System Of Variational Inequalities Research Articles

Trading VIX futures under mean reversion with regime switching

This paper studies the optimal VIX futures trading problems under a regime-switching model. We consider the VIX as mean reversion dynamics with dependence on the regime that switches among a finite number of states. For the trading strategies, we analyze the timings and sequences of the investor’s market participation, which leads to several corresponding coupled system of variational inequalities. The numerical approach is developed to solve these optimal double stopping problems by using projected-successive-over-relaxation (PSOR) method with Crank–Nicolson scheme. We illustrate the optimal boundaries via numerical examples of two-state Markov chain model. In particular, we examine the impacts of transaction costs and regime-switching timings on the VIX futures trading strategies.

A Class of Delay Differential Variational Inequalities

In the paper, we introduce a class of delay differential variational inequalities consisting of a system of delay differential equations and variational inequalities. The existence conclusion of Caratheodory’s weak solution for delay differential variational equalities is obtained. Furthermore, an algorithm for solving the delay differential variational inequality is shown, and the convergence analysis for the algorithm is given. Finally, a numerical example is given to verify the validity of the algorithm.

Dynamic Spatial Auction Market Models with General Cost Mappings

The deregulation of electricity markets in Europe has deeply changed the organization of this sector. Vertically integrated generating companies have been unbundled to create competition and to increase the competitiveness of electricity markets. Directive 96/92/EC was issued by the European Commission to liberalize electricity markets and to pave the way for the creation of the Internal Electricity Market. In particular, this Directive aimed at promoting the competition in the activities of electricity generation and wholesale through the creation of a “marketplace” and the maximization of transparency and efficiency. Competition in European day-ahead electricity markets has been established through auction markets where electricity producers and consumers offer/bid prices and volumes. This paper suggests a dynamic equilibrium model for a system of auction markets linked by transmission lines and subject to energy balance and transmission constraints, such as those characterizing restructured electricity markets. This model is treated as a system of variational inequalities with arbitrary monotone mappings. An inexact splitting type method is proposed to find its solution. Numerical experiments are conducted on the Italian day-ahead electricity market.

Trading VIX Futures Under Mean Reversion with Regime Switching

This paper studies the optimal VIX futures trading problems under a regime-switching model. We consider the VIX as mean reversion dynamics with dependence on the regime that switches among a finite number of states. For the trading strategies, we analyze the timings and sequences of the investor’s market participation, which leads to several corresponding coupled system of variational inequalities. The numerical approach is developed to solve these optimal double stopping problems by using projected-successive-over-relaxation (PSOR) method with Crank-Nicolson scheme. We illustrate the optimal boundaries via numerical examples of two-state Markov chain model. In particular, we examine the impacts of transaction costs and regime-switching timings on the VIX futures trading strategies.

Iterative algorithms for solving variational inequalities and fixed point problems for asymptotically nonexpansive mappings in Banach spaces

The purpose of this paper is to study some iterative algorithms for finding a common element of the set of solutions of systems of variational inequalities for inverse-strongly accretive mappings and the set of fixed points of an asymptotically nonexpansive mapping in uniformly convex and 2-uniformly smooth Banach space or uniformly convex and q-uniformly smooth Banach space. Strong convergence theorems are obtained under suitable conditions. We also give some numerical examples to support our main results. The results obtained in this paper improve and extend the recent ones announced by many others in the literature.

Some New Algorithms for Solving a System of General Variational Inequalities

Some New Algorithms for Solving a System of General Variational Inequalities

Optimal Switching in Finite Horizon under State Constraints

We study an optimal switching problem with a state constraint: the controller is only allowed to choose strategies that keep the controlled diffusion in a closed domain. We prove that the value function associated with this problem is the limit of value functions associated with unconstrained switching problems with penalized coefficients, as the penalization parameter goes to infinity. This convergence allows one to set a dynamic programming principle for the constrained switching problem. We then prove that the value function is a solution to a system of variational inequalities (SVI) in the constrained viscosity sense. We finally prove that uniqueness for our SVI cannot hold, and we give a weaker characterization of the value function as the maximal solution to this SVI. All our results are obtained without any regularity assumption on the constraint domain.

Perfect Plasticity with Damage and Healing at Small Strains, Its Modeling, Analysis, and Computer Implementation

The quasistatic, Prandtl--Reuss perfect plasticity at small strains is combined with a gradient, reversible (i.e., admitting healing) damage which influences both the elastic moduli and the yield stress. Existence of weak solutions of the resulting system of variational inequalities is proved by a suitable fractional-step discretization in time with guaranteed numerical stability and convergence. After finite-element approximation, this scheme is computationally implemented and illustrative two-dimensional simulations are performed. The model allows, e.g., for application in geophysical modeling of reoccurring rupture of lithospheric faults. Resulting incremental problems are solved in MATLAB by quasi-Newton method to resolve the elastoplasticity component of the solution, while the damage component is obtained by solving a quadratic programming problem.

Strong convergence theorem for a system of generalized mixed equilibrium problems and finite family of Bregman nonexpansive mappings in Banach spaces

In this paper we introduce a general iterative algorithm for finding the common element of the set of common fixed points of a finite family of Bregman nonexpansive mappings and the set of solutions of systems of generalized mixed equilibrium problems. As an application, we find a solution for system of mixed variational inequality.

Convergence of an Iterative Algorithm for Systems of Variational Inequalities and Nonlinear Mappings in Banach Spaces

In this paper, we consider the problem of convergence of an iterative algorithm for a general system of variational inequalities, a nonexpansive mapping and an -strictly pseudo-contractive mapping. Strong convergence theorems are established in the framework of real Banach spaces.

Hybrid extragradient method with regularization for triple hierarchical variational inequalities with general mixed equilibrium and split feasibility constraints

In this paper, we introduce a hybrid extragradient iterative algorithm with regularization for solving the triple hierarchical variational inequality problem (THVIP) (defined over the common fixed point set of finitely many nonexpansive mappings and a strictly pseudocontraction) with constraints of a general mixed equilibrium problem (GMEP), a split feasibility problem (SFP) and a general system of variational inequalities (GSVI). The iterative algorithm is based on Korpelevich's extragradient method, viscosity approximation method, Mann's iteration method, hybrid steepest descent method and gradient-projection method (GPM) with regularization. It is proven that, under very mild conditions, the sequences generated by the proposed algorithm converge strongly to a unique solution of the THVIP. We also give the applications of our results for solving some special cases of the THVIP. The results presented in this paper improve and extend some corresponding ones in the earlier and recent literature.

Nash-Type Equilibria for Systems of Szulkin Functionals

In this paper the solutions of some systems of variational inequalities are obtained as Nash-type equilibria of the corresponding systems of Szulkin functionals. This is achieved by an iterative scheme based on Ekeland’s variational principle, whose convergence is proved via the vector technique involving inverse-positive matrices. An application to periodic solutions for a system of two second order ordinary differential equations with singular ϕ-Laplacians is included.

A new technique for convergence theorem of fixed point problem of quasi-nonexpansive mapping

For the purpose of this paper, we use the method different from the relaxed extragradient method for finding a common element of the set of fixed points of a quasi-nonexpansive mapping, the set of solutions of equilibrium problems, and the set of solutions of a modified system of variational inequalities without demiclosed condition of W and $W_{\omega}:= (1-\omega )I+\omega W$ , where W is a quasi-nonexpansive mapping and $\omega\in (0,\frac{1}{2} )$ in the framework of Hilbert space. By using our main result, we obtain a strong convergence theorem involving a finite family of nonspreading mappings and another corollary. Moreover, we give a numerical example to encourage our main theorem.

The generalized viscosity implicit rules of nonexpansive mappings in Hilbert spaces

AbstractThe generalized viscosity implicit rules of nonexpansive mappings in Hilbert spaces are established. The strong convergence theorems of the rules are proved under certain assumptions imposed on the sequences of parameters. The results presented in this paper extend and improve the main results of Refs. (Moudafi in J. Math. Anal. Appl. 241:46-55, 2000; Xu et al. in Fixed Point Theory Appl. 2015:41, 2015). Moreover, applications to a more general system of variational inequalities, the constrained convex minimization problem and K-mapping are included.

Convergence analysis of hybrid projection with Cesàro mean method for the split equilibrium and general system of finite variational inequalities

In this paper, we present a new iterative algorithm for finding a common element of three solution sets; (i) split equilibrium problem; (ii) general system of finite variational inequality problem; and (iii) fixed point problem. This algorithm is modified by hybrid method based on Cesàro mean in real Hilbert spaces. Furthermore, a strong convergent theorem is established and this theorem is the generalization of many previously known results in this research area. Finally, we study the rate of convergence of the iterative algorithm and some illustrative numerical examples are presented.

Mann-type hybrid steepest-descent method for three nonlinear problems

We introduce a Mann-type hybrid steepest-descent iterative algorithm for finding a common element of the set of solutions of a general mixed equilibrium problem, the set of solutions of general system of variational inequalities, the set of solutions of finitely many variational inequalities, and the set of common fixed points of finitely many nonexpansive mappings and a strict pseudocontraction in a real Hilbert space. We derive the strong convergence of the iterative algorithm to a common element of these sets, which also solves some hierarchical variational inequality.

Strong convergence of a general iterative algorithm for a finite family of accretive operators in Banach spaces

AbstractThe purpose of this paper is to present a new iterative scheme for finding a common solution to a variational inclusion problem with a finite family of accretive operators and a modified system of variational inequalities in infinite-dimensional Banach spaces. Under mild conditions, a strong convergence theorem for approximating this common solution is proved. The methods in the paper are novel and different from those in the early and recent literature.

Hybrid viscosity methods for equilibrium problems, variational inequalities, and fixed point problems

In this paper, we introduce a hybrid viscosity iterative algorithm for finding a common element of the set of solutions of a general mixed equilibrium problem, the set of solutions of general system of variational inequalities, the set of common fixed points of one finite family of nonexpansive mappings, and another infinite family of nonexpansive mappings in a real Hilbert space. This hybrid viscosity iterative algorithm is based on viscosity approximation method, Mann’s iterative method, projection method, strongly positive bounded linear Operator, and -mapping approaches. We study the strong convergence of the proposed algorithm to a common element under appropriate assumptions, which also solves an optimization problem. The result presented in this paper improves and extends some known results in the literature.

American Option Valuation under Continuous-Time Markov Chains

This paper is concerned with the solution of the optimal stopping problem associated to the value of American options driven by continuous-time Markov chains. The value-function of an American option in this setting is characterised as the unique solution (in a distributional sense) of a system of variational inequalities. Furthermore, with continuous and smooth fit principles not applicable in this discrete state-space setting, a novel explicit characterisation is provided of the optimal stopping boundary in terms of the generator of the underlying Markov chain. Subsequently, an algorithm is presented for the valuation of American options under Markov chain models. By application to a suitably chosen sequence of Markov chains, the algorithm provides an approximate valuation of an American option under a class of Markov models that includes diffusion models, exponential Lévy models, and stochastic differential equations driven by Lévy processes. Numerical experiments for a range of different models suggest that the approximation algorithm is flexible and accurate. A proof of convergence is also provided.

American Option Valuation under Continuous-Time Markov Chains

This paper is concerned with the solution of the optimal stopping problem associated to the value of American options driven by continuous-time Markov chains. The value-function of an American option in this setting is characterised as the unique solution (in a distributional sense) of a system of variational inequalities. Furthermore, with continuous and smooth fit principles not applicable in this discrete state-space setting, a novel explicit characterisation is provided of the optimal stopping boundary in terms of the generator of the underlying Markov chain. Subsequently, an algorithm is presented for the valuation of American options under Markov chain models. By application to a suitably chosen sequence of Markov chains, the algorithm provides an approximate valuation of an American option under a class of Markov models that includes diffusion models, exponential Lévy models, and stochastic differential equations driven by Lévy processes. Numerical experiments for a range of different models suggest that the approximation algorithm is flexible and accurate. A proof of convergence is also provided.