Existence Of Equilibrium Control For A Class Of Time-inconsistent LQ Control Problem

Time-inconsistent control problem has become a hot spot subject in the crossdisciplinary of mathematics and finance.In this paper,we will discuss the time-inconsistent LQ control about time-dependent discounting factors.For any given(t,x)∈[0,T),consider the following controlled system:Then,we introduce the following cost functional:As discussed earlier the non-exponential discount functions Q,S,M,G,q,m,G,g in the cost functional(4)render the underlying LQ problem generally time-inconsistent.Timeinconsistent LQ control problem is not only very important in itself,but also a breakthrough for further research on time-inconsistent control problem.About the solution of time-inconsistent control problem,Bjork and Murgoci introduce the precise definition of an equilibrium control u(see[60]).Let u:[0,T]×Rn→Rm be a measurable mapping and satisfies u(·,X(·))∈u[0,T],we define The control u is an equilibrium control if This definition reflects the impact of environmental change on the problem,and it is closer to the actual situation.It’s well known that,a fundamental theory of linear-quadratic(LQ)optimal control is the equivalence between the solvability of the optimal control problem,the two-point boundary value problem,the Fredholm equation and the Riccati equation.We know that the time-inconsistent control problem is not suitable for the optimality principle and the solution is feedback.Therefore,the common methods for solving the optimal control problem cannot be directly applied to solve the time-inconsistent control problem,which makes it very difficult to study the solvability of the time-inconsistent control problem.In order to solve the problem,we introduce linear equilibrium control and equilibrium two-point boundary value problem,equilibrium Fredholm equation,equilibrium Riccati equation.Additionally,we extend the equivalence of the four problems to a general timeinconsistent deterministic LQ problem.Then,by studying the solvability of the Riccati equation,we show the existence of the linear equilibrium for the time-inconsistent LQ problem in the infinite time zone(T=+∞)and finite time zone,and in the case that the discount factor Q is semi-positive definite or indefinite.These results and research methods lay a foundation for in-depth research on time-inconsistent control problem.