Time-inconsistent Stochastic Linear-quadratic Control Problems With Poisson Jumps

In the thesis,we consider a continuous-time, n-dimensional nonhomogeneous linear controlled system Where A is a bounded deterministic function on [t,T]with value in Rn×n. The other pa-rameters B, Cj,Dj are all essentially bounded adapted processes on [t,T] with values in Rl×n,Rn×n,Rn×1,respectively；ξj and ηj are essentially bounded adapted processes on [t,T]×R0with values in Rn×n, Rn×l; R0:=R\{0}. b and σj are stochastic processes in LF2(t,T;Rn). The process u∈LF2(t,T;Rn) is the control. X is the state process with value in Rn,xt∈Rn is the initial state.It is obviously that for any control u∈LF2(t,T;Rl.),the system(1)has a unique solution X∈LF2(Ω;C(t,T;Rn)).LF2(t,T;Rl)means the set of {fs}s∈[t,T]-adapted processes f={fs: t≤s≤T) with E[∫tT|fs|2ds]<+∞, LF2(Ω;C(t,T;Rn))means the set of continuous {fs}s∈[t,T]-adapted processes f={fs: t≤s≤T} with E[sups∈[t,T]|fs|2]<+∞The objective functional is as follows where u∈LF2(t,T;Rl), X=Xt,xt,u, Et[·]=E[·|Ft].Q and R are both given essentially bounded adapted processes on [t,T] with values in Sn and Sl,respectively；G,h,μ1,μ2are all constants in Sn,Sn,Rn×n,Rn.We assume that Q≥0,R≥0,G≥0(here "≥" means a symmetric matrix is positive semidefinite; Sl denotes the set of symmetric l×l real matrices).Problem(D). Our aim is to look for equilibrium control u*∈Lf2(t, T; Rl), such thatThe first two terms in the objective functional (2) are standard in a linear-quadratic control problem, the term-1/2<hEt[XT], Et[XT]> stems from the variance term in a mean-variance portfolio choice model [16,27], the last term-<μ1xt+μ2,Et[XT]> is motivated by a state-dependent utility function in economics [6]. Since these two terms are time-inconsistent, this lead to time-inconsistency of the model. In the case of time inconsistency, it is obviously inappropriate to use the concept of "optimality", because the control is not always optimal on the full time interval. Here we adopt the notion of equilibrium control, from the perspective of the concept, the decision made by the controller at every instant of time is playing a game against all the decisions made by his future incarnations. The "equilibrium" control at every moment is considered as the best.In this paper, we study a general time-inconsistent stochastic linear-quadratic control problems with Poisson jumps, and derive a general sufficient condition for equilibrium con-trols (Theorem2.1). When the coefficients are all deterministic and n=1, we find an explicit equilibrium control (3.7). The coefficients in (3.7) is provided by (3.8),(3.9).