Time-Inconsistent Optimal Control Problems and the Equilibrium HJB Equation

A general time-inconsistent optimal control problem is considered for stochastic differential equations with deterministic coefficients. Under suitable conditions, a Hamilton-Jacobi-Bellman type equation is derived for the equilibrium value function of the problem. Well-posedness and some properties of such an equation is studied, and time-consistent equilibrium strategies are constructed. As special cases, the linear-quadratic problem and a generalized Merton's portfolio problem are investigated.

where the maps g(·) and h(·) explicitly depend on the initial time t in some general way. The optimal control problem associated with (1.1) and (1.8), called Problem (N), will not be time-consistent, or time-inconsistent, in general, meaning that a restriction of an optimal control for a specific initial pair on a later time interval might not be optimal for that corresponding initial pair. Some concrete examples will be presented in the next section.
The purpose of this paper is to obtain time-consistent optimal controls (which should be more properly called equilibrium control) for Problem (N) mentioned above. Let us now briefly describe our approach. Inspired by [27,28], we introduce a sequence of multi-person hierarchical differential games as follows. For any N > 1, let Π be a partition of the time interval [0, T ] defined by Π : 0 = t 1 < t 2 < · · · < t N = T, with the mesh size Π given by Π = max 1≤k≤N (t k − t k−1 ).
The differential game, denoted by Problem (G Π ), associated with partition Π consists of N players. The k-th player controls the system on [t k−1 , t k ) by taking his/her control u k (·) ∈ U[t k−1 , t k ]. The cost functional is constructed in a sophisticated way, by using some techniques of forward-backward stochastic differential equations (FBSDEs, for short) found in [16,17]. The interaction among the players are as follows: (i) The terminal pair (t k , X(t k )) of Player k is the initial pair of Player (k + 1); (ii) All the player know that each player tries to find an optimal control for his/her own problem; and (iii) Each player will discount the future costs in his/her own way, regardless of the fact that the later players will control the system. Under certain conditions, each player will have an optimal control, denoted byū k (·) ∈ U[t k−1 , t k ], for his/her own problem, as well as his/her own value function V k (· , ·) defined on [t k−1 , t k ] × lR n . Definē (1.9) and (1.10) We may callū Π (·) and V Π (· , ·) the Nash equilibrium control and Nash equilibrium value function of Problem (G Π ), respectively. When the following limits exist for someū(·) ∈ U[0, T ] and V : [0, T ] × lR n → lR, we call them a time-consistent equilibrium control and a time-consistent equilibrium value function of Problem (N), respectively. As a major contribution of this paper, we have derived the equilibrium Hamilton-Jacobi-Bellman equation which can be used to characterize the equilibrium value function V (· , ·), and it recovers the result for time-inconsistent deterministic linearquadratic problem presented in [27]. The well-posedness of such an HJB equation will be established for the case that the diffusion of the state equation does not contain the control. The general case is open at the moment, and we expect to present some more complete results in our future publications. As important and interesting special cases, we will construct equilibrium controls for stochastic LQ problem with general discounting and for generalized Merton's portfolio problem.

Two Examples of Time-Inconsistent Optimal Control Problems
In this section, we present two interesting examples of optimal control problems which are time-inconsistent.
Interestingly, in the case that (2.9) holds, one has Thus, in this case, (2.13) does not hold. As a matter of fact, the problem is time-consistent and (2.13) should not be true.

Some Preliminaries
For convenience, let us rewrite the state equation and the cost functional below.

(3.2)
Clearly, our cost functional covers the non-exponential/hyperbolic discounting situations. By comparing the above state equation and cost functional, it seems that we may consider a little more general state equation of the following form: x, s, X(s), u(s))ds + σ(t, x, s, X(s), u(s))dW (s), s ∈ [t, T ], However, according to [28] (see also [27]), we know that when an equilibrium pair (see below for definition) is constructed, the eventual effective state equation will take the following form: Therefore, it suffices to consider the state equation of form (3.1).
In what follows, we let T > 0 be a fixed time horizon, and U ⊆ lR m be a closed subset, which could be either bounded or unbounded (it is allowed that U = lR m ). We will use K > 0 as a generic constant which can be different from line to line. Let S n be the set of all (n × n) symmetric real matrices. Denote Recall from Section 1 that Note that in the case U is bounded, for different q ≥ 1, all the U q [t, T ] are the same as U[t, T ]. We introduce the following standing assumptions.
(H1) The maps b : [0, T ] × lR n × U → lR n , σ : [0, T ] × lR n × U → lR n×d are continuous and there exist constants L > 0 and k ≥ 0 such that where |x| ∨ |y| = max{|x|, |y|}, and T ] × lR n → lR are continuous, and there exist constants L > 0 and q ≥ 0 such that Let us make a couple of remarks on (H1). First of all, if x → b(t, x, u) is uniformly Lipschitz, then the first two conditions of (3.5) hold. On the other hand, we point out that the first condition in (3.5) merely implies that x → b(t, x, u) is locally Lipschitz, and the second condition in (3.5) alone does not imply the global Lipschitz condition for the map x → b(t, x, u). A simple example that the first and the second conditions in (3.5) are satisfied but x → b(t, x, u) is not uniform Lipschitz is the following: It is clear that the above map is not uniformly Lipschitz in x, the first condition in (3.5) holds with k = 2, and we can check that Note that under (3.5), one has and |σ(t, x, u)| 2 ≤ L 2 1 + |x| + |u| 2 ≤ 3L 2 1 + |x| 2 + |u| 2 .
Now, for (H2), we note that the nonnegativity of g(·) and h(·) can be replaced by the condition that both g(·) and h(·) are bounded from below. The following result is concerning the well-posedness of the state equation.
Problem (N). For any given initial pair (t, x) ∈ [0, T ) × lR n , find aū(·) ∈ U[t, T ] such that J(t, x; u(·)). (3.11) From the examples presented in the previous section, we know that the above Problem (N) is timeinconsistent, in general. Our goal is to find time-consistent equilibrium controls and characterize the equilibrium value function, which will be made precise below.

(3.15)
This is a multi-valued map. Suppose we can define a map ψ : D(ψ) ⊆ D(H) → U such that H(τ, t, x, p, P ) ≡ lH(τ, t, x, ψ(τ, t, x, p, P ), p, P ) = inf u∈U lH(τ, t, x, u, p, P ) > −∞, ∀(τ, t, x, p, P ) ∈ D(ψ). (3.16) The set D(ψ) is called the domain of ψ, which consists of all points (τ, t, x, p, P ) ∈ D(H) such that the infimum in (3.16) is achieved at ψ(τ, t, x, p, P ). It is clear that ψ(·) is actually a selection of arg min lH(·), i.e., ψ(τ, t, x, p, P ) ∈ arg min lH(τ, t, x, · , p, P ), ∀(τ, t, x, p, P ) ∈ D(ψ). The map ψ(·) will play an important role later. Therefore, let us say a little bit more on it. Note that when U is bounded (since it is assumed to be closed, it is compact in this case), one has However, when U is unbounded, say, U = lR m , one might have To say something about the case of (3.18), let us present the following simple lemma. For any ε > 0, let Then there exists a u ε ∈ U such that Proof. First of all, fix a u 0 ∈ U . For any minimizing sequence u k ∈ U of f ε (·), we may assume that Thus, u k is bounded. Consequently, by the closeness of U , we may assume that u k → u ε ∈ U exists, which attains the infimum of f ε (·). Next, it is clear that f ε (·) decreases as ε decreases, and Now, for any δ > 0, there exists a u δ ∈ U such that On the other hand,f Hence, letting ε → 0, we getf 0 ≤f + δ.
The following example shows that sometimes, ψ can also be continuous.
with σ(· , ·) and R(· , ·) being continuous, and R(τ, t) > 0 for all (τ, t) ∈ D[0, T ]. Then lH(τ, t, x, u, p, P ) = pu + 1 2 Consequently, Clearly, both H and ψ are continuous. Also, we see that even if all the coefficients are very smooth, we cannot guarantee that H and ψ are as smooth as the coefficients, in general.
Here is another example which shows that H and ψ could be as smooth as the coefficients.
From the above discussion, we see that the situation concerning the map ψ(·) is very complicated. For the simplicity of presentation below, we adopt the following assumption.
(H3) The map ψ : D[0, T ] × lR n × lR n × S n → U is well-defined and has needed regularity.
We will address more general situations concerning ψ(·) in our future publications. Now, let us recall a standard verification theorem for Problem (C) stated in Section 1, which will be used below. For our later purposes, it suffices to consider Problem (C) with the discount rate δ = 0. We will denote The proof of the following result can be found in [8].
is a classical solution to the following Hamilton-Jacobi-Bellman equation: where lH 0 (t, x, u, p, P ) = b(t, x, u), p +tr a(t, x, u)P + g 0 (t, x, u).
We make some remarks on the above verification theorem. First of all, to guarantee (3.21) to have a classical solution V 0 (· , ·), one may pose different conditions. A typical one is the uniform ellipticity condition: for some δ > 0. This condition implies that n ≤ d and σ(t, x, u) stays full rank for all (t, x, u). Thus, it does not include the case that (x, u) → σ(t, x, u) is linear, which is the case for LQ problems. On the other hand, for a standard LQ problem with deterministic coefficients, when the Riccati equation admits a solution P (·), the function V 0 (t, x) = P (t)x, x is a classical solution to the corresponding HJB equation for which the uniform ellipticity condition fails. The similar situation happens for the classical Merton's portfolio problem. This observation shows that there are quite different conditions under which the corresponding HJB equation admits a classical solution. In the following sections, from time to time, we will simply say that the relevant HJB equation has a classical solution without getting into detailed sufficient conditions for that. Likewise, suppose there exists a map ψ 0 : [0, T ] × lR n × lR n × S n → U such that Then the pair (X(·),ū(·)) appears in Proposition 3.6 is just a state-control pair satisfying the following and we do not have to have the Lipschitz continuity of the map In the following section, from time to time, we will just say some process is a solution to the relevant closed-loop system without mentioning if the drift and diffusion are Lipschitz continuous, etc.

Time-Consistent Equilibria via Multi-Person Differential Games
In this section, we are going to search time-consistent solution to Problem (N). Inspired by [27,28], we will take an approach of multi-person differential games.
To begin with, let us first introduce some necessary notions. Let P[0, T ] be the set of all partitions and with the mesh size Π defined by the following: We introduce the following important definition.
In what follows, the following definition which is equivalent to the above is more convenient to use.

Multi-Person Differential Games.
We now consider an N -person differential game, called Problem (G Π ), as briefly described in Section 1. Throughout this section, we assume that (H1)-(H3) hold. Let us start with Player N who controls the system on [t N −1 t N ). More precisely, for each (t, x) ∈ [t N −1 , t N ] × lR n , consider the following controlled SDE: We pose the following problem for Player N : The above defines the value function V Π (· , ·) on [t N −1 , t N ] × lR n , and in the caseū N (·) exists, by (4.12), we have Under proper conditions, V Π (· , ·) is the classical solution to the following HJB equation: By the definition of ψ : 17)), we may also write (4.15) as follows x ∈ lR n . (4.16) With such a solution V Π (· , ·) of (4.15) (or (4.16)), let us assume that the following closed-loop system admits a unique solutionX Then under (H3) and Proposition 3.6, an optimal controlū N (·) of Problem (C N ) for the initial pair (t N −1 , x) admits the following feedback representation: (4.18) andX N (·) ≡X N (· ; t N −1 , x) is the corresponding optimal state process.
Next, we consider an optimal control problem for Player (N − 1) on [t N −2 , t N −1 ). Naturally, for any initial To determine the suitable cost functional, we note that Player (N − 1) can only control the system on [t N −2 , t N −1 ) and Player N will take over at t N −1 to control the system thereafter. Moreover, Player (N − 1) knows that Player N will play optimally based on the initial pair (t N −1 , X N −1 (t N −1 )) of Player N , which is the terminal pair of Player (N − 1). Hence, the sophisticated cost functional of Player (N − 1) should be Note that although Player (N − 1) knows that Player N will control the system on [t N −1 , t N ], he/she still "discounts" the future costs in his/her own way (see t N −2 appearing in the running cost on [t N −1 , t N ] and in the terminal cost at t N ). Now, if we denote then the cost functional (4.20) can be written as We see that the optimal control problem associated with the state equation (4.19) and the cost functional (4.21) looks like a standard one. But, the map x → h N −1 (x) seems to be a little too implicit, which is difficult for us to pass to the limits later on. We now would like to make it more explicit in some sense. Inspired by the idea of Four Step Scheme introduced in [16,17] for FBSDEs with deterministic coefficients, we proceed as follows. For the optimal state processX N which is equivalent to the following: Note that t N −2 appears in the drift of BSDE and in the terminal condition. This BSDE admits a unique adapted solution (Y N (·), Z N (·)) ≡ Y N (· ; x), Z N (· ; x) ( [17,29]), uniquely depending on x ∈ lR n . Further, one has It is seen that (4.17) and (4.23) form an FBSDE. By [16] (see also [17,29]), we have the following represen- as long as Θ N (· , ·) is a classical solution to the following PDE: 25) or equivalently, x ∈ lR n . (4.26) . We point out that in general, (4.27) With the above representation Θ N (· , ·) of Y N (·), we can rewrite the cost functional (4.21) as follows: We now pose the following problem: The above defines the value function By the definition of the map ψ(·) again (see (3.16)-(3.17)), we may also write the above as x ∈ lR n . (4.31) From (4.27), we see that in general, For any x ∈ lR n , suppose the following admits a unique solutionX N −1 (·): which, again by Proposition 3.6, is an optimal control of Problem (C N −1 ) with the initial pair (t N −2 , x). Now, for the optimal pair we make a natural extension on [t N −1 , t N ] as follows: We refer to such a pair Then (4.34) can be written compactly as Also, one has x ∈ lR n . (4.38) We point out that in general it may happen that, which means that the the sophisticated equilibrium pair might not be an optimal pair (for the given initial pair).
Similar to the above, in order to state an optimal control problem for Player (N − 2) on [t N −3 , t N −2 ], we introduce the following BSDE on [t N −2 , t N ]: x) be the adapted solution of this BSDE. Then (4.37) and (4.40) form an FBSDE. Similar to the above, we have as long as Θ N −1 (· , ·) is the solution to the following PDE: x ∈ lR n . (4.42) Having the above preparation, we now consider, for any (t, x) ∈ [t N −3 , t N −2 ) × lR n , the state equation and the (sophisticated) cost functional We pose the following problem for Player (N − 2): Together with the previous definition, we see that V Π (· , ·) is now well-defined on [t N −3 , t N ] × lR n . Under proper conditions, V Π (· , ·) is a classical solution to the following HJB equation: Further, by the definition of the map ψ(·), we may also write the above as x ∈ lR n . (4.47) Also, similar to (4.27), we have that in general, The above procedure can be continued recursively. By induction, we can construct sophisticated cost functional J k (t, x; u k (·)) for Player k, and with the value function V Π (· , ·) satisfying the following HJB equations on the time intervals associated with the partition Π: and for k = 1, 2, · · · , N − 1, x ∈ lR n , (4.51) where, for k = 1, 2, · · · , N − 1, Θ k+1 (· , ·) is the solution to the following (linear) PDE: x ∈ lR n . (4.52) Then for any given x ∈ lR n , letX Π (·) be the solution to the following closed-loop system: and denoteū According to our construction, we have where u k (·) ⊕ Ψ Π (·) [t k ,T ] is defined the same way as (4.8)-(4.9). Similar to (4.39), for k = 1, 2, · · · , N − 1, we have in general that Since the involved N players in Problem (G Π ) interact through the initial/terminal pairs (t k , X(t k )), k = 1, 2, · · · , N − 1, one should actually denote Hence, (4.56) means that if we letū then (ū 1 (·), · · · ,ū N (·)) is a Nash equilibrium of the N -person non-cooperative differential game associated with J k x; u 1 (·), · · · , u N (·) , 1 ≤ k ≤ N (defined in (4.58)). 20

The formal limits.
We now would like to look at the situation when Π → 0. Suppose we have the following: uniformly for (t, x) in any compact sets, for some V (· , ·). Under (H3), we also have uniformly for (t, x) in any compact sets, for  By (4.56), we have Thus, passing to the limits, we have (4.5). Also, we have the following: for some R(r) with R(r) → 0 as r → 0. Hence, by Definition 4.1, Ψ(· , ·) is a time-consistent equilibrium strategy, and V (· , ·) is a time-consistent equilibrium value function of Problem (N).
In the rest of this subsection, we will formally pass to the limits to find the equations that can be used to characterize the equilibrium value function V (· , ·). To this end, let us first write the equations for Θ k+1 (· , ·) in the integral forms: For k = 1, 2, · · · , N − 1, one has Let us look at V Π (· , ·).
Let us make some remarks on (4.77).
(i) It is an interesting feature of (4.77) that both Θ(τ, t, x) and Θ(t, t, x) appear in the equation where the later is the restriction of the former on τ = t. On one hand, although the equation is fully nonlinear, due to the fact that Θ(t, t, x) is different from Θ(τ, t, x), the existing theory for fully nonlinear parabolic equations cannot apply directly. On the other hand, it is seen that if Θ(t, t, x) is obtained from an independent way, then (4.77) is actually a linear equation for Θ(τ, t, x) with τ can be purely regarded as a parameter.
(ii) In the case that D(ψ) is not equal to D[0, T ] × lR n × lR n × S n , the condition has to be regarded as a part of the solution. We will see that for some interesting special cases, the above condition can come automatically. More generally, we may also write (4.75) as since the set arg min lH t, t, x, Θ x (t, t, x), Θ xx (t, t, x) might contain more than one point.
It is not hard to see that the above actually amounts to defining g ε (τ, t, x, u) = g(τ, t, x, u) + ε|u| 2 .
We may refer to the corresponding problem as a regularized problem. If the corresponding equilibrium value function is denoted by V ε (· , ·), then it is expected that However, in general, if (X ε (·),ū ε (·)) is an equilibrium pair for the regularized problem, we might not have the limit ofū ε (·) as ε → 0. In this case, we should be satisfied by the above characterization of the equilibrium value function V (· , ·), andū ε (·) can be regarded as some kind of "near equilibrium control".
(iv) For the case σ(t, x, u) = σ(t, x), i.e., the control does not enter the diffusion of the state equation, ψ(·) is independent of P and ψ(t, x, p) ∈ arg min p, b(t, x, ·) +g(τ, t, x, ·) . (4.82) Then the equilibrium HJB equation can be written as (4.83) We will carefully discuss this case in the next section. Note that for a deterministic problem, namely Problem (N) for an ordinary differential equation system, we may take for ε > 0 to regularize the problem. Then the corresponding equilibrium HJB equation reads (4.84) It is expected that Θ ε (· , · , ·) → Θ(· , · , ·) in some sense, as ε → 0, with (4.85) At the moment, it is not clear to us how one can define viscosity solution to the above equation.

(5.2)
Consider the following linear abstract backward evolution equation: Under some mild conditions, the above is well-posed, and we have the following variation of constant formula: where E(· , · ; v(·)) is called the backward evolution operator generated by L(· , v(·)). Consequently, the (timeconsistent) equilibrium value function V (t, ·) = Θ(t, t, ·) should be the solution to the following nonlinear functional integral equation: T t E(s, t; V (·))G(t, s, V (s))ds, t ∈ [0, T ]. (5.5) We call (5.5) the equilibrium HJB integral equation for Problem (N). Once a solution V (· , ·) of (5.5) is found, we can, in principle, construct a (time-consistent) equilibrium control and an equilibrium pair for Problem (N). Of course, if we like, we may also solve the equilibrium HJB equation (4.77), which actually is not necessary as far as the construction of a time-consistent equilibrium pair is concerned.
The well-posedness of (5.5) seems to be difficult for the general case. At the moment, we do not have a complete solution for that and hopefully, we can present some satisfactory results for the equilibrium HJB integral equation (5.5) in our future publications. On the other hand, in the rest of this section, we are going to present a well-posedness result for an interesting special case of (5.5), from which one can get some taste of the problem. The main hypothesis that we will assume below is the following: namely, the control does not enter the diffusion of the state equation. As we discussed in Section 4, in this case, our equilibrium HJB equation reads The essential feature of (5.7) is that Θ xx (t, t, x) does not appear in the equation (although Θ x (t, t, x) still appears there). This leads to the well-posedness problem much more accessible. Further, from Example 3.5, we see that there are cases for which ψ is as smooth as the coefficients and b(t, x, ψ(t, t, x, p)) is bounded. Therefore, the case that we are going to consider below, although very special, includes a big class of problems.
To avoid heavy notations, let us consider the following equation To investigate the well-posedness of (5.8) above, let us make some preparations. Let C α (lR n ) be the space of all continuous functions ϕ : lR n → lR such that Further, let C 1+α (lR n ) and C 2+α (lR n ) be the spaces of all functions ϕ : lR n → lR such that respectively. Next, let B([0, T ]; C α (lR n )) be the set of all measurable functions f : [0, T ] × lR n → lR such that for each t ∈ [0, T ], f (t, ·) ∈ C α (lR n ) and Also, we let C([0, T ]; C α (lR n )) be the set of all continuous functions that are also in B([0, T ]; C α (lR n )). Thus, Similarly, we define B([0, T ]; C k+α (lR n )) and C([0, T ]; C k+α (lR n )), respectively, for k = 1, 2.
We introduce the following hypotheses for the above equation (5.8).
Further, a(t, x) −1 exists for all (t, x) ∈ [0, T ] × lR n and there exist constants λ 0 , λ 1 > 0 such that We point out here that some of the conditions assumed in (P) can be substantially relaxed. However, we prefer not to get into those generalities for the sake of simplicity in our presentation. Note also that typically, the ellipticity condition of a(t, x) looks like for some δ > 0. It is clear that when a(· , ·) is assumed to be bounded, then the above is equivalent to (5.11). The number λ 0 in (5.11) will be used below.
For any v(· , ·) ∈ C([0, T ]; C 1+α (lR n )), we consider the following linear PDE, parameterized by τ ∈ [0, T ): where the differential operator L[t, v(·)] is defined by the following: In what follows, we let We have the following result whose proof follows a relevant one found in [9], with some minor modifications.
From the above procedure, we see that K > 0 in the above is an absolute constant, independent of (τ, t) ∈ D[0, T ]. Hence, in particular, we have (denoting .
Recall that in Section 4, by assuming the convergence of Θ Π (· , · , ·) and V Π (· , ·), we get the equilibrium HJB equation for Θ(· , · , ·) and then V (· , ·) is characterized by an equilibrium HJB integral equation. We now want to show that under conditions ensuring (P), we do have the expected convergence. This will make our whole procedure satisfactorily complete for certain cases, at least. For the sake of simplicity, we assume that all the involved functions are bounded and continuously differentiable up to a needed order with bounded derivatives. When (5.6) holds, for k = 0, 1, · · · , N − 1, we have Since (5.38) is well-posed, we have Also, by the assumed uniform Lipschitz continuity of τ → (h(τ, x), h y (τ, y), g(τ, t, x, u)), we have Next, by Proposition 5.1, we have − b s, y, ψ(s, s, y, V x (s, y)) , Θ x (τ, s, y) x (s, y)) − g τ, s, y, ψ(s, x, V x (s, y)) dyds.

(5.44)
This yields that From this, our expected convergence follows.

Some Special Cases
In this section, we are going to look at several important special cases. We will mainly look at the corresponding forms of our equilibrium HJB equations.

A Linear-Quadratic Problem
Let us look at the LQ problem. Then lH(τ, t, x, u, p, P ) = p, This yields ψ(τ, t, x, p, P ) = − R(τ, t) and lH(τ, t, x, ψ(t, t, x,p,P ), p, P ) t, x,p,P ), ψ(t, t, x,p,P ) Hence, the equilibrium HJB equation takes the following form: Although having a little bit complicated looking, the above has a quadratic structure which can help us to study the well-posedness of it. To see that, let with some undetermined map P : D[0, T ] → S n . Plugging the above into (6.3), we see that the map P (· , ·) should satisfy the following equation: with the terminal condition P (τ, T ) = G(τ ).
Note that if P (t, t) and R(t, t) are replaced by P (τ, t) and R(τ, t), respectively, the above becomes a standard Riccati equation with a parameter τ . The appearance of P (t, t) and R(t, t) makes the above non-standard.

34
We may rewrite the above as follows (suppressing t in P (τ, t), etc., for simplicity): Then the equation for P (· , ·) can be written as follows: Applying Itô's formula to s → P (τ, s)Φ(s, t)x, Φ(s, t)x on [t, T ], we have which leads to (6.8) Note that although P (τ, t) is a deterministic function, the above representation is stochastic. From the above, we have In particular, taking τ = t and denoting P (t) = P (t, t), one has Combining the above, we end up with the following system for the function P (·): (6.11) We refer to the above as a Riccati-Volterra integral equation system for the corresponding (time-inconsistent) LQ problem. Note that the above is actually a coupled forward-backward stochastic Volterra integral equation system (FBSVIE, for short). Some relevant results concerning backward stochastic Volterra integral equations (BSVIEs, for short) can be found in [25,26]. If Φ(· , ·), P (·) is a solution to the above, then the time-consistent equilibrium control is given bȳ In the case that A 1 (·) = 0, B 1 (·) = 0, (6.13) the above (6.11) is reduced to the following Riccati-Volterra integral equation system (for a deterministic time-inconsistent LQ problem):  (6.14) and the time-consistent equilibrium control is given by (6.12) with a simpler Γ(·). This recovers the case presented in [27] where the well-posedness of (6.14) was established.
For (6.11), we have the following result. Further, suppose Then (6.11) admits a unique solution.
Then (suppressing t) Consequently, with K > 0 being an absolute constant. Hence, we obtain Hence, the map p(·) → P (·) is contractive on X [τ, T ] as long as T − τ > 0 small. Then a usual argument applies to obtain a unique fixed point of p(·) → P (·) on X [0, T ]. This proves the well-posedness of (6.11).
To conclude this section, let us make a remark on the condition (6.16). It is not hard to show that when m = n and B 1 (·) −1 exists and bounded, then (6.16) holds. Apparently, this is a restrictive condition. We hope that in our future publications, such a condition can be removed.
The maximum of (u, c) → lH(t, s, x, u, c, p, P ) is attained at

Appendix
In this appendix, we present some detailed calculations.
Example 2.1. Recall that we are considering the following one-dimensional controlled linear SDE: with cost functional where σ > 0 is a constant and g(t) is a deterministic non-constant, continuous and positive function. For such a linear-quadratic optimal control problem on [t, T ] (with deterministic coefficients) and with t ∈ [0, T ) fixed, the Riccati equation takes the following form: (note that t ∈ [0, T ) is a parameter) P s (s, t) = P (s, t) 2 − σ 2 P (s, t), s ∈ [t, T ], P (T, t) = g(t).
Let us solve the above Riccati equation. By separation of variables, we have Integrating from s to T , one has Hence, P (s, t) = σ 2 g(t)e σ 2 (T −s) σ 2 + g(t)(e σ 2 (T −s) − 1) , s ∈ [t, T ], and the optimal control is given byū Thus, the closed-loop SDE reads dX(s) = −P (s, t)X(s)ds + σX(s)dW (s), s ∈ [t, T ], Consequently, the optimal state process is given bȳ Consequently, the optimal control also admits the following open-loop form: .
Thus, (7.31) will not be true. When (7.34) holds, the problem is referred to as the (classical) Merton's portfolio problem. In this case, (with a = λ−δ