scholarly journals Equilibrium strategies for time-inconsistent stochastic switching systems

2019 ◽  
Vol 25 ◽  
pp. 64 ◽  
Author(s):  
Hongwei Mei ◽  
Jiongmin Yong
Keyword(s):  
Optimal Control ◽  
Regime Switching ◽  
Hjb Equation ◽  
Equilibrium Strategy ◽  
Cost Functional ◽  
State Dependent ◽  
Global Optimal ◽  
Time Inconsistent ◽  
The Cost ◽  
Exponential Discounting

An optimal control problem is considered for a stochastic differential equation containing a state-dependent regime switching, with a recursive cost functional. Due to the non-exponential discounting in the cost functional, the problem is time-inconsistent in general. Therefore, instead of finding a global optimal control (which is not possible), we look for a time-consistent (approximately) locally optimal equilibrium strategy. Such a strategy can be represented through the solution to a system of partial differential equations, called an equilibrium Hamilton–Jacob–Bellman (HJB) equation which is constructed via a sequence of multi-person differential games. A verification theorem is proved and, under proper conditions, the well-posedness of the equilibrium HJB equation is established as well.

scholarly journals Time-inconsistent stochastic optimal control problems and backward stochastic Volterra integral equations

2021 ◽  
Author(s):  
Jiongmin Yong ◽  
Hanxiao Wang
Keyword(s):  
Optimal Control ◽  
Mesh Size ◽  
Equilibrium Strategy ◽  
Time Interval ◽  
Well Posedness ◽  
Cost Functional ◽  
New Class ◽  
Time Inconsistent ◽  
Hamilton Jacobi Bellman ◽  
The Cost

An optimal control problem is considered for a stochastic differential equation with the cost functional determined by a backward stochastic Volterra integral equation (BSVIE, for short). This kind of cost functional can cover the general discounting (including exponential and non-exponential) situation with a recursive feature. It is known that such a problem is time-inconsistent in general. Therefore, instead of finding a global optimal control, we look for a time-consistent locally near optimal equilibrium strategy. With the idea of multi-person differential games, a family of approximate equilibrium strategies is constructed associated with partitions of the time intervals. By sending the mesh size of the time interval partition to zero, an equilibrium Hamilton--Jacobi--Bellman (HJB, for short) equation is derived, through which the equilibrium valued function and an equilibrium strategy are obtained. Under certain conditions, a verification theorem is proved and the well-posedness of the equilibrium HJB is established. As a sort of Feynman-Kac formula for the equilibrium HJB equation, a new class of BSVIEs (containing the diagonal values $Z(r,r)$ of $Z(\cd\,,\cd)$) is naturally introduced and the well-posedness of such kind of equations is briefly discussed.

scholarly journals Linear quadratic stochastic optimal control problems with operator coefficients: open-loop solutions

2019 ◽  
Vol 25 ◽  
pp. 17 ◽  
Author(s):  
Qingmeng Wei ◽  
Jiongmin Yong ◽  
Zhiyong Yu
Keyword(s):  
Optimal Control ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Optimal Control Problems ◽  
Mean Field ◽  
Open Loop ◽  
Control Problems ◽  
Linear Quadratic ◽  
Cost Functional ◽  
The Cost

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 (LQ, for short) optimal control problems for mean-field stochastic differential equations. Open-loop solvability of the problem is characterized as the solvability of a system of linear coupled forward-backward stochastic differential equations (FBSDE, for short) with operator coefficients, together with a convexity condition for the cost functional. Under proper conditions, the well-posedness of such an FBSDE, which leads to the existence of an open-loop optimal control, is established. Finally, as applications of our main results, a general mean-field LQ control problem and a concrete mean-variance portfolio selection problem in the open-loop case are solved.

scholarly journals Optimal Portfolio Selection of Mean-Variance Utility with Stochastic Interest Rate

2020 ◽  
Vol 2020 ◽  
pp. 1-10
Author(s):  
Shuang Li ◽  
Shican Liu ◽  
Yanli Zhou ◽  
Yonghong Wu ◽  
Xiangyu Ge
Keyword(s):  
Optimal Control ◽  
Interest Rate ◽  
Differential Equations ◽  
Risk Aversion ◽  
Hjb Equation ◽  
Time Inconsistency ◽  
Equilibrium Strategy ◽  
Stochastic Interest Rate ◽  
Stochastic Interest ◽  
Mean Variance

In order to tackle the problem of how investors in financial markets allocate wealth to stochastic interest rate governed by a nested stochastic differential equations (SDEs), this paper employs the Nash equilibrium theory of the subgame perfect equilibrium strategy and propose an extended Hamilton-Jacobi-Bellman (HJB) equation to analyses the optimal control over the financial system involving stochastic interest rate and state-dependent risk aversion (SDRA) mean-variance utility. By solving the corresponding nonlinear partial differential equations (PDEs) deduced from the extended HJB equation, the analytical solutions of the optimal investment strategies under time inconsistency are derived. Finally, the numerical examples provided are used to analyze how stochastic (short-term) interest rates and risk aversion affect the optimal control strategies to illustrate the validity of our results.

BILINEAR OPTIMAL CONTROL FOR A WAVE EQUATION

1999 ◽  
Vol 09 (01) ◽  
pp. 45-68 ◽  
Author(s):  
MIN LIANG
Keyword(s):  
Optimal Control ◽  
Wave Equation ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Optimality System ◽  
Cost Functional ◽  
Quadratic Cost ◽  
Uniqueness Of The Solution ◽  
Initial Boundary Value ◽  
The Cost

We consider the problem of optimal control of a wave equation. A bilinear control is used to bring the state solutions close to a desired profile under a quadratic cost of control. We establish the existence of solutions of the underlying initial boundary-value problem and of an optimal control that minimizes the cost functional. We derive an optimality system by formally differentiating the cost functional with respect to the control and evaluating the result at an optimal control. We establish existence and uniqueness of the solution of the optimality system and thus determine the unique optimal control in terms of the solution of the optimality system.

Optimal Pole Assignment for Discrete Linear Regulator With Deterministic Disturbances

1988 ◽  
Vol 110 (4) ◽  
pp. 433-436
Author(s):  
Tsu-Tian Lee ◽  
Shiow-Harn Lee
Keyword(s):  
Optimal Control ◽  
Closed Loop ◽  
Pole Assignment ◽  
External Input ◽  
Circular Region ◽  
Cost Functional ◽  
Regulator Problem ◽  
Input Disturbance ◽  
Linear Regulator ◽  
The Cost

This paper presents the solution of the linear discrete optimal control to achieve poles assigned in a circular region and, meanwhile, accommodate deterministic disturbances. It is shown that by suitable manipulations, the problem can be reduced to a standard discrete quadratic regulator problem that simultaneously ensures: 1) closed-loop-poles all inside a circle centered at (β, 0) with a radius α, where |β| + α ≤ 1, 2) minimizes the cost functional, and 3) accommodates external input disturbance.

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