Linear-quadratic McKean-Vlasov stochastic control problems with random coefficients on finite and infinite horizon, and applications *

Huyen Pham (1) , Matteo Basei (1)

1 LPMA - Laboratoire de Probabilités et Modèles Aléatoires

We propose a simple and original approach for solving linear-quadratic mean-field stochastic control problems. We study both finite-horizon and infinite-horizon problems , and allow notably some coefficients to be stochastic. Our method is based on a suitable extension 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 two applications with explicit solutions: the first one deals with a portfolio liquidation problem with trade crowding, and the second one considers an economic model of substitutable production goods. MSC Classification: 49N10, 49L20, 93E20.

Mots clés: Mean-field SDEs; LQ optimal control; weak martingale optimality principle; Riccati equation; liquidation with trade crowding; exhaustible resource production

Domaines: Probabilités [math.PR]; Optimisation et contrôle [math.OC]; Finance quantitative [q-fin.CP]

Huyen Pham, Matteo Basei. Linear-quadratic McKean-Vlasov stochastic control problems with random coefficients on finite and infinite horizon, and applications *. 2017. ⟨hal-01648491v1⟩