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.