Linear quadratic optimal control of conditional McKean-Vlasov equation with random coefficients and applications *

We consider the optimal control problem for a linear conditional McKean-Vlasov equation with quadratic cost functional. The coefficients of the system and the weigh-ting matrices in the cost functional are allowed to be adapted processes with respect to the common noise filtration. Semi closed-loop strategies are introduced, and following the dynamic programming approach in [32], we solve the problem and characterize time-consistent optimal control by means of a system of decoupled backward stochastic Riccati differential equations. We present several financial applications with explicit solutions, and revisit in particular optimal tracking problems with price impact, and the conditional mean-variance portfolio selection in incomplete market model.


Introduction and problem formulation
Let us formulate the linear quadratic optimal control of conditional (also called stochastic) McKean-Vlasov equation with random coefficients (LQCMKV in short form).Consider the controlled stochastic McKean-Vlasov dynamics in R d given by Here W, W 0 are two independent one-dimensional Brownian motions on some probability space (Ω, F, P), F 0 = (F 0 t ) 0≤t≤T is the natural filtration generated by W 0 , F = (F t ) 0≤t≤T is the natural filtration generated by (W, W 0 ), augmented with an independent σ-algebra G, ξ 0 ∈ L 2 (G; R d ) is a square-integrable G-measurable random variable with values in R d , E[X t |W 0 ] denotes the conditional expectation of X t given the whole σ-algebra F 0 T of W 0 , and the control process α is an F 0 -progressively measurable process with values in A equal either to R m or to L(R d ; R m ) the set of Lipschitz functions from R d into R m .This distinction of the control sets will be discussed later in the introduction, but for the moment, one may interpret roughly the case when A = R m as the modeling for open-loop control and the case when A = L(R d ; R m ) as the modeling for closed-loop control.When A = R m , we require that α satisfies the square-integrability condition L 2 (Ω × [0, T ]), i.e., E[ T 0 |α t | 2 dt] < ∞, and we denote by A the set of control processes.The coefficients b t (x, x, a), σ t (x, x, a), σ 0 t (x, x, a), 0 ≤ t ≤ T , are F 0 -adapted processes with values in R d , for any x, x ∈ R d , a ∈ A, and of linear form: where b 0 , γ, γ 0 are F 0 -adapted processes vector-valued in R d , satisfying a square-integrability condition L 2 (Ω × [0, T ]): ] < ∞, B, B, D, D, D 0 , D0 are essentially bounded F 0 -adapted processes matrix-valued in R d×d , and C, F , F 0 are essentially bounded F 0 -adapted processes matrix-valued in R d×m .For any α ∈ A, there exists a unique strong solution X = X α to (1.1), which is F-adapted, and satisfies the squareintegrability condition S 2 (Ω × [0, T ]): for some positive constant C α depending on α: when A = R m , C α depends on α via E[ T 0 |α t | 2 dt] < ∞, and when A = L(R d ; R m ), C α depends on α via its Lipschitz constant.
The cost functional to be minimized over α ∈ A is: where {f t (x, x, a), 0 ≤ t ≤ T }, is an F 0 -adapted real-valued process, g(x, x) is a F 0 Tmeasurable random variable, for any x, x ∈ R d , a ∈ A, of quadratic form: where Q, Q are essentially bounded F 0 -adapted processes, with values in S d the set of symmetric matrices in R d×d , P , P are essentially bounded F 0 T -measurable random matrices in S d , N is an essentially bounded F 0 -adapted process, with values in S m , M is an F 0adapted process with values in R d , satisfying a square integrability condition L 2 (Ω × [0, T ]), L is an F 0 T -measurable square integrable random vector in R d , and ⊺ denotes the transpose of any vector or matrix.
The above control formulation of stochastic McKean-Vlasov equations provides a unified framework for some important classes of control problems.In particular, it is motivated in particular by the asymptotic formulation of cooperative equilibrium for a large population of particles (players) in mean-field interaction under common noise (see, e.g., [17], [19]) and also occurs when the cost functional involves the first and second moment of the (conditional) law of the state process, for example in (conditional) mean-variance portfolio selection problem (see, e.g., [27], [7], [10]).When A = L(R d ; R m ), this corresponds to the problem of a (representative) agent, using a control α based on her/his current private state X t at time t, and of the information brought by the common noise F 0 t , typically the conditional mean E[X t |W 0 ], which represents, in the large population equilibrium interpretation, the limit of the empirical mean of the state of all the players when their number tend to infinity from the propagation of chaos.In other words, the control α may be viewed as a semi closed-loop control, i.e., closed-sloop w.r.t. the state process, and open-loop w.r.t. the common noise W 0 , or alternatively as a F 0 -progressively measurable random field control α = {α t (x), 0 ≤ t ≤ T, x ∈ R d }.This class of semi closed-loop control extends the class of closed-loop strategies for the LQ control of McKean-Vlasov equations (or mean-field stochastic differential equations) without common noise W 0 , as recently studied in [28] where the controls are chosen at any time t in linear form w.r.t. the current state value X t and the deterministic expected value E[X t ].When A = R m , the LQCMKV problem may be viewed as a special partial observation control problem for a state dynamics like in (1.1) where the controls are of open-loop form, and adapted w.r.t. an observation filtration F I = F 0 generated by some exogenous random factor process I driven by W 0 .In the case where σ = 0, we see that the process X is F 0 -adapted, hence E[X t |W 0 ] = X t , and the LQCMKV problem is reduced to the classical LQ control problem (see, e.g., [40]) with random coefficients, with open-loop controls for A = R m or closed-loop controls for A = L(R d ; R m ).Note that this distinction between open-loop and closed-loop strategies for LQ control problems has been recently introduced in [36] where closed-loop controls are assumed of linear form w.r.t. the current state value, while it is considered here a priori only Lipschitz w.r.t. the current state value.
Optimal control of McKean-Vlasov equation is a rather new topic in the area of stochastic control and applied probability, and addressed, e.g., in [4], [11], [8], [15], [31].In this McKean-Vlasov context, the class of linear quadratic optimal control, which provides a typical case for solvable applications, has been studied in several papers, among them [24], [39], [25], [35], where the coefficients are assumed to be deterministic.It is often argued that due to the presence of the law of the state in a nonlinear way (here for the LQ problem, the square of the expectation), the problem is time-inconsistent in the sense that an optimal control viewed from today is no more optimal when viewed from tomorrow, and this would prevent a priori the use of the dynamic programming method.To tackle time inconsistency, one then focuses typically on either pre-commitment strategies, i.e. controls that are optimal for the problem viewed at the initial time, but may be not optimal at future date, or game-equilibrium strategies, i.e., control decisions considered as a game against all the future decisions the controller is going to make.
In this paper, we shall focus on the optimal control for the initial value V 0 of the LQCMKV problem with random coefficients, but following the approach developed in [32], we emphasize that time consistency can be actually restored for pre-commitment strategies, provided that one considers as state variable the conditional law of the state process instead of the state itself, therefore making possible the use of the dynamic programming method.We show that the dynamic version of the LQCMKV control problem defined by a random field value function, has a quadratic structure with respect to the conditional law of the state process, leading to a characterization of the optimal control in terms of a decoupled system of backward stochastic Riccati equations (BSREs) whose existence and uniqueness are obtained in connection with a standard LQ control problem.The main ingredient for such derivation is an Itô's formula along a flow of conditional measures and a suitable notion of differentiability with respect to probability measures.We illustrate our results with several financial applications.We first revisit the optimal trading and benchmark tracking problem with price impact for general price and target processes, and obtain closed-form solutions extending some known results in the literature.We next solve a variation of the mean-variance portfolio selection problem in an incomplete market with random factor.Our last example considers an interbank systemic risk model with random factor in a common noise environment.
The paper is organized as follows.Section 2 gives some key preliminaries: we reformulate the LQCMKV problem into a problem involving the conditional law of the state process as state variable for which a dynamic programming verification theorem is stated and time consistency holds.We also recall the Itô's formula along a flow of conditional measures.Section 3 is devoted to the characterization of the optimal control by means of a system of BSREs in the case of both a control set A = R m and A = L(R d ; R m ).We develop in Section 4 the applications.
We end this introduction with some notations.
Notations.We denote by P 2 (R d ) the set probability measures µ on R d , which are square integrable, i.e., µ 2 2 := R d |x| 2 µ(dx) < ∞.For any µ ∈ P 2 (R d ), we denote by L 2 µ (R q ) the set of measurable functions ϕ : R d → R q , which are square integrable with respect to µ, by L 2 µ⊗µ (R q ) the set of measurable functions ψ : R d × R d → R q , which are square integrable with respect to the product measure µ ⊗ µ, and we set We also define L ∞ µ (R q ) (resp.L ∞ µ⊗µ (R q )) as the subset of elements ϕ ∈ L 2 µ (R q ) (resp.L 2 µ⊗µ (R q )) which are bounded µ (resp.µ ⊗ µ) a.e., and ϕ ∞ is their essential supremum.For any random variable X on (Ω, F, P), we denote by L(X) its probability law (or distribution) under P, by L(X|W 0 ) its conditional law given F 0 T , and we shall assume w.l.o.g. that G is rich enough in the sense that

Preliminaries
For any α ∈ A, and X α = (X α t ) 0≤t≤T the solution to (1.1), we define ρ α t = L(X α t |W 0 ) as the conditional law of X α t given F 0 T for 0 ≤ t ≤ T .Since X α is F-adapted, and W 0 is a (P, F)-Wiener process, we notice that ρ α t (dx and thus {ρ α t , 0 ≤ t ≤ T } admits an F 0 -progressively measurable modification (see, e.g., Theorem 2.24 in [5]), that will be identified with itself in the sequel, and is valued in P 2 (R d ) by (1.3), namely: Moreover, we mention that the process ρ α = (ρ α t ) 0≤t≤T has continuous trajectories as it is valued in P 2 (C([0, T ]; R d ) the set of square integrable probability measures on the space C([0, T ]; R d ) of continuous functions from [0, T ] into R d .Now, by the law of iterated conditional expectations, and recalling that α ∈ A is F 0progressively measurable, we can rewrite the cost functional as where we used in the second equality the fact that {f t (x, x, a), x, From the quadratic forms of f, g in (1.4), the random fields ft (µ, a) and ĝ(µ), (t, µ, a) where we define the functions on We shall make the following assumptions on the coefficients of the model: (H2) One of the two following conditions holds: (i) N is uniformly positive definite i.e.N t ≥ δI m , 0 ≤ t ≤ T , a.s.for some δ > 0; (ii) P or Q is uniformly positive definite, and F is uniformly nondegenerate, i.e. |F t | ≥ δ, 0 ≤ t ≤ T , a.s., for some δ > 0.
Let us define the dynamic formulation of the stochastic McKean-Vlasov control problem.For any t ∈ [0, T ], ξ ∈ L 2 (G; R d ), and α ∈ A, there exists a unique strong solution, denoted by {X t,ξ,α s , t ≤ s ≤ T }, to the equation (1.1) starting from ξ at time t, and by noting that X t,ξ,α is also unique in law, we see that the conditional law of X t,ξ,α s given F 0 T depends on ξ only through its law L(ξ) = L(ξ|W 0 ) (recall that G is independent of W 0 ).Then, recalling also that G is rich enough, the relation defines for any t ∈ [0, T ], µ ∈ P 2 (R d ), and α ∈ A, an F 0 -progressively measurable continuous process (up to a modification) {ρ t,µ,α s , t ≤ s ≤ T }, with values in P 2 (R d ), and as a consequence of the pathwise uniqueness of the solution {X t,ξ,α s , t ≤ s ≤ T }, we have the flow property for the conditional law (see Lemma 3.1 in [32] for details): (2.4) We then consider the conditional cost functional which is well-defined by (2.1) and under the boundedness assumptions on the weighting matrices of the quadratic cost function.We next define the F 0 -adapted random field value function so that which may take a priori for the moment the value −∞.We shall see later that the Assumptions (H1) and (H2) will ensure that V 0 is finite and there exists an optimal control.The dynamic counterpart of (2.5) is given by where A t (α) = {β ∈ A : β s = α s , s ≤ t}, and the second equality in (2.6) follows from the flow property (2.4) and the observation that ρ β t = ρ α t for β ∈ A t (α).By using general results in [22] for dynamic programming, one can show (under the condition that the random field v(µ) is finite) that the process {v t (ρ α t )+ t 0 fs (ρ α s , α s )ds, 0 ≤ t ≤ T } is a (P, F 0 )-submartingale, for any α ∈ A, and α * ∈ A is an optimal control for V 0 if and only if {v t (ρ We shall use a converse result, namely a dynamic programming verification theorem, which takes the following formulation in our context.

Lemma 2.1 Suppose that one can find an
for some positive constant C, and nonnegative F 0 -adapted process Then α is an optimal control for V 0 , i.e.V 0 = J(α), and Moreover, α is time consistent in the sense that Proof.By the local submartingale property in condition (ii), there exists a nondecreasing sequence of F 0 -stopping times (τ n ) n , τ n ր T a.s., such that From the quadratic form of f in (1.4), we easily see that for all n, for some positive constant C α depending on α (when ] < ∞, and when A = L(R d ; R m ), C α depends on α via its Lipschitz constant).Together with the quadratic growth condition of w, and from (2.1), one can then apply dominated convergence theorem by sending n to infinity into (2.8), and get where we used the terminal condition (i), and the expression (2.2) of the cost functional.Since α is arbitrary in A, this shows that w 0 (L(ξ 0 )) ≤ V 0 .The equality is obtained with the local martingale property for α in condition (iii).
From the flow property (2.4), and since ρ β t = ρ α t for β ∈ A t (α), we notice that the local submartingale and martingale properties in (ii) and (iii) are formulated on the interval [t, T ] as: By the same arguments as for the initial date, this implies that V α t = J t (ρ α t , α) = w t (ρ α t ), which means that α is an optimal control over [t, T ], once we start at time t from the initial state ρ α t , i.e., the time consistency of α. 2 The practical application of Lemma 2.1 consists in finding a random field {w t (µ), µ ∈ P 2 (R d ), 0 ≤ t ≤ T }, smooth (in a sense to be precised), so that one can apply an Itô's formula to {w t (ρ α t ) + t 0 fs (ρ α s , α s )ds, 0 ≤ t ≤ T }, and check that the finite variation term is nonnegative for any α ∈ A (the local submartingale condition), and equal to zero for some α ∈ A (the local martingale condition).For this purpose, we need a notion of derivative with respect to a probability measure, and shall rely on the one introduced by P.L. Lions in his course at Collège de France [29].We briefly recall the basic definitions and refer to [14] for the details, see also [12], [21].This notion is based on the lifting of functions u defined on In this case, the Fréchet derivative viewed as an element DU (X) of L 2 (G; R d ) by Riesz's theorem can be represented as is differentiable in the above lifted sense.Its derivative, interpreted thus as a mapping , and for any compact set K of P 2 (R d ), we have We next need an Itô's formula along a flow of conditional measures proved in [16] for processes with common noise.In our context, for the flow of the conditional law Then, for all t ∈ [0, T ], we have where for (t, µ, a) ∈ [0, T ] × P 2 (R d ) × A, L a t u(µ), D a t u(µ) are the F 0 t -measurable random functions in L 2 µ (R) defined by and The dynamic programming verification result in Lemma 2.1 and Itô's formula (2.9) are valid for a general stochastic McKean-Vlasov equation (beyond the LQ framework), and by combining with an Itô-Kunita type formula for random field processes, similar to the one in [26], one could apply it to {w t (ρ α t ) + t 0 fs (ρ α s , α s )ds, 0 ≤ t ≤ T } in order to derive a form of stochastic Hamilton-Jacobi-Bellman, i.e., a backward stochastic partial differential equation (BSPDE) for w t (µ), as done in [30] for controlled diffusion processes with random coefficients.We postpone this general approach for further study and, in the next sections, return to the important special case of LQCMKV problem[s] for which we show that BSPDE[s] are reduced to backward stochastic Riccati equations (BSRE) as in the classical LQ framework.

Backward stochastic Riccati equations
We search for an F 0 -adapted random field solution to the LQCMKV problem in the quadratic form for some F 0 -adapted processes (K, Λ, Y, χ), with values in S d × S d × R d × R, and in the backward SDE form for some F 0 -adapted processes K, Λ, Z Let us denote, for any α ∈ A, by S α the F 0 -adapted process equal to S α t = w t (ρ α t ) + t 0 fs (ρ α s , α s )ds, 0 ≤ t ≤ T , and observe then by Itô's formula (2.9) that it is of the form (The second-order derivatives terms w.r.t.k, ℓ and y do not appear since the functions v 2 , Var and v 1 are linear, respectively, in k, ℓ and y, respectively).From the derivatives expression of Var, v 2 and v 1 in (3.3), we then have We now distinguish between the cases when the control set

Control set
From the linear form of b t , σ t , σ 0 t in (1.2), and the quadratic form of ft in (2.3), after some straightforward calculations, we have: (3.5)Then, after square completion under the condition that Γ t (k, ℓ) is positive definite in S m , we have Therefore, whenever holds for all 0 ≤ t ≤ T , we have which implies that D α t ≥ 0, 0 ≤ t ≤ T , for all α ∈ A, i.e. S α t = w t (ρ α t ) + t 0 fs (ρ α s , α s )ds, 0 ≤ t ≤ T satisfies the (P, F 0 )-local submartingale property for all α ∈ A. We are then led to consider the system of BSDEs: s., the matrix process Γ(K, Λ) with values in S m is positive definite a.s., and the following relation The following verification result makes the connection between the system (3.7) and the LQCMKV1 control problem.
where X * = X α * is the state process with the feedback control ât (., K t , Λ t , Y t ), is an optimal control for the LQCMKV1 problem, i.e., V 0 = J(α * ), and we have ) is essentially bounded, and so α * ∈ A. Finally, from (3.6) we see that D α * = 0, which gives the (P, F 0 )-local martingale property of S α * , and we conclude by the dynamic programming verification Lemma 2.1. 2 Let us now show, under assumptions (H1) and (H2), the existence of a solution to the BSDE (3.7) satisfying the integrability conditions of Proposition 3.1.We point out that this system is decoupled: ∈ S d is linear, with essentially bounded coefficients.Since the terminal condition P is also essentially bounded, it is known by standard results for linear BSDEs that there exists a unique solution (K, Z K ) with values in S d × S d , s.t.K is essentially bounded and Z K lies in L 2 (Ω × [0, T ]).Moreover, since P and Φ t (0, 0) = Q t are nonnegative under (H1), we also obtain by standard comparison principle for BSDE that K t is nonnegative, for all 0 ≤ t ≤ T .
(ii) Given K, we next consider the BSDE for (Λ, Z Λ ) with generator: , and terminal condition P + P .This is a backward stochastic Riccati equation (BSRE), and it is well-known (see, e.g., [9]) that it is associated with a stochastic standard LQ control problem (without McKean-Vlasov dependence) with controlled linear dynamics: and quadratic cost functional where Under the condition that N K is positive definite, we can rewrite this cost functional after square completion as By noting that QK t ≥ Q t + Qt , it follows that the symmetric matrices QK and P + P are nonnegative under condition (H1), and assuming furthermore that N K is uniformly positive definite, we obtain from [37] the existence and uniqueness of a solution (Λ, Z Λ ) to this BSRE, with Λ being nonnegative and essentially bounded, and ).This implies, in particular, that Γ −1 (K, Λ) is well-defined and essentially bounded.Since K is nonnegative under (H1), notice that the uniform positivity condition on N K is satisfied under (H2): this is clear when N is uniformly positive definite (as usually assumed in LQ problem), and holds also true when F is uniformly nondegenerate, and K is uniformly positive definite, which occurs when P or Q is uniformly positive definite from comparison principle for the linear BSDE for K.
To sum up, we have proved the following result: Theorem 3.1 Under assumptions (H1) and (H2), there exists a unique solution (K, Z K ), (Λ, Z Λ ), (Y, Z Y ), (χ, Z χ ) to the BSDE (3.7) satisfying the integrability condition of Proposition 3.1, and consequently we have an optimal control for the LQCMKV1 problem given by (3.8).
and by the same arguments as in Proposition 3.1, we have the following verification result making the connection between the system (3.10) and the LQCMKV2 control problem.
and χ lies in S 1 (Ω × [0, T ]).Then, the control process α * with values in L(R d ; R m ) and defined by where X * = X α * is the state process with the feedback control ât (., ., K t , Λ t , Y t ), is an optimal control for the LQCMKV2 problem, i.e., V 0 = J(α * ), and we have Let us now discuss the existence of a solution to the BSDE (3.10) satisfying the integrability conditions of Proposition 3.2.As for (3.7), this system is decoupled.The difference w.r.t to the LQCMKV1 problem is in the BSDE for (K, Z K ), where the generator (k, z) ∈ In general, it is not in the class of BSREs related to LQ control problem, but existence can be obtained in some particular cases: (1) The coefficients B, C, D, F , D 0 , F 0 , Q, P , N are deterministic.In this case, the BSRE for K is reduced to a matrix Riccati ordinary differential equation: This problem is associated to the LQ problem with controlled linear dynamics where the control process α is an F-adapted process with values in R m , and the cost functional to be minimized over α is It was solved in [38] under assumption (H1) and the condition (H2)(i) that N is uniformly positive definite, and this gives the existence and uniqueness of K ∈ C 1 ([0, T ]; S d ), which is nonnegative.
(2) D ≡ F ≡ 0. In this case, the BSRDE for (K, Z K ) is associated to the LQ problem with controlled linear dynamics where the control process α is an F 0 -adapted process with values in R m , and the cost functional to be minimized over α is It is then known from [37] that under assumptions (H1) and (H2)(i), there exists a unique pair (K, Z K ) solution to the BSRDE, with K nonnegative, and essentially bounded.
(3) N ≡ 0, P is uniformly positive, m = d, and F is invertible with F −1 bounded.In this case, the BSDE for K is reduced to the linear BSDE: for which it is known that there exists a unique solution (K, Z K ), with K positive, and essentially bounded.
It is an open question whether existence of a solution for K to the BSRE (3.10) holds in the general case.Anyway, once a solution K exists, and is given, the BSDEs for the pairs (Λ, Z Λ ), (Y, Z Y ), (χ, Z χ ) are the same as in (3.7), and then their existence and uniqueness are obtained under the same conditions.

Trading with price impact and benchmark tracking
We consider an agent trading in a financial market with an inventory X t , i.e., a number of shares held at time t in a risky stock, governed by where the control α, a real-valued F 0 -progressively measurable process in L 2 (Ω × [0, T ]), represents the trading rate.Given a real-valued F 0 -adapted stock price process (S t ) 0≤t≤T in L 2 (Ω × [0, T ]), a real-valued F 0 -adapted target process (I t ) 0≤t≤T in L 2 (Ω × [0, T ]), and a terminal benchmark H as a square integrable F 0 T -measurable random variable, the objective of the agent is to minimize over control processes α a cost functional of the form: where η > 0, q ≥ 0, and λ ≥ 0 are constants.Such formulation is connected with optimal trading and hedging problems in presence of liquidity frictions like price impact, and widely studied in the recent years: when S ≡ = 0, the cost functional in (4.1) arises in option hedging in presence of transient price impact, see, e.g., [34], [3], [6], and is also related to the problem of optimal VWAP execution (see [23], [20]), or benchmark tracking, see [13].When q = 0, the minimization of the cost functional in (4.1) corresponds to the optimal execution problem arising in limit order book (LOB), as originally formulated in [2] in a particular Bachelier model for S, and has been extended (with general shape functions in LOB) in the literature, but mostly by assuming the martingale property of the price process, see, e.g., [1], [33].By rewriting the cost functional after square completion as with αt = α t + St 2η , we see that this problem fits into the LQCMKV1 framework (with b 0 t = − St 2η , without McKean-Vlasov dependence but with random coefficients), and Assumptions (H1), (H2) are satisfied.From Theorem 3.1, the optimal control is then given by where Λ is solution to the (ordinary differential) Riccati equation and Y is solution to the linear BSDE The solution to the Riccati equation is Λ t η = q/η q/η sinh( q/η(T − t)) + λ/η cosh( q/η(T − t)) λ/η sinh( q/η(T − t)) + q/η cosh( q/η(T − t)) , 0 ≤ t ≤ T, while the solution to the linear BSDE is given by By integrating the function Λ/η, we have and plugging into the expectation form of Y , the optimal control in (4.2) is then expressed as where and a kernel The optimal trading rule in (4.5) is decomposed in two parts: (i) The first term α * ,IH prescribes the agent to trade optimally towards a weighted average ÎH t , rather than the current target position I. Indeed, ÎH is a convex combination of the expected future of the terminal random target H, and of a weighted average of the running target I (notice that K(t, .) is a nonnegative kernel integrating to one over [t, T ]).The rate towards this target is at a speed proportional to its distance w.r.t the current investor's position, and the coefficient of proportionality is determined by the costs parameters η, q, λ and the time to maturity T −t.We retrieve the interpretation and results obtained in [6] in the limiting cases where λ = 0 (no constraint on the terminal position), and λ = ∞ (constraint on the terminal position X T = H).In the case where q = 0, we have Λ t /η = λ/(η + λ(T − t)), ÎH t = E[H|F 0 t ], and we retrieve, in particular, the expression α * ,IH = −X * t /(T − t), of optimal trading rate when H = 0, and λ → ∞ corresponding to the optimal execution problem with terminal liquidation X T = 0.
(ii) The second term α * ,S related to the stock price, is an incentive to buy or sell depending on whether the weighted average of expected future value of the stock is larger or smaller than its current value.In particular, when the price process is a martingale, then which is nonpositive for nonnegative price S t , hence meaning that due to the price impact, one must sell.Moreover, in the limiting case where λ → ∞, i.e., the terminal inventory X T is constrained to achieve the target H, then α * ,S is zero: we retrieve the result that the optimal trading rate does not depend on the price process when it is a martingale, see [1], [33].
On the other hand, by applying Itô's formula to (4.2), and using (4.3)-(4.4),we have which implies the notable property: t 0 (X * s − I s )ds, 0 ≤ t ≤ T, is a martingale.

Conditional Mean-variance portfolio selection in incomplete market
We consider an agent who can invest in a financial market model with one bond of price process S 0 and one risky asset of price process S governed by dS 0 t = S 0 t r(I t )dt dS t = S t ((b + r)(I t )dt + σ(I t )dW t ), where I is a factor process with dynamics governed by a Brownian motion W 0 , assumed to be non correlated with the Brownian motion W driving the asset price process S, and r the interest rate, b the excess rate of return, and σ the volatility are measurable bounded functions of I, with σ(I t ) ≥ ε for some ε > 0. We shall assume that the natural filtration generated by the observable factor process I is equal to the filtration F 0 generated by W 0 .Notice that the market is incomplete as the agent cannot trade in the factor process.The investment strategy of the agent is modeled by a random field F 0 -progressively measurable process α = {α t (x), 0 ≤ t ≤ T, x ∈ R} (or equivalently as a F 0 -progressively measurable process with values in L(R; R)) where α t (x) with values in R, is Lipschitz in x, and represents the amount invested in the stock at time t, when the current wealth is X t = x, and based on the past observations F 0 t of the factor process.The evolution of the controlled wealth process is then given by dX t = r(I t )X t dt + α t (X t ) b(I t )dt + σ(I t )dW t , 0 ≤ t ≤ T, X 0 = x 0 ∈ R. (4.6)The objective of the agent is to minimize over investment strategies a criterion of the form: where λ is a positive F 0 T -measurable random variable.In the absence of random factors in the dynamics of the price process, hence in a complete market model, and when λ is constant, the above criterion reduces to the classical mean-variance portfolio selection, as studied e.g. in [27].Here, in presence of the random factor, we consider the expectation of a conditional mean-variance criterion, and also allow the risk-aversion parameter λ to depend reasonably on the random factor environment.By rewriting the cost functional as we then see that this conditional mean-variance portfolio selection problem fits into the LQCMKV2 problem, and more specifically into the case (3) of the discussion following Proposition 3.2.The optimal control is then given from (3.11) by where X * is the optimal wealth process in (4.6) controlled by α * , K is the solution to the linear BSDE with ᾱt (X t ) = α t (X t ) − q(E[X t |W 0 ] − X t ).Assuming that q 2 ≤ η, this model fits into the LQCMKV2 problem, and more specifically into the case (2) of the discussion following Proposition 3.2.The optimal control is then given from (3.11) by where X * is the optimal log-monetary reserve in (4.11) controlled by α * , K is the solution to the BSRE: Λ is the solution to the BSRE The nonnegative solution K to the BSRE is, in general, not explicit, while the solution for (Λ, Y ) is obviously equal to Λ ≡ 0 ≡ Y .From (4.12), it is then clear that E[α * t (X * t )|W 0 ] = 0, so that the conditional mean of the optimal log-monetary reserve is governed from (4.11) by dE[X * t |W 0 ] = σ(I t )ρ(I t )dW 0 t .