Characterizations of equilibrium controls in time inconsistent mean-field stochastic linear quadratic problems. I

Abstract

In this paper, a class of time inconsistent linear quadratic optimal control problems of mean-field stochastic differential equations (SDEs) is considered under Markovian framework. Open-loop equilibrium controls and their particular closed-loop representations are introduced and characterized via variational ideas. Several interesting features are revealed and a system of coupled Riccati equations is derived. In contrast with the analogue optimal control problems of SDEs, the mean-field terms in state equation, which is another reason of time inconsistency, prompts us to define above two notions in new manners. An interesting result, which is almost trivial in the counterpart problems of SDEs, is given and plays significant role in the previous characterizations. As application, the uniqueness of open-loop equilibrium controls is discussed.Keywords. Mean-field linear quadratic optimal control problems, time inconsistency, equilibrium controls, system of Riccati equations.

Extended backward stochastic Volterra integral equations and their applications to time-Inconsistent stochastic recursive control problems

“…They characterized an open-loop equilibrium control by using a variational method, which is a natural generalization of the stochastic maximum principle of Peng [16] to the time-inconsistent problem. This approach was further developed in [12,34,29,24,25,2,1,8].…”

“…Compared with the above papers, we provide a necessary and sufficient condition for an open-loop equilibrium control in a time-inconsistent stochastic recursive control problem with general discounting. We also refer to relevant works of Wang [24,25], where the author characterized open-loop equilibrium controls in a linear-quadratic time-inconsistent mean-field control problem (which is different from our setting) by a system of conditions named as first-order, secondorder equilibrium conditions.…”

Non-exponential discounting portfolio management with habit formation

“…One is the open-loop control, which is mainly used in linear-quadratic control problem (see [13], [13], [26]) and the main way to get an open-loop control is stochastic maximum principle. Alia [2] found the open-loop control of Merton's problem with non-constant discounting.…”

Controlled Markov chains with non-exponential discounting and distribution-dependent costs

“…Because of the non-exponential discounting feature, one has to modify the Nash optimality appropriately. Previous works on dealing with the two time inconsistencies together include [22,30,36]. To the best of our knowledge, most of the papers are concerned with a special class of linear stochastic differential equations with quadratic-type costs.…”