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The objective of this paper is two-fold. The first objective is to complete the former work of Bensoussan and Frehse [2]. One big limitation of this paper was the fact that they are systems of PDE. on a bounded domain. One can expect solutions to be bounded, since one looks for smooth solutions. This is a very important property for the development of the method. It is true also that solutions which exist in a bounded domain may fail to exist on R n , because of the lack of bounds. We give conditions so that the results of [2] can be extended to R n. The second objective is to consider the BSDE (Backward stochastic differential equations) version of the system of PDE. This is the objective of a more recent work of Xing and Z̆itković [8]. They consider systems of BSDE with quadratic growth, which is a well-known open problem in the BSDE literature. Since the BSDE are Markovian, the problem is equivalent to the analytic one. However, because of this motivation the analytic problem is in R n and not on a bounded domain. Xing and Z̆itković developed a probabilistic approach. The connection between the analytic problem and the BSDE is not apparent. Our objective is to show that the analytic approach can be completely translated into a probabilistic one. Nevertheless probabilistic concepts are also useful, after their conversion into the analytic framework. This is in particular true for the uniqueness result. [ABSTRACT FROM AUTHOR]

In this paper, we solve a continuous-time portfolio choice problem of an investor under a Markov jump linear system that effectively captures stochasticity in asset returns, price impacts, and market resilience. Specifically, the investor chooses his portfolio to maximize the expected excess returns netting risks and trading costs due to the price impacts in a Markov switching market. We show that the value function and the optimal feedback trading strategy can be expressed in terms of the solution of a coupled matrix Riccati differential system. The presence of the market resilience and the Markov switching results in the coupled Riccati system being non-canonical. We manage to establish a set of transparent sufficient conditions to ensure the well-posedness of the coupled system. Our numerical results indicate that the investor trades more (less) aggressively in a liquidity regime where the temporary price impact is low (high) and permanent price impact is high (low) with low (high) market resilience. [ABSTRACT FROM AUTHOR]

In his lectures at College de France , P.L. Lions introduced the concept of Master equation, see [8] for Mean Field Games. It is introduced in a heuristic fashion, from the prospective as a system of partial differential equations, that the equation is associated to a Nash equilibrium for a large, but finite, number of players. The method, also explained in [3] , composed of a formalism of derivations. The interest of this equation is that it contains interesting particular cases, which can be studied directly, in particular the system of HJB–FP (Hamilton–Jacobi–Bellman, Fokker–Planck) equations obtained as the limit of the finite Nash equilibrium game, when the trajectories are independent, see [6] . Usually, in mean field theory, one can bypass the large Nash equilibrium, by introducing the concept of representative agent, whose action is influenced by a distribution of similar agents, and obtains directly the system of HJB–FP equations of interest, see for instance [1] . Apparently, there is no such approach for the Master equation. We show here that it is possible. We first do it for the Mean Field type control problem, for which we interpret completely the Master equation. For the Mean Field Games itself, we solve a related problem, and obtain again the Master equation. [ABSTRACT FROM AUTHOR]