48 research outputs found
Linear-Quadratic Optimal Control Problems for Mean-Field Stochastic Differential Equations
Linear-quadratic optimal control problems are considered for mean-field stochastic differential equations with deterministic coefficients. By a variational method, the optimality system is derived, which is a linear mean-field forward-backward stochastic differential equation. Using a decoupling technique, two Riccati differential equations are obtained which are uniquely solvable under certain conditions. Then a feedback representation is obtained for the optimal control
Linear-Quadratic Optimal Control Problems for Mean-Field Stochastic Differential Equations --- Time-Consistent Solutions
Linear-quadratic optimal control problems are considered for mean-field
stochastic differential equations with deterministic coefficients.
Time-inconsistency feature of the problems is carefully investigated. Both
open-loop and closed-loop equilibrium solutions are presented for such kind of
problems. Open-loop solutions are presented by means of variational method with
decoupling of forward-backward stochastic differential equations, which lead to
a Riccati equation system lack of symmetry. Closed-loop solutions are presented
by means of multi-person differential games, the limit of which leads to a
Riccati equation system with a symmetric structure.Comment: 51 page
Linear Quadratic Stochastic Optimal Control Problems with Operator Coefficients: Open-Loop Solutions
An optimal control problem is considered for linear stochastic differential
equations with quadratic cost functional. The coefficients of the state
equation and the weights in the cost functional are bounded operators on the
spaces of square integrable random variables. The main motivation of our study
is linear quadratic optimal control problems for mean-field stochastic
differential equations. Open-loop solvability of the problem is investigated,
which is characterized as the solvability of a system of linear coupled
forward-backward stochastic differential equations (FBSDE, for short) with
operator coefficients. Under proper conditions, the well-posedness of such an
FBSDE is established, which leads to the existence of an open-loop optimal
control. Finally, as an application of our main results, a general mean-field
linear quadratic control problem in the open-loop case is solved.Comment: to appear in ESAIM Control Optim. Calc. Var. The original publication
is available at www.esaim-cocv.org (https://doi.org/10.1051/cocv/2018013
Partial Information Differential Games for Mean-Field SDEs
This paper is concerned with non-zero sum differential games of mean-field
stochastic differential equations with partial information and convex control
domain. First, applying the classical convex variations, we obtain stochastic
maximum principle for Nash equilibrium points. Subsequently, under additional
assumptions, verification theorem for Nash equilibrium points is also derived.
Finally, as an application, a linear quadratic example is discussed. The unique
Nash equilibrium point is represented in a feedback form of not only the
optimal filtering but also expected value of the system state, throughout the
solutions of the Riccati equations.Comment: 7 page