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.
Highlights
Stochastic differential games model the interaction between players whose objective functions depend on the evolution of a certain continuous-time stochastic process
They constitute a generalization of the well-known 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
This paper presents a fixed-point policy-iteration-type algorithm to solve systems of quasi-variational inequalities resulting from a Verification Theorem for symmetric nonzero-sum stochastic impulse games
Summary
Stochastic differential games model the interaction between players whose objective functions depend on the evolution of a certain continuous-time stochastic process. This class is broad enough to include many interesting applications; no less than the competing central banks problem (whether in its linear form [1] or others considered in the single bank formulation [4,19,30,32]), the cash management problem [10] (reducing its dimension by a simple change of variables) and the generalization of many impulse control problems to the two-player case For this class of games, an iterative algorithm is presented which substantially improves [3, Alg.2] by harnessing the symmetry of the problem, removing the strong dependence on the initial guess and dispensing with the relaxation scheme altogether. The latter give insight and motivate further research into this field
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