17,132 research outputs found
Maximum Principle for Forward-Backward Doubly Stochastic Control Systems and Applications
The maximum principle for optimal control problems of fully coupled
forward-backward doubly stochastic differential equations (FBDSDEs in short) in
the global form is obtained, under the assumptions that the diffusion
coefficients do not contain the control variable, but the control domain need
not to be convex. We apply our stochastic maximum principle (SMP in short) to
investigate the optimal control problems of a class of stochastic partial
differential equations (SPDEs in short). And as an example of the SMP, we solve
a kind of forward-backward doubly stochastic linear quadratic optimal control
problems as well. In the last section, we use the solution of FBDSDEs to get
the explicit form of the optimal control for linear quadratic stochastic
optimal control problem and open-loop Nash equilibrium point for nonzero sum
differential games problem
Closed-Loop Solvability of Linear Quadratic Mean-Field Type Stackelberg Stochastic Differential Games
This paper is devoted to a Stackelberg stochastic differential game for a
linear mean-field type stochastic differential system with a mean-field type
quadratic cost functional in finite horizon. The coefficients in the state
equation and weighting matrices in the cost functional are all deterministic.
Closed-loop Stackelberg equilibrium strategies are introduced which require to
be independent of initial states. Follower's problem is solved firstly, which
is a stochastic linear quadratic optimal control problem. By converting the
original problem into a new one whose optimal control is known, the closed-loop
optimal strategy of the follower is characterized by two coupled Riccati
equations as well as a linear mean-field type backward stochastic differential
equation. Then the leader turns to solve a stochastic linear quadratic optimal
control problem for a mean-field type forward-backward stochastic differential
equation. Necessary conditions for the existence of closed-loop optimal
strategies for the leader is given by the existence of two coupled Riccati
equations with a linear mean-field type backward stochastic differential
equation. The solvability of Riccati equations of leader's optimization problem
is discussed in the case where the diffusion term of the state equation does
not contain the control process of the follower. Moreover, leader's value
function is expressed via two backward stochastic differential equations and
two Lyapunov equations.Comment: 44 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
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