System Of Quasi-variational Inequalities Research Articles

Nonzero-Sum Stochastic Impulse Games with an Application in Competitive Retail Energy Markets

We study a nonzero-sum stochastic differential game with both players adopting impulse controls, on a finite time horizon. The objective of each player is to maximize her total expected discounted profits. The resolution methodology relies on the connection between Nash equilibrium and the corresponding system of quasi-variational inequalities (QVIs in short). We prove, by means of the weak dynamic programming principle for the stochastic differential game, that the equilibrium expected payoff of each player is a constrained viscosity solution to the associated QVIs system in the class of linear growth functions. We also introduce a family of equilibrium expected payoffs converging to our equilibrium expected payoff of each player, and which is characterized as the unique constrained viscosity solutions of an approximation of our QVIs system. This convergence result is useful for numerical purpose. We apply a probabilistic numerical scheme which approximates the solution of the QVIs system to the case of the competition between two electricity retailers. We show how our model reproduces the qualitative behavior of electricity retail competition.

Nonzero-Sum Stochastic Differential Games Between an Impulse Controller and a Stopper

We study a two-player nonzero-sum stochastic differential game, where one player controls the state variable via additive impulses, while the other player can stop the game at any time. The main goal of this work is to characterize Nash equilibria through a verification theorem, which identifies a new system of quasivariational inequalities, whose solution gives equilibrium payoffs with the correspondent strategies. Moreover, we apply the verification theorem to a game with a one-dimensional state variable, evolving as a scaled Brownian motion, and with linear payoff and costs for both players. Two types of Nash equilibrium are fully characterized, i.e. semi-explicit expressions for the equilibrium strategies and associated payoffs are provided. Both equilibria are of threshold type: in one equilibrium players’ intervention are not simultaneous, while in the other one the first player induces her competitor to stop the game. Finally, we provide some numerical results describing the qualitative properties of both types of equilibrium.

A Fixed-Point Policy-Iteration-Type Algorithm for Symmetric Nonzero-Sum Stochastic Impulse Control Games

Nonzero-sum stochastic differential games with impulse controls offer a realistic and far-reaching modelling framework for applications within finance, energy markets, and other areas, but the difficulty in solving such problems has hindered their proliferation. Semi-analytical approaches make strong assumptions pertaining to very particular cases. To the author’s best knowledge, the only numerical method in the literature is the heuristic one we put forward in Aïd et al (ESAIM Proc Surv 65:27–45, 2019) to solve an underlying system of quasi-variational inequalities. Focusing on symmetric games, this paper presents a simpler, more precise and efficient fixed-point policy-iteration-type algorithm which removes the strong dependence on the initial guess and the relaxation scheme of the previous method. A rigorous convergence analysis is undertaken with natural assumptions on the players strategies, which admit graph-theoretic interpretations in the context of weakly chained diagonally dominant matrices. A novel provably convergent single-player impulse control solver is also provided. The main algorithm is used to compute with high precision equilibrium payoffs and Nash equilibria of otherwise very challenging problems, and even some which go beyond the scope of the currently available theory.

Markovian strategies with continuous and impulse controls for a differential game model of revolution

ABSTRACT This paper is concerned with a piecewise-deterministic differential game model of political regime changes. We modify and study the model proposed by Boucekkine et al. in [7]. The original model does not allow all players to take full controls as the situation progresses. Hence, it does not lead to closed-loop strategies. We fix the problem by deriving and using a system of quasi-variational inequalities associated with the differential game, and proving a criterion for the regime change. As a result, we find Markovian strategies for all players. A numerical example for illustration of the method is given. Implications of the results to political changes in a society are discussed. Some results are extended to more general models that incorporate gradual and abrupt changes, as well as continuous and impulse controls.

A Fixed-Point Policy-Iteration-Type Algorithm for Symmetric Nonzero-Sum Stochastic Impulse Control Games

Nonzero-sum stochastic differential games with impulse controls offer a realistic and far-reaching modelling framework for applications within finance, energy markets, and other areas, but the difficulty in solving such problems has hindered their proliferation. Semi-analytical approaches make strong assumptions pertaining to very particular cases. To the author's best knowledge, the only numerical method in the literature is the heuristic one we put forward in [ABM+19] to solve an underlying system of quasi-variational inequalities. Focusing on symmetric games, this paper presents a simpler, more precise and efficient fixed-point policy-iteration-type algorithm which removes the strong dependence on the initial guess and the relaxation scheme of the previous method. A rigorous convergence analysis is undertaken with natural assumptions on the players strategies, which admit graph-theoretic interpretations in the context of weakly chained diagonally dominant matrices. A novel provably convergent single-player impulse control solver is also provided. The main algorithm is used to compute with high precision equilibrium payoffs and Nash equilibria of otherwise very challenging problems, and even some which go beyond the scope of the currently available theory.

A policy iteration algorithm for nonzero-sum stochastic impulse games

This work presents a novel policy iteration algorithm to tackle nonzero-sum stochastic impulse games arising naturally in many applications. Despite the obvious impact of solving such problems, there are no suitable numerical methods available, to the best of our knowledge. Our method relies on the recently introduced characterisation of the value functions and Nash equilibrium via a system of quasi-variational inequalities. While our algorithm is heuristic and we do not provide a convergence analysis, numerical tests show that it performs convincingly in a wide range of situations, including the only analytically solvable example available in the literature at the time of writing.

Overlapping Domain Decomposition Method for a Noncoercive System of Quasi-Variational Inequalities Related to the Hamilton–Jacobi–Bellman Equation

This paper deals with numerical analysis of an overlapping Schwarz method on nonmatching grids for a noncoercive system of quasi-variational inequalities related to the Hamilton---Jacobi---Bellman equation. We prove that the discrete alternating Schwarz sequences on every subdomain converge monotonically into the unique solution of the discrete problem and geometrically in uniform norm. Optimally L?-error estimates are also established.

On a New Class of Systems of Generalized Quasi-Variational Inequalities

We introduce new types of systems of generalized quasi-variational inequalities and we prove the existence of the solutions by using results of pair equilibrium existence for free abstract economies. We consider the fuzzy models and we also introduce the random free abstract economy and the random equilibrium pair. The existence of the solutions for the systems of quasi-variational inequalities comes as consequences of the existence of equilibrium pairs for the considered free abstract economies.

Level-Set Approach for Reachability Analysis of Hybrid Systems under Lag Constraints

This study aims at characterizing a reachable set of a hybrid dynamical system with a lag constraint in the switch control. The setting does not consider any controllability assumptions and uses a level-set approach. The approach consists of the introduction of an adequate hybrid optimal control problem with lag constraints on the switch control whose value function allows a characterization of the reachable set. The value function is in turn characterized by a system of quasi-variational inequalities (SQVI). We prove a comparison principle for the SQVI which shows uniqueness of its solution. A class of numerical finite difference scheme for solving the system of inequalities is proposed, and the convergence of the numerical solution toward the value function is studied using the comparison principle. Some numerical examples illustrating the method are presented. Our study is motivated by an industrial application. We are interested in the maximum range of hybrid vehicles.

Some Parallel Algorithms for a New System of Quasi Variational Inequalities

In this paper, we introduce a new system of quasi variational inequalities. The projection technique is used to establish the equivalence between this new system of quasi variational inequalities and the fixed point problem. The fixed point formulation enables us to suggest some parallel projection iterative methods for solving the system of quasi variational inequalities. Convergence analysis of the proposed methods is investigated. Several special cases are discussed. Results proved in this paper continue to hold for these problems.

Domain decomposition method for a system of quasivariational inequalities

We propose a domain decomposition method for a system of quasivariational inequalities related to the HJB equation. The monotone convergence of the algorithm is also established.

An iterative scheme for a system of quasi variational inequalities

An iterative scheme for a system of quasi variational inequalities

Optimal Control of Hybrid Systems and a System of Quasi-Variational Inequalities

Optimal control of a class of hybrid dynamical systems is studied using the method of dynamic programming. It is proved that the value function is a discontinuous viscosity solution of the corresponding Hamilton--Jacobi--Bellman (HJB) equation which appears as a system of quasi-variational inequalities (SQVI) coupled by a nonlocal operator with a variable boundary condition. The comparison theorems are established for the sub- and super-solutions of the SQVI.

Optimal sale strategies in illiquid markets

The value of a position in a risky asset when optimally sold in an illiquid market is considered. The optimization problem is described as a stochastic impulse control problem, and it is shown that it is related to solving a system of quasi-variational inequalities. Existence of a solution to these inequalities are proved. A numerical implementation of the valuation algorithm is discussed and two numerical examples are presented. Further, two examples where the stochastic impulse control problem can be reduced to deterministic optimization problems are also given.

On the existence of solutions to systems of vector quasi-optimization problems

Some sufficient conditions for the existence of solutions to systems of vector quasi-optimization problems are given. These systems also include the systems of quasi-variational inequalities, quasi-equilibrium problems etc. As special case, we obtain some extensions for minimax theorems, general vector optimization problems, general vector quasi-optimization problems, quasi-equilibrium problems... concerning vector functions. From these we generalize some well-known results obtained by Blum and Oettli [4], Park [19], Chan and Pang[6], Parida and Sen [18], Browder and Minty [16], Ky Fan [8], etc.

Robust Optimal Switching Control for Nonlinear Systems

We formulate a robust optimal control problem for a general nonlinear system with finitely many admissible control settings and with costs assigned to switching of controls. We formulate the problem both in an L2-gain/dissipative system framework and in a game-theoretic framework. We show that, under appropriate assumptions, a continuous switching-storage function is characterized as a viscosity supersolution of the appropriate system of quasi-variational inequalities (the appropriate generalization of the Hamilton--Jacobi--Bellman--Isaacs equation for this context) and that the minimal such switching-storage function is equal to the continuous switching lower-value function for the game. Finally, we show how a prototypical example with one-dimensional state space can be solved by a direct geometric construction.

Comparison Theorems for Viscosity Solutions of a System of Quasivariational Inequalities with Application to Optimal Control with Switching Costs

In this paper we prove a comparison principle between a viscosity sub- and supersolution for a system of quasivariational inequalities and apply it to show that a continuous lower value vector function of an optimal switching-cost control problem is characterized as the minimal, nonnegative, continuous, viscosity supersolution of the SQVI.

The system of quasi-variational inequalities attached to the two-armed bandit problem

We give a characterization of the optimal value function of switching and stopping problems for two-parameter stochastic processes, particularly, bi-Markov processes in terms of the system of quasivariational inequalities, under the condition termed (F5) in Mazziotto and Millet [10]. We also give the construction of an optimal solution for such problems by the arguments of the one-parameter optimal stopping problem

Differential games with switching strategies

A zero-sum differential game of infinite horizon is considered. Positive switching costs are associated with each player. We prove that under a condition, which is different from the Isaacs' condition, the Elliot-Kalton value of the game always exists. The value is the unique viscosity solution of corresponding Isaacs' type system of equations, which appears to be a system of quasi-variational inequalities with bilateral obstacles. The discussion of the case that the switching costs approach to zero shows that the Isaacs' condition still has to be assumed to guarantee that the limit of the values corresponding to positive switching costs converge to the value corresponding to zero switching costs—the classical Elliot-Kalton value of the game.

A System of Quasi Variational Inequalities and Its Application to Reaction Diffusion Equations

On considere un systeme d'inequations quasi variationnelles qu'on applique a l'analyse des systemes de diffusion-reaction avec deux types de conditions aux limites (conditions de Dirichlet et de Neumann). Sous les conditions de coercivite on obtient les solutions stationnaires comme les points minimaux de fonctionnelles convexes. On traite le probleme de commande a retroaction a priori