非零和随机冲量游戏的一种策略迭代算法

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 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 to solve an underlying system of quasi-variational inequalities. Focusing on symmetric games, this paper presents a simpler, more precise…

Nonzero-sum stochastic games and mean-field games with impulse controls

We consider a general class of nonzero-sum $N$-player stochastic games with impulse controls, where players control the underlying dynamics with discrete interventions. We adopt a verification approach and provide sufficient conditions for the Nash equilibria (NEs) of the game. We then consider the limit situation of $N \to \infty$, that is, a suitable mean-field game (MFG) with impulse controls. We show that under appropriate technical conditions, the existence of unique NE solution to the MFG, which is an $ε$-NE approximation to the $N$-player game, with $ε=O\left(\frac{1}{\sqrt{N}}\right)$.…

Nonzero-sum stochastic differential games between an impulse controller and a stopper

We study a two-player nonzero-sum stochastic differential game where one player controls the state variable via additive impulses while the other player can stop the game at any time. The main goal of this work is characterize Nash equilibria through a verification theorem, which identifies a new system of quasi-variational inequalities whose solution gives equilibrium payoffs with the correspondent strategies. Moreover, we apply the verification theorem to a game with a one-dimensional state variable, evolving as a scaled Brownian motion, and with linear payoff and costs for both players. Two…

Nonzero-sum stochastic impulse games with an application in competitive retail energy markets

We study a nonzero-sum stochastic differential game with both players adopting impulse controls, on a finite time horizon. The objective of each player is to maximize her total expected discounted profits. The resolution methodology relies on the connection between Nash equilibrium and the corresponding system of quasi-variational inequalities (QVIs in short). We prove, by means of the weak dynamic programming principle for the stochastic differential game, that the value function of each player is a constrained viscosity solution to the associated QVIs system in the class of linear growth fun…

Finite Horizon Robust Impulse Control in a Non-Markovian Framework and Related Systems of Reflected BSDEs

We consider a robust impulse control problem in finite horizon where the underlying uncertainty stems from an impulsively and continuously controlled functional stochastic differential equation (FSDE) driven by Brownian motion. We assume that the controller acts upon the system by impulses while the adversary player (nature) acts through continuous controls. We look for a weak solution which leads us to consider a system of sequentially interconnected, obliquely reflected backward stochastic differential equations (RBSDEs) with stochastic Lipschitz coefficients. We show existence of solutions …

有限时域零和随机微分脉冲控制博弈值的数值逼近

Numerical approximations of the value of zero-sum stochastic differential impulse controls game in finite horizon

本文研究有限时域零和随机微分脉冲控制博弈，基于经典Policy Iteration算法求Nash均衡数值解及最优策略对，并严格证明逼近方案收敛至粘性解。