errors of historical data. We adapt the generalized Holt–Winters formulation so that it can consider the starting values of the local components of level, trend and seasonality as decision variables of the nonlinear programming problem associated with this forecasting procedure. A spreadsheet model is used to solve the problems of optimization efficiently. We show that our approach produces accurate forecasts with little data per product.     

Divide and Recombine Approaches for Fitting Smoothing Spline Models with Large Datasets

Spline smoothing is a widely used nonparametric method that allows data to speak for themselves. Due to its complexity and flexibility, fitting smoothing spline models is usually computationally intensive which may become prohibitive with large datasets. To overcome memory and CPU limitations, we propose four divide and recombine (D&R) approaches for fitting cubic splines with large datasets. We consider two approaches to divide the data: random and sequential. For each approach of division, we consider two approaches to recombine. These D&R approaches are implemented in parallel without communication. Extensive simulations show that these D&R approaches are scalable and have comparable performance as the method that uses the whole data. The sequential D&R approaches are spatially adaptive which lead to better performance than the method that uses the whole data when the underlying function is spatially inhomogeneous.     

A Trust-Region Algorithm for Global Optimization

We consider the global minimization of a bound-constrained function with a so-called funnel structure. We develop a two-phase procedure that uses sampling, local optimization, and Gaussian smoothing to construct a smooth model of the underlying funnel. The procedure is embedded in a trust-region framework that avoids the pitfalls of a fixed sampling radius. We present a numerical comparison to three popular methods and show that the new algorithm is robust and uses up to 20 times fewer local minimizations steps.     

Asymptotic risks of Viterbi segmentation

We consider the maximum likelihood (Viterbi) alignment of a hidden Markov model (HMM). In an HMM, the underlying Markov chain is usually hidden and the Viterbi alignment is often used as the estimate of it. This approach will be referred to as the Viterbi segmentation. The goodness of the Viterbi segmentation can be measured by several risks. In this paper, we prove the existence of asymptotic risks. Being independent of data, the asymptotic risks can be considered as the characteristics of the model that illustrate the long-run behavior of the Viterbi segmentation.     

A smoothing-type algorithm for solving system of inequalities

In this paper we consider system of inequalities. By constructing a new smoothing function, the problem is approximated via a family of parameterized smooth equations. A Newton-type algorithm is applied to solve iteratively the smooth equations so that a solution of the problem concerned is found. We show that the algorithm is globally and locally quadratically convergent under suitable assumptions. Preliminary numerical results are reported.     

On the Legendre-based filters of Persson and Strang

In this paper we consider the Legendre-based method of Persson and Strang and its modification for smoothing uniformly spaced data. The considered Legendre-based methods are numerically compared with the classical Savitzky-Golay method for filtering uniformly spaced data. We show how the Legendre-based filters can be extended to irregularly spaced data.     

A reliability model for multivariate exponential distributions

In this paper, we consider a counting process approach for characterizing a system having dependent component failure rates. We study the transient state probabilities and related reliability properties based on a series of Poisson shocks. We also show that the proposed infinitesimal generator representation can be used to characterize the bivariate exponential distributions of Freund, Marshall-Olkin, Block-Basu and Friday-Patil.     

Numerical discretization-based kernel type estimation methods for ordinary differential equation models

We consider the problem of parameter estimation in both linear and nonlinear ordinary differential equation(ODE) models. Nonlinear ODE models are widely used in applications. But their analytic solutions are usually not available. Thus regular methods usually depend on repetitive use of numerical solutions which bring huge computational cost. We proposed a new two-stage approach which includes a smoothing method(kernel smoothing or local polynomial fitting) in the first stage, and a numerical discretization method(Eulers discretization method, the trapezoidal discretization method,or the Runge–Kutta discretization method) in the second stage. Through numerical simulations, we find the proposed method gains a proper balance between estimation accuracy and computational cost.Asymptotic properties are also presented, which show the consistency and asymptotic normality of estimators under some mild conditions. The proposed method is compared to existing methods in term of accuracy and computational cost. The simulation results show that the estimators with local linear smoothing in the first stage and trapezoidal discretization in the second stage have the lowest average relative errors. We apply the proposed method to HIV dynamics data to illustrate the practicability of the estimator.     

Sur le choix du paramètre d'ajustement dans le lissage par fonctions spline

Summary We consider the problem of approximating an unknown functionf, known with error atn equally spaced points of the real interval [a, b].To solve this problem, we use the natural polynomial smoothing splines. We show that the eigenvalues associated to these splines converge to the eigenvalues of a differential operator and we use this fact to obtain an algorithm, based on the Generalized Cross Validation method, to calculate the smoothing parameter.With this algorithm, we divide byn the time used by classical methods.

     

Bayesian estimation of finite time ruin probabilities

In this paper, we consider Bayesian inference and estimation of finite time ruin probabilities for the Sparre Andersen risk model. The dense family of Coxian distributions is considered for the approximation of both the inter‐claim time and claim size distributions. We illustrate that the Coxian model can be well fitted to real, long‐tailed claims data and that this compares well with the generalized Pareto model. The main advantage of using the Coxian model for inter‐claim times and claim sizes is that it is possible to compute finite time ruin probabilities making use of recent results from queueing theory. In practice, finite time ruin probabilities are much more useful than infinite time ruin probabilities as insurance companies are usually interested in predictions for short periods of future time and not just in the limit. We show how to obtain predictive distributions of these finite time ruin probabilities, which are more informative than simple point estimations and take account of model and parameter uncertainty. We illustrate the procedure with simulated data and the well‐known Danish fire loss data set. Copyright © 2009 John Wiley & Sons, Ltd.     

On the dimension of the attractor for a perturbed 3d Ladyzhenskaya model

We consider the so-called Ladyzhenskaya model of incompressible fluid, with an additional artificial smoothing term ?Δ³. We establish the global existence, uniqueness, and regularity of solutions. Finally, we show that there exists an exponential attractor, whose dimension we estimate in terms of the relevant physical quantities, independently of ? > 0.