A Type Of Time Inconsistency LQ Problem

Time-inconsistent control problem has become a hotspot subject in the cross-disciplinary of mathematics and finance.In this paper,we will discuss the time-inconsistent LQ con-trol about two kinds of state-dependent discounting factors,undoubtedly with extensive applications and profound theoretical significance.For any given（t,x）∈[0,T),consider the following controlled system:（?）（s）＝AX（s）+Bu（s）,（?）S ∈（t,T],X（t）=x,（6）here u（·）∈ L²（t,T;R^k）,and for any initial pair（t,x）∈[0,T）and control u∈L²（t,T:R^k）,X_t,x^u represents the corresponding solution of system（6）.Then,we introduce the fol-lowing cost functionals（?）（7）and（?）（8）respectively.Our aim is the cost functional minimization,namely,problem（P_i）（i=1,2）:fund a control （?）:[0,T]× Rⁿ→R^k such that for any given（t,x,v）∈[0,T)x Rⁿ x R^k,one has（?）（9）where（?）（10）.To this end,the above problem is firstly treated as the limit of a class of quasi LQ control problems,and the solvability of which is equivalent to a class of two-point boundary value problem or Riccati equation ones.Then,we discuss the solvability of problem（P_i）by means of two-point boundary value problem and establish the maximum principle.Meanwhile,we also investigate problem（P_i）via the Principle of optimal,so as to establish a generalized Bellman optimality principle,namely,the equilibrium prin-ciple,derive the equilibrium HJB equation,and obtain the verification principle.These conclusions are of great significance to establish general theories.