A mean-field approach for the optimal control of discrete-time linear systems with multiplicative noises.

• Autor(a) principal: Barbieri, Fabio
• Data de Publicação: 2020
• Tipo de documento: Tese
• Idioma: eng
• Título da fonte: Biblioteca Digital de Teses e Dissertações da USP
• Texto Completo: https://www.teses.usp.br/teses/disponiveis/3/3139/tde-01072021-111725/
• topic: Controle estocástico; Controle ótimo; Intertemporal restrictions; Linear systems; Optimal control; Otimização de carteira de investimentos; Portfolio optimisation; Restrições intertemporais; Sistemas lineares; Solução estabilizadora; Stabilising solution; Stochastic control

In this work, we study the stochastic multi-period optimal control for discrete-time linear systems subject to multiplicative noises. Initially, we consider a multi-period mean-variance trade-off performance criterion for the finite-horizon case with and without constraints, and then, its infinite-horizon case with the long-run as well as the discount factor criteria. We adopt the mean-field approach to tackle the problems and get their solutions in terms of a set of two generalised coupled algebraic Riccati equations (GCARE for short). For the finite-horizon case, we derive the optimal control law for a general multi-period mean-variance problem and obtain the optimal control strategy for the constrained problems using the Lagrangian multipliers approach. From the general unrestricted result, we obtain a sufficient condition for a closed-form solution for one of the constrained problems considered in this work. For the infinite-horizon case, we establish sufficient conditions for the existence of the maximal solution, necessary and sufficient conditions for the existence of the mean-square stabilising solution to the GCARE, and derive the optimal control laws for the discounted and long-run problems. When particularised to the portfolio selection problem, we show that our results match some of the results available in the literature. A numerical example illustrates the obtained optimal controls for the multi-period portfolio selection problem in which is desired to optimise the sum of the mean-variance trade-off costs of a portfolio against a benchmark along the time.