A Fixed-Point Policy-Iteration-Type Algorithm for Symmetric Nonzero-Sum Stochastic Impulse Control Games
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A Fixed-Point Policy-Iteration-Type Algorithm for Symmetric Nonzero-Sum Stochastic Impulse Control Games Diego Zabaljauregui1
© The Author(s) 2020
Abstract 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. Keywords Stochastic differential games · Nonzero-sum games · Impulse control · Nash equilibrium · Quasi-variational inequality · Howard’s algorithm · Fixed point policy iteration · Weakly chained diagonally dominant matrix
Introduction Stochastic differential games model the interaction between players whose objective functions depend on the evolution of a certain continuous-time stochastic process. The subclass of impulse games focuses on the case where the players only act at discrete
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Diego Zabaljauregui [email protected] Department of Statistics, London School of Economics and Political Science, London, UK
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Applied Mathematics & Optimization
(usually random) points in time by shifting the process. In doing so, each of them incurs into costs and possibly generates “gains” for the others at the same time. They constitute a generalization of the well-known (single-player) optimal impulse control problems [33, Chpt.7-10], which have found a wide range of applications in finance, energy markets and insurance [5,9,13,23,31], among plenty of other fields. From a deterministic numerical viewpoint, an impulse control problem entails the resolution of a differential quasi-variational inequality (QVI) to compute the value function and, when possible, retrieve an optimal strategy. Policy-iteration-type algorithms [4,15,16] undoubtedly occupy an ubiquitous place in this respect, especially in the infinite horizon case. The presence of a second player makes matters much more challenging, as one needs to find two optimal (or equilibrium) payoffs dependent on one
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