Solution of oligopoly market equilibrium problem using modified newton method

The paper aims to find the solution of oligopoly market equilibrium problem through system of nonlinear equations. We propose modified newton method to obtain the solution of system of nonlinear equations. We show that our proposed method has higher order of convergence. Keywords: System of nonlinear equations, modified newton method, oligopolistic market equilibrium problem.Comment: arXiv admin note: substantial text overlap with arXiv:2209.0100

Computing Perfect Stationary Equilibria in Stochastic Games

The notion of stationary equilibrium is one of the most crucial solution concepts in stochastic games. However, a stochastic game can have multiple stationary equilibria, some of which may be unstable or counterintuitive. As a refinement of stationary equilibrium, we extend the concept of perfect equilibrium in strategic games to stochastic games and formulate the notion of perfect stationary equilibrium (PeSE). To further promote its applications, we develop a differentiable homotopy method to compute such an equilibrium. We incorporate vanishing logarithmic barrier terms into the payoff functions, thereby constituting a logarithmic-barrier stochastic game. As a result of this barrier game, we attain a continuously differentiable homotopy system. To reduce the number of variables in the homotopy system, we eliminate the Bellman equations through a replacement of variables and derive an equivalent system. We use the equivalent system to establish the existence of a smooth path, which starts from an arbitrary total mixed strategy profile and ends at a PeSE. Extensive numerical experiments further affirm the effectiveness and efficiency of the method

Competitive multi-player stochastic games with applications to multi-person financial contracts

Competitive Multi-Player Stochastic Games with Applications to Multi-Person Financial Contracts Ivan Guo Abstract In the financial market, almost all traded derivatives only involve two parties. The aim of this thesis is to design and evaluate financial contracts involving multiple parties. This is done by utilising and extending concepts from game theory, financial mathematics and backward stochastic differential equations. The thesis is divided into two parts: multi-player stochastic competitive games and multi-person financial contracts. The first part of the thesis proposes two novel classes of multi-period multi-player stopping games: the multi-player redistribution game and the multi-player affine game. Both formulations are generalisations of the classic two-player Dynkin game, with a focus on designing the dependence between the payoffs of all players and their stopping decisions. These games are shown to be weakly unilaterally competitive, and sufficient conditions are given for the existence of optimal equilibria (a new solution concept motivated by financial applications), individual values and coalition values. The second part of the thesis introduces the notion of multi-person financial contracts by extending the two-person game option. These contracts may involve an arbitrary number of parties and each party is allowed to make a wide array of decisions, which then determines the settlement date as well as the payoffs. The generalised Snell envelope is introduced for the valuation of multi-person contracts and sufficient conditions for the existence of unique and additive arbitrage prices are provided. Finally, a new class of multi-dimensional reflected backward stochastic differential equations are proposed to model multi-person affine game options under market friction

Competitive multi-player stochastic games with applications to multi-person financial contracts

Competitive Multi-Player Stochastic Games with Applications to Multi-Person Financial Contracts Ivan Guo Abstract In the financial market, almost all traded derivatives only involve two parties. The aim of this thesis is to design and evaluate financial contracts involving multiple parties. This is done by utilising and extending concepts from game theory, financial mathematics and backward stochastic differential equations. The thesis is divided into two parts: multi-player stochastic competitive games and multi-person financial contracts. The first part of the thesis proposes two novel classes of multi-period multi-player stopping games: the multi-player redistribution game and the multi-player affine game. Both formulations are generalisations of the classic two-player Dynkin game, with a focus on designing the dependence between the payoffs of all players and their stopping decisions. These games are shown to be weakly unilaterally competitive, and sufficient conditions are given for the existence of optimal equilibria (a new solution concept motivated by financial applications), individual values and coalition values. The second part of the thesis introduces the notion of multi-person financial contracts by extending the two-person game option. These contracts may involve an arbitrary number of parties and each party is allowed to make a wide array of decisions, which then determines the settlement date as well as the payoffs. The generalised Snell envelope is introduced for the valuation of multi-person contracts and sufficient conditions for the existence of unique and additive arbitrage prices are provided. Finally, a new class of multi-dimensional reflected backward stochastic differential equations are proposed to model multi-person affine game options under market friction

Self-Evaluation Applied Mathematics 2003-2008 University of Twente

This report contains the self-study for the research assessment of the Department of Applied Mathematics (AM) of the Faculty of Electrical Engineering, Mathematics and Computer Science (EEMCS) at the University of Twente (UT). The report provides the information for the Research Assessment Committee for Applied Mathematics, dealing with mathematical sciences at the three universities of technology in the Netherlands. It describes the state of affairs pertaining to the period 1 January 2003 to 31 December 2008