Quadratic Differential Equations Research Articles

An incomplete equilibrium with a stochastic annuity

We prove the global existence of an incomplete, continuous-time finite-agent Radner equilibrium in which exponential agents optimise their expected utility over both running consumption and terminal wealth. The market consists of a traded annuity, and along with unspanned income, the market is incomplete. Set in a Brownian framework, the income is driven by a multidimensional diffusion and in particular includes mean-reverting dynamics. The equilibrium is characterised by a system of fully coupled quadratic backward stochastic differential equations, a solution to which is proved to exist under Markovian assumptions. We also show that the equilibrium allocations lead to Pareto-optimal allocations only in exceptional situations.

Systems of Ergodic BSDEs Arising in Regime Switching Forward Performance Processes

We introduce and solve a new type of quadratic backward stochastic differential equation (BSDE) systems defined in an infinite time horizon, called ergodic BSDE systems. Such systems arise naturall...

Fast Solution Methods for Convex Quadratic Optimization of Fractional Differential Equations

In this paper, we present numerical methods suitable for solving convex quadratic Fractional Differential Equation (FDE) constrained optimization problems, with box constraints on the state and/or control variables. We develop an Alternating Direction Method of Multipliers (ADMM) framework, which uses preconditioned Krylov subspace solvers for the resulting sub-problems. The latter allows us to tackle a range of Partial Differential Equation (PDE) optimization problems with box constraints, posed on space-time domains, that were previously out of the reach of state-of-the-art preconditioners. In particular, by making use of the powerful Generalized Locally Toeplitz (GLT) sequences theory, we show that any existing GLT structure present in the problem matrices is preserved by ADMM, and we propose some preconditioning methodologies that could be used within the solver, to demonstrate the generality of the approach. Focusing on convex quadratic programs with time-dependent 2-dimensional FDE constraints, we derive multilevel circulant preconditioners, which may be embedded within Krylov subspace methods, for solving the ADMM sub-problems. Discretized versions of FDEs involve large dense linear systems. In order to overcome this difficulty, we design a recursive linear algebra, which is based on the Fast Fourier Transform (FFT). We manage to keep the storage requirements linear, with respect to the grid size $N$, while ensuring an order $N \log N$ computational complexity per iteration of the Krylov solver. We implement the proposed method, and demonstrate its scalability, generality, and efficiency, through a series of experiments over different setups of the FDE optimization problem.

On the uniqueness of solutions to quadratic BSDEs with non-convex generators and unbounded terminal conditions

We prove a uniqueness result of the unbounded solution for a quadratic backward stochastic differential equation whose terminal condition is unbounded and whose generator $g$ may be non-Lipschitz continuous in the state variable $y$, non-convex (non-concave) in the state variable $z$, and instead satisfies a strictly quadratic condition and an additional assumption. The key observation is that if the generator is strictly quadratic, then the quadratic variation of the first component of the solution admits an exponential moment. Typically, a Lipschitz perturbation of some convex (concave) function satisfies the additional assumption mentioned above. This generalizes some results obtained in \cite{BriandHu2006PTRF} and \cite{BriandHu2008PTRF}.

APPROXIMATE SOLUTION TO FRACTIONAL RICCATI DIFFERENTIAL EQUATIONS

In this paper, quadratic Riccati differential equation of fractional order has been solved by employing the optimal homotopy asymptotic method (Optimal HAM) with application to random processes, optimal control and diffusion problems. Optimal HAM uses simple computations with quite acceptable approximate solutions which have close agreement with exact solutions as compared to other techniques. To illustrate the efficiency and reliability of the method, some examples are provided and the results are discussed with tables.

Expected Utility Maximization Problem Under State Constraints and Model Uncertainty

We study a general robust utility maximization problem from terminal wealth and consumption under state constraints. Our framework includes financial models with constrained portfolios, labor income and large investor models. We state the existence and the uniqueness of the consumption–investment strategy by studying the associated quadratic backward stochastic differential equation. We characterize the optimal control by using the duality method and deriving a dynamic maximum principle.

Identifying weak focus and center in a convection model

In this paper, we consider a 3-dimensional convection model and identify its weak focus and center. This model is described by quadratic polynomial differential equations, having three equilibria, two of them may be center-focus type. Center manifold theory and normal form theory are applied to prove that at least three limit cycles can bifurcate from a Hopf critical point around one of the two equilibria.

Recasting the mass-action rate equations of open chemical reaction networks into a universal quadratic format

Recasting the rate equations of mass-action chemical kinetics into universal formats is a potentially useful strategy to rationalize typical features that are observed in the space of the species concentrations. For example, a remarkable feature is the appearance of the so-called slow manifolds (subregions of the concentration space where the trajectories bundle), whose detection can be exploited to simplify the description of the slow part of the kinetics via model reduction and to understand how the chemical network approaches the stationary state. Here we focus on generally open chemical reaction networks with continuous injection of species at constant rates, that is, the situation of idealized biochemical networks and microreactors under well-mixing conditions and externally controllable input of chemicals. We show that a unique format of pure quadratic ordinary differential equations can be achieved, regardless of the nonlinearity of the kinetic scheme, by means of a suitable change and extension of the set of dynamical variables. Then we outline some possible employments of such a format, with special emphasis on a low-computational-cost strategy to localize the slow manifolds which are indeed observed also for open systems.

A computational method for solving differential equations with quadratic nonlinearity by using Bernoulli polynomials

In this paper, a matrix method is developed to solve quadratic non-linear differential equations. It is assumed that the approximate solutions of main problem which we handle primarily, is in terms of Bernoulli polynomials. Both the approximate solution and the main problem are written in matrix form to obtain the solution. The absolute errors are applied to numeric examples to demonstrate efficiency and accuracy of this technique. The obtained tables and figures in the numeric examples show that this method is very sufficient and reliable for solution of non-linear equations. Also, a formula is utilized based on residual functions and mean value theorem to seek error bounds.

Coupled and mixed coupled hybrid fixed point principles in a partially ordered Banach algebra and PBVPs of nonlinear coupled quadratic differential equations

Coupled and mixed coupled hybrid fixed point principles in a partially ordered Banach algebra and PBVPs of nonlinear coupled quadratic differential equations

An Application of Picard Iteration Method to Fractional Quadratic Ricccati Differential Equations

In this paper, we present an algorithm of the Picard's Iteration method to solve Fractional Quadratic Riccati differential equations with Caputo-derivatives. The non-linear terms are easily and simply expanded using the traditional method of expansion. Four (4) illustrative examples are given to verify the reliability and efficiency of the method. The approximate solutions obtained compare favorably with the exact solutions and the approximate solutions obtained by other numerical methods in the literature.

An Application of Picard Iteration Method to Fractional Quadratic Ricccati Differential Equations

In this paper, we present an algorithm of the Picard's Iteration method to solve Fractional Quadratic Riccati differential equations with Caputo-derivatives. The non-linear terms are easily and simply expanded using the traditional method of expansion. Four (4) illustrative examples are given to verify the reliability and efficiency of the method. The approximate solutions obtained compare favorably with the exact solutions and the approximate solutions obtained by other numerical methods in the literature.

Quadratic Differential Equations in Three Variables without Multivalued Solutions: Part I

For ordinary differential equations in the complex domain, a central problem is to understand, in a given equation or class of equations, those whose solutions do not present multivaluedness. We consider autonomous, first-order, quadratic homogeneous equations in three variables, and begin the classification of those which do not have multivalued solutions.

Phase portraits of Abel quadratic differential systems of second kind with symmetries

ABSTRACTWe provide normal forms and the global phase portraits on the Poincaré disk of the Abel quadratic differential equations of the second kind having a symmetry with respect to an axis or to the origin. Moreover, we also provide the bifurcation diagrams for these global phase portraits.

Algorithms for synthesizing management solutions based on OLAP-technologies

OLAP technologies are a convenient means of analyzing large amounts of information. An attempt was made in their work to improve the synthesis of optimal management decisions. The developed algorithms allow forecasting the needs and accepted management decisions on the main types of the enterprise resources. Their advantage is the efficiency, based on the simplicity of quadratic functions and differential equations of only the first order. At the same time, the optimal redistribution of resources between different types of products from the assortment of the enterprise is carried out, and the optimal allocation of allocated resources in time. The proposed solutions can be placed on additional specially entered coordinates of the hypercube representing the data warehouse.

A stability approach for solving multidimensional quadratic BSDEs

We establish an existence and uniqueness result for a class of multidimensional quadratic backward stochastic differential equations (BSDEs). This class is characterized by constraints on some uniform a priori estimates on solutions of a sequence of approximated BSDEs. We also present effective examples of applications. Our approach relies on the strategy developed by Briand and Elie in [Stochastic Process. Appl. 123 2921–2939] concerning scalar quadratic BSDEs.

Quadratic backward stochastic differential equations driven by [formula omitted]-Brownian motion: Discrete solutions and approximation

In this paper, we consider backward stochastic differential equations driven by G-Brownian motion (GBSDEs) under quadratic assumptions on coefficients. We prove the existence and uniqueness of solution for such equations. On the one hand, a priori estimates are obtained by applying the Girsanov type theorem in the G-framework, from which we deduce the uniqueness. On the other hand, to prove the existence of solutions, we first construct solutions for discrete GBSDEs by solving corresponding fully nonlinear PDEs, and then approximate solutions for general quadratic GBSDEs in Banach spaces.

Modeling of a closed mono-branch labor market conditions

Market modeling issues are currently acute as they are responsible for efficient operation of a labor market to achieve full employment and a high rate of economic growth. The aim of the study is to construct a theoretically reasonable closed mono-branch mathematical model that describes the behavior of economic agents at a labor market using a system of differential equations. The model is constructed on the following hypotheses: 1) a market is a closed system with a constant number of the unemployed (applicants) and employees; 2) employees and applicants can be divided into three conventional categories: low-skilled workers the demand for whom is low or absent, the number of these people is equal to the number of vacant positions at an enterprise; average-skilled workers, who may later join to a category of low-skilled or highly qualified workers; highly qualified workers and employers are mostly interested in them. To analyze the dynamic of average-skilled workers the staff training coefficient is implemented to select these employees. Shares of the each category representatives have been chosen as modeling variables. The staff training coefficient, as well as the number of employees and the unemployed and the amount of subjects of each category have been accepted to be constant according to the initial hypotheses. The dynamic of the variables is described by the system of three nonlinear differential equations. Consideration of the system peculiarities makes it possible to find the exact solution of the system in quadratures i.e. to determine the quantitative structure of each of the subjects of the labor market at any moment of time. Particular attention is paid to the asymptotic properties of solutions: the equilibrium points of the system have been found and their stability has been investigated. The research results have revealed that a proactive employer hires highly qualified workers and pays particular attention to human resource policy and it proves the model adequacy. The economic interpretation of the obtained mathematic results in the terms of the initial task has classified possible situations at a labor market and has made a conclusion about the dynamics of each category of the economic agents depending on the initial conditions. Further studies will be devoted to the system complication by including new parameters, e.g. salary impact factor, training costs spent on average-skilled employees, lag, etc. The model may be interesting for both the scholars studying labor market conditions and human resource managers. Keywords labor market conditions, dynamic model of a labor market, a closed system, employees, the unemployed, staff, staff turn-over, skills of employees, system of differential equations, asymptotic features.

A Global Stochastic Maximum Principle for Fully Coupled Forward-Backward Stochastic Systems

We study a stochastic optimal control problem for fully coupled forward-backward stochastic control systems with a nonempty control domain. For our problem, the first-order and second-order variational equations are fully coupled linear forward-backward stochastic differential equations. Inspired by Hu [Probab. Uncertain. Quant. Risk, 2 (2017), pp. 1--20], we develop a new decoupling approach by introducing an adjoint equation which is a quadratic backward stochastic differential equation. By revealing the relations among the terms of the first-order Taylor expansions, we estimate the orders of them and derive a global stochastic maximum principle which includes a completely new term. Applications to stochastic linear quadratic control problems are investigated.

Dhage iteration method for approximating positive solutions of quadratic functional differential equations

Dhage iteration method for approximating positive solutions of quadratic functional differential equations