A Constructive Approach to Existence of Equilibria in Time-Inconsistent Stochastic Control Problems

We extend the construction of equilibria for linear-quadratic and mean-variance portfolio problems available in literature to a large class mean-field time-inconsistent stochastic control continuous time. Our approach relies on time discretization problem via $n$-person games, which are characterized maximum principle using backward differential equations. The existence is proved by applying weak convergence arguments solutions games. A numerical implementation provided approximating games finite Markov chains.