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 stron...

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Vydáno v:Applied mathematics & optimization Ročník 84; číslo 2; s. 1751 - 1790
Hlavní autor: Zabaljauregui, Diego
Médium: Journal Article
Jazyk:angličtina
Vydáno: New York Springer US 01.10.2021
Springer Nature B.V
Témata:
Algorithms
Calculus of Variations and Optimal Control; Optimization
Control
Convergence
Differential games
Game theory
Graph theory
Iterative methods
Mathematical and Computational Physics
Mathematical Methods in Physics
Mathematics
Mathematics and Statistics
Numerical and Computational Physics
Numerical methods
Simulation
Systems Theory
Theoretical
Weakly chained diagonally dominant matrix
Nonzero-sum games
Nash equilibrium
Quasi-variational inequality
Impulse control
Fixed point policy iteration
Stochastic differential games
Howard’s algorithm
ISSN:0095-4616, 1432-0606
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Shrnutí: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.
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ISSN:0095-4616
1432-0606
DOI:10.1007/s00245-020-09694-z