A fixed-point policy-iteration-type algorithm for symmetric nonzero-sum stochastic impulse games

09/09/2019 ∙ by Diego Zabaljauregui, et al. ∙ 0

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 very particular cases. To the author's best knowledge, the only numerical method in the literature is the heuristic one we put forward to solve an underlying system of quasi-variational inequalities. Focusing on symmetric games, this paper presents a simpler and more 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 provably convergent single-player impulse control solver, often outperforming classical policy iteration, is also provided. The main algorithm is used to compute with high precision equilibrium payoffs and Nash equilibria of otherwise too challenging problems, and even some for which results go beyond the scope of all the currently available theory.

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