Continuous-time mean–variance portfolio selection with value-at-risk and no-shorting constraints

An investment problem is considered with dynamic mean–variance(M-V) portfolio criterion under discontinuous prices which follow jump–diffusion processes according to the actual prices of stocks and the normality and stability of the financial market. The short-selling of stocks is prohibited in this mathematical model. Then, the corresponding stochastic Hamilton–Jacobi–Bellman(HJB) equation of the problem is presented and the solution of the stochastic HJB equation based on the theory of stochastic LQ control and viscosity solution is obtained. The efficient frontier and optimal strategies of the original dynamic M-V portfolio selection problem are also provided. And then, the effects on efficient frontier under the value-at-risk constraint are illustrated. Finally, an example illustrating the discontinuous prices based on M-V portfolio selection is presented.     

Neural network-based mean–variance–skewness model for portfolio selection

In this study, a novel neural network-based mean–variance–skewness model for optimal portfolio selection is proposed integrating different forecasts and trading strategies, as well as investors’ risk preference. Based on the Lagrange multiplier theory in optimization and the radial basis function (RBF) neural network, the model seeks to provide solutions satisfying the trade-off conditions of mean–variance–skewness. The feasibility of the RBF network-based mean–variance–skewness model is verified with a simulation experiment. The experimental results show that, for all examined investor risk preferences and investment assets, the proposed model is a fast and efficient way of solving the trade-off in the mean–variance–skewness portfolio problem. In addition, we also find that the proposed approach can also be used as an alternative tool for evaluating various forecasting models.     

Fuzzy mean–variance–skewness portfolio selection models by interval analysis

In portfolio selection problem, the expected return, risk, liquidity etc. cannot be predicted precisely. The investor generally makes his portfolio decision according to his experience and his economic wisdom. So, deterministic portfolio selection is not a good choice for the investor. In most of the recent works on this problem, fuzzy set theory is widely used to model the problem in uncertain environments. This paper utilizes the concept of interval numbers in fuzzy set theory to extend the classical mean–variance (MV) portfolio selection model into mean–variance–skewness (MVS) model with consideration of transaction cost. In addition, some other criteria like short and long term returns, liquidity, dividends, number of assets in the portfolio and the maximum and minimum allowable capital invested in stocks of any selected company are considered. Three different models have been proposed by defining the future financial market optimistically, pessimistically and in the combined form to model the fuzzy MVS portfolio selection problem. In order to solve the models, fuzzy simulation (FS) and elitist genetic algorithm (EGA) are integrated to produce a more powerful and effective hybrid intelligence algorithm (HIA). Finally, our approaches are tested on a set of stock data from Bombay Stock Exchange (BSE).     

Uncertain exit time multi-period mean–variance portfolio selection with endogenous liabilities and Markov jumps

This paper considers an uncertain exit time multi-period mean–variance portfolio selection problem with endogenous liabilities in a Markov jump market, where assets and liabilities of the balance sheet are simultaneously optimized. The random returns of assets and liabilities depend on the states of the financial market. By applying the Lagrange duality method, the Khatri–Rao matrix product technique and the dynamic programming approach, the explicit expressions for the mean–variance efficient strategy and efficient frontier are derived. In addition, the optimal balance sheet structures in both cases with and without boundary constraints are studied. Moreover, some degenerate cases are discussed, and some results in the literature are recovered as degenerate cases under our setting. Furthermore, a numerical example based on real data from the Chinese stock market is provided to illustrate the results obtained in this paper, and some interesting findings are presented.     

Numerical solution of nonlinear system of Klein–Gordon equations by cubic B-spline collocation method

A technique to approximate the solutions of nonlinear Klein–Gordon equation and Klein–Gordon-Schrödinger equations is presented separately. The approach is based on collocation of cubic B-spline functions. The above-mentioned equations are decomposed into a system of partial differential equations, which are further converted to an amenable system of ODEs. The obtained system has been solved by SSP-RK54 scheme. Numerical solutions are presented for five examples, to show the accuracy and usefulness of proposed approach. The approximate solutions of both the equations are computed without using any transformation and linearization. The technique can be applied with ease to solve linear and nonlinear PDEs and also reduces the computational work.     

The mean–variance cardinality constrained portfolio optimization problem: An experimental evaluation of five multiobjective evolutionary algorithms

This paper compares the effectiveness of five state-of-the-art multiobjective evolutionary algorithms (MOEAs) together with a steady state evolutionary algorithm on the mean–variance cardinality constrained portfolio optimization problem (MVCCPO). The main computational challenges of the model are due to the presence of a nonlinear objective function and the discrete constraints. The MOEAs considered are the Niched Pareto genetic algorithm 2 (NPGA2), non-dominated sorting genetic algorithm II (NSGA-II), Pareto envelope-based selection algorithm (PESA), strength Pareto evolutionary algorithm 2 (SPEA2), and e-multiobjective evolutionary algorithm (e-MOEA). The computational comparison was performed using formal metrics proposed by the evolutionary multiobjective optimization community on publicly available data sets which contain up to 2196 assets.     

Numerical solutions of nonlinear problems of continua—II. Survey of incompressibility constraints and software aspects

In Part I [Comput. Struct. 19, 865–877 (1984)], a comprehensive survey is provided for the formulation and solution techniques of finite element applications in nonlinear continuum mechanics problems. In this paper we extend the survey to cover incompressibility constraints and software aspects. Various methods of incorporating incompressibility constraints in the finite element formulation are discussed. The extension of these methods to account for nonlinear structural analysis is discussed. An emphasis is put on the numerical aspects of various techniques. In the software aspects, a discussion of various tools for solving nonlinear continuum mechanics problems is given. These are the general purpose programs, the special purpose programs, and the programming systems. A practical working practical example is outlined to aid in the assessment of various tools in view of specific problem requirements.     

Differential quadrature solution of nonlinear Klein–Gordon and sine-Gordon equations

Differential quadrature method (DQM) is proposed to solve the one-dimensional quadratic and cubic Klein–Gordon equations, and two-dimensional sine-Gordon equation. We apply DQM in space direction and also blockwise in time direction. Initial and derivative boundary conditions are also approximated by DQM. DQM provides one to obtain numerical results with very good accuracy using considerably small number of grid points. Numerical solutions are obtained by using Gauss–Chebyshev–Lobatto (GCL) grid points in space intervals, and GCL grid points in each equally divided time blocks.     

Mean–CVaR portfolio selection: A nonparametric estimation framework

In this paper, we use Conditional Value-at-Risk (CVaR) to measure risk and adopt the methodology of nonparametric estimation to explore the mean–CVaR portfolio selection problem. First, we obtain the estimated calculation formula of CVaR by using the nonparametric estimation of the density of the loss function, and formulate two nonparametric mean–CVaR portfolio selection models based on two methods of bandwidth selection. Second, in both cases when short-selling is allowed and forbidden, we prove that the two nonparametric mean–CVaR models are convex optimization problems. Third, we show that when CVaR is solved for, the corresponding VaR can also be obtained as a by-product. Finally, we present a numerical example with Monte Carlo simulations to demonstrate the usefulness and effectiveness of our results, and compare our nonparametric method with the popular linear programming method.     

Numerical solution of unsteady Navier–Stokes equations on curvilinear meshes

The objective of the present work is to extend our FDS-based third-order upwind compact schemes by Shah et al. (2009) [8] to numerical solutions of the unsteady incompressible Navier–Stokes equations in curvilinear coordinates, which will save much computing time and memory allocation by clustering grids in regions of high velocity gradients. The dual-time stepping approach is used for obtaining a divergence-free flow field at each physical time step. We have focused on addressing the crucial issue of implementing upwind compact schemes for the convective terms and a central compact scheme for the viscous terms on curvilinear structured grids. The method is evaluated in solving several two-dimensional unsteady benchmark flow problems.     

Wei–Yao–Liu conjugate gradient projection algorithm for nonlinear monotone equations with convex constraints

In this paper, the Wei–Yao–Liu (WYL) conjugate gradient projection algorithm will be studied for nonlinear monotone equations with convex constraints, which can be viewed as an extension of the WYL conjugate gradient method for solving unconstrained optimization problems. These methods can be applied to solving large-scale nonlinear equations due to the low storage requirement. We can obtain global convergence of our algorithm without requiring differentiability in the case that the equation is Lipschitz continuous. The numerical results show that the new algorithm is efficient.     

Optimal mean–variance control for discrete-time linear systems with Markovian jumps and multiplicative noises

In this paper, we consider the stochastic optimal control problem of discrete-time linear systems subject to Markov jumps and multiplicative noises under two criteria. The first one is an unconstrained mean–variance trade-off performance criterion along the time, and the second one is a minimum variance criterion along the time with constraints on the expected output. We present explicit conditions for the existence of an optimal control strategy for the problems, generalizing previous results in the literature. We conclude the paper by presenting a numerical example of a multi-period portfolio selection problem with regime switching in which it is desired to minimize the sum of the variances of the portfolio along the time under the restriction of keeping the expected value of the portfolio greater than some minimum values specified by the investor.     

Numerical solution of the nonlinear Schrödinger equation,starting from the scattering data

Starting from the scattering data, the initial-value problem for the focusing nonlinear Schrödinger equation is solved numerically by following the path of the inverse scattering transform. The numerical results of an extensive experimentation suggest that: (a) our method is very effective, whenever the scattering data are analytically known; (b) the split-step Fourier method is not really effective if the exact solution is not smooth enough.     

Numerical solution of linear Fredholm integral equations using sine–cosine wavelets

If we divide the interval [0,1] into N sub-intervals, then sine–cosine wavelets on each sub-interval can approximate any function. This ability helps us to obtain a more accurate approximation of piecewise continuous functions, and, hence, we can obtain more accurate solutions of integral equations. In this article we use a combination of sine–cosine wavelets on the interval [0,1] to solve linear integral equations. We convert the integral equation into a system of linear equations. Numerical examples are given to demonstrate the applicability of the proposed method.     

Numerical solution of nonlinear two-dimensional Fredholm integral equations of the second kind using Sinc Nyström method

In this paper, a double-exponential (DE) Sinc Nyström method is utilized to solve nonlinear two-dimensional Fredholm integral equations of the second kind. Using the DE transformation, the Sinc quadrature rule for a definite integral is extended to double integral over a rectangular region. Therefore, a nonlinear Fredholm integral equation is reduced to a system of nonlinear algebraic equations, which is solved using the Newton iteration method. Convergence analysis shows that the proposed method can converge exponentially. Several numerical examples are provided to demonstrate the high efficiency and accuracy of the proposed method.     

Robust portfolio optimization via solution to the Hamilton–Jacobi–Bellman equation

We consider a problem of dynamic stochastic portfolio optimization modelled by a fully non-linear Hamilton–Jacobi–Bellman (HJB) equation. Using the Riccati transformation, the HJB equation is transformed to a simpler quasi-linear partial differential equation. An auxiliary quadratic programming problem is obtained, which involves a vector of expected asset returns and a covariance matrix of the returns as input parameters. Since this problem can be sensitive to the input data, we modify the problem from fixed input parameters to worst-case optimization over convex or discrete uncertainty sets both for asset mean returns and their covariance matrix. Qualitative as well as quantitative properties of the value function are analysed along with providing illustrative numerical examples. We show application to robust portfolio optimization for the German DAX30 Index.     

Chebyshev polynomials for the numerical solution of fractal–fractional model of nonlinear Ginzburg–Landau equation

Engineering with Computers - This paper introduces a new version for the nonlinear Ginzburg–Landau equation derived from fractal–fractional derivatives and proposes a computational...     

A novel group decision model based on mean–variance–skewness concepts and interval-valued fuzzy sets for a selection problem of the sustainable warehouse location under uncertainty

Recently, sustainable warehouse location has been regarded as one of the most critical and significant decision problems for long-term planning in the supply chain. This strategic decision can be effected by different quantitative and qualitative evaluation criteria via three dimensions of the sustainability. Main theme of the paper is to select the most optimal location decision from a number of potential sustainable warehouse candidates. For this purpose, this paper presents a novel multi-criteria decision-making model by a group of supply chain experts or decision makers with interval-valued fuzzy setting and asymmetric uncertainty information. Concepts of mean, variance and skewness are introduced into the proposed group decision model, and their mathematical relations are defined based on a fuzzy possibilistic statistical approach. Then, new relations in this model are presented for obtaining ideal solutions under uncertainty with two high and low values of the possibilistic mean and possibilistic standard deviation, along with the possibilistic cube root of skewness. In addition, novel separation measures and new fuzzy ranking index of hybridized relative closeness coefficients are presented to provide final preference order of warehouse location candidates under uncertain conditions. Finally, a sustainable warehouse location selection problem in a pharmaceutical company is presented and solved by the proposed group decision model to demonstrate its applicability and suitability.

     

Satisfiability of ECTL⁎ with constraints

We show that satisfiability and finite satisfiability for ${\mathsf{ECTL}}^{⁎}$ with equality-, order-, and modulo-constraints over $\mathbb{Z}$ are decidable. Since ${\mathsf{ECTL}}^{⁎}$ is a proper extension of ${\mathsf{CTL}}^{⁎}$ this greatly improves the previously known decidability results for certain fragments of ${\mathsf{CTL}}^{⁎}$, e.g., the existential and positive fragments and EF. We also show that our choice of local constraints is necessary for the result in the sense that, if we add the possibility to state non-local constraints over $\mathbb{Z}$, the resulting logic becomes undecidable.     

Distributed fault diagnosis for continuous-time nonlinear systems: The input–output case

In this paper, some new results on distributed fault diagnosis of continuous-time nonlinear systems with partial state measurements are proposed. By exploiting an overlapping decomposition framework, the dynamics of a nonlinear uncertain large-scale dynamical system is described as the interconnections of several subsystems. Each subsystem is monitored by a Local Fault Diagnoser: a set of local estimators, based on the nominal local dynamic model and on an adaptive approximation of the interconnection and of the fault function, allows to derive a local fault decision. A consensus-based protocol is used in order to improve the detectability and the isolability of faults affecting variables shared among different subsystems because of the overlapping decomposition. A sufficient condition ensuring the convergence of the estimation errors is derived. Finally, possibly non-conservative time-varying threshold functions guaranteeing no false-positive alarms and theoretical results dealing with detectability and isolability sufficient conditions are presented.