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Abstract We consider linear–quadratic mean field Stackelberg differential games with the adapted open-loop information structure of the leader. There are one leader and N followers, where N is arbitrarily large. The leader holds a dominating position in the game in the sense that the leader first chooses and then announces the optimal strategy to which the N followers respond by playing a Nash game, i.e., choosing their optimal strategies noncooperatively and simultaneously based on the leader's observed strategy. In our setting, the followers are coupled with each other through the mean field term included in their cost functions, and are strongly influenced by the leader's open-loop strategy included in their cost functions and dynamics. From the leader's perspective, he is coupled with the N followers through the mean field term included in his cost function. To circumvent the complexity brought about by the coupling nature among the leader and the followers with large N , which makes the use of the direct approach almost impossible, our approach in this paper is to characterize an approximated stochastic mean field process by solving a local optimal control problem of the followers with leader's control taken as an exogenous stochastic process. We show that for each fixed strategy of the leader, the followers' local optimal decentralized strategies lead to an ϵ -Nash equilibrium. The paper then solves the leader's local optimal control problem, as a nonstandard constrained optimization problem, with constraints being induced by the approximated mean field process determined by Nash followers (which also depend on the leader's control). We show that the local optimal decentralized controllers for the leader and the followers constitute an (ϵ 1 , ϵ 2) -Stackelberg–Nash equilibrium for the original game, where ϵ 1 and ϵ 2 both converge to zero as N → ∞. Numerical examples are provided to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we provide a closed-form solution to an optimal portfolio execution problem with stochastic price impact and stochastic net demand pressure. Specifically, each trade of an investor has temporary and permanent price impacts, both of which are driven by a continuous-time Markov chain; whereas the net demand pressure from other inventors is modelled by an Ornstein—Uhlenbeck process. The investor optimally liquidates his portfolio to maximize his expected revenue netting his cumulative inventory cost over a finite time. Such a problem is first reformulated as an optimal stochastic control problem for a Markov jump linear system. Then, we derive the value function and the optimal feedback execution strategy in terms of the solutions to coupled differential Riccati equations. Under some mild conditions, we prove that the coupled system is well-posed, and establish a verification theorem. Financially, our closed-form solution shows that the investor optimally liquidates his portfolio towards a dynamic benchmark. Moreover, the investor trades aggressively (conservatively) in the state of low (high) price impact. [ABSTRACT FROM AUTHOR]