Characterization of Markovian equilibria in a class of differential games

Abstract

We consider an alternative method to the classical one for the determination of Markov perfect Nash equilibria. The approach shown in the paper is based on the study of a quasi-linear system of partial di erential equations instead of the Hamilton-Jacobi-Bellman system. The simpler structure of the former allows us to determine existence and uniqueness of Nash equilibria in non-renewable resource games under some assumptions. When closed-form solutions are not available, we give a method to obtain numerical solutions.JEL classiÿcation: C73; Q32

New Approach to Stochastic Optimal Control

“…Remark 3.1 There is in the literature a different but closely related system of PDEs which characterize the vector of adjoint variables under some regularity assumptions, see (17) below. This system was obtained for the first time in Bismut [7] and later in 2 Elliot [18].…”

“…Clearly, the results remain valid now for C 1 solutions and Assumption A2 is not needed. In Rincón-Zapatero et al [16] and Rincón-Zapatero [17], an extension to differential games is provided.…”

“…The idea to systematically obtain a system of PDEs for the optimal control dates back to the paper by Bourdache-Siguerdidjane and Fliess [15] for deterministic control problems. The method was later extended to differential games in Rincón-Zapatero et al [16] and Rincón-Zapatero [17].…”

Euler–Lagrange equations of stochastic differential games: application to a game of a productive asset

“…Further concrete applications of the use of the EL equations system in differential game theory can be found in Shimomura (1991), Dockner and Sorger (1996), and Sorger (1998), all of them in the deterministic counterpart of the model we study in this paper. In Rincón-Zapatero et al (1998) and Rincón-Zapatero (2004), this approach has been made systematic. These two papers also provide sufficient conditions of optimality which are independent of the value function.…”

On a PDE Arising in One-Dimensional Stochastic Control Problems

“…Though the initial idea of obtaining a system of PDEs for the optimal control appears in [1] in connection with deterministic control problems, the main antecedents of this paper are: [2] and [3] in deterministic differential games; [4], in stochastic control problems, where the diffusion parameter of the state process is independent of the control variables; [5] in the Merton problem; and [6] in a model of optimal liquidation in illiquid markets. In all these papers, the use of the PDE for optimal control has proved to be useful.…”