We consider a general nonzero-sum impulse game with two players. The main mathematical contribution of the paper is a verification theorem which provides, under some regularity conditions, a suitable system of quasi-variational inequalities for the payoffs and the strategies of the two players at some Nash equilibrium. As an application, we study an impulse game with a one-dimensional state variable, following a real-valued scaled Brownian motion, and two players with linear and symmetric running payoffs. We fully characterize a family of Nash equilibria and provide explicit expressions for the corresponding equilibrium strategies and payoffs. We also prove some asymptotic results with respect to the intervention costs. Finally, we consider two further non-symmetric examples where a Nash equilibrium is found numerically.

We propose a simple and original approach for solving linear-quadratic meanfield stochastic control problems. We study both finite-horizon and infinite-horizon problems, and allow notably some coefficients to be stochastic. Extension to the common noise case is also addressed. Our method is based on a suitable version of the martingale formulation for verification theorems in control theory. The optimal control involves the solution to a system of Riccati ordinary differential equations and to a linear mean-field backward stochastic differential equation; existence and uniqueness conditions are provided for such a system. Finally, we illustrate our results through an application to the production of an exhaustible resource.MSC Classification: 49N10, 49L20, 93E20.1 closed-loop controls. Our approach is simpler in the sense that it does not rely on the notion of derivative in the Wasserstein space, and considers the larger class of open-loop controls. We are able to obtain analytical solutions via the resolution of a system of two Riccati equations and the solution to a linear mean-field backward stochastic differential equation.We first consider linear-quadratic McKean-Vlasov (LQMKV) control problems in finite horizon, where we allow some coefficients to be stochastic. We prove, by means of a weak martingale optimality principle, that there exists, under mild assumption on the coefficients, a unique optimal control, expressed in terms of the solution to a suitable system of Riccati equations and SDEs. We then provide some alternative sets of assumptions for the coefficient. We also show how the results adapt to the case where several independent Brownian motions are present. We also consider problem with common noise: Here, a similar formula holds, now considering conditional expectations. We then study the infinite-horizon case, characterizing the optimal control and the value function. Finally, we propose a detailed application, dealing with an infinitehorizon model of production of an exhaustible resource with a large number of producers and random price process.We remark that in the infinite-horizon case some additional assumptions on the coefficients are required. On the one hand, having a well-defined value function requires a lower bound on the discounting parameter. On the other hand, we here deal with an infinite-horizon SDE, and the existence of a solution is a non-trivial problem. Finally, the admissibility of the optimal control requires a further condition of the discounting coefficient.The literature on McKean-Vlasov control problems is now quite important, and we refer to the recent books by Bensoussan, Frehse and Yam [3] and Carmona and Delarue [5], and the references therein. In this McKean-Vlasov framework, linear-quadratic (LQ) models provide an important class of solvable applications, and have been studied in many papers, including [4,17,10,13,8], however mostly for constant or deterministic coefficients, with the exception of [15] on finite horizon, and [12], which deals with stochastic coeff...

We characterize the value of swing contracts in continuous time as the unique viscosity solution of a Hamilton-Jacobi-Bellman equation with suitable boundary conditions. The case of contracts with penalties is straightforward, and in that case only a terminal condition is needed. Conversely, the case of contracts with strict constraints gives rise to a stochastic control problem with a nonstandard state constraint. We approach this problem by a penalty method: we consider a general constrained problem and approximate the value function with a sequence of value functions of appropriate unconstrained problems with a penalization term in the objective functional. Coming back to the case of swing contracts with strict constraints, we finally characterize the value function as the unique viscosity solution with polynomial growth of the HJB equation subject to appropriate boundary conditions.

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 → ∞, that is, a suitable mean-field game (MFG) with impulse controls. We show that under appropriate technical conditions, the MFG is an -NE approximation to the N -player game, with = O 1 √ N . As an example, we analyze in details a class of stochastic games which extends the classical cash management problem to the game setting. In particular, we characterize the NEs for its two-player case and compare the results to the single-player case, showing the impact of competition on the player's optimal strategy, with sensitivity analysis of the model parameters.

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