Bruno Bouchard Ceremade

Ngoc Minh Dang
email: dang@ceremade.dauphine.fr

Generalized stochastic target problems for pricing and partial hedging under loss constraints -Application in optimal book liquidation

Keywords: Stochastic target problems, state constraints, pricing under risk constraint, book liquidation. Mathematical subject classifications: Primary 49L25, 60J60; secondary 49L20, 35K55

We consider a singular with state constraints version of the stochastic target problems studied in [14], [16] and more recently [1], among others. This provides a general framework for the pricing of contingent claims under risk constraints. Our extended version perfectly suits to market models with proportional transaction costs and to order book liquidation issues. Our main result is a PDE characterization of the associated pricing function. As an example of application, we discuss the evaluation of VWAP-guaranteed type book liquidation contracts, for a general class of risk functions.

Introduction

Stochastic target technics have been originally introduced in mathematical finance by Soner and Touzi [START_REF] Soner | Super-replication under Gamma constraint[END_REF] in order to provide a PDE characterization of the super-hedging price of an European claim under gamma constraints. The classical super-hedging problem takes the general form: find the minimal Y φ (0) such that there exists a control φ, in a suitable admissibility set A, satisfying Y φ (T ) ≥ g(X φ (T )) Pa.s., where g is the payoff function of an European claim, φ stands for the financial strategy, Y φ for the wealth process and X φ for the stock price process, which may be influenced by the financial strategy, as in large investor models for instance. In general, such a problem is treated in mathematical finance via the dual formulation approach which allows one to relate the minimal Y φ (0) to a stochastic control problem in standard form. However, this approach heavily relies on the fact that the wealth dynamics is linear in the control and that the stocks prices are not influenced by the trading strategy. In particular, it does not apply to large investor models or to more general dynamics or constraints, such as gamma constraints. This was the motivation of Soner and Touzi for introducing the so-called stochastic target approach.

Their main discovery is a dynamic programming principle which is directly written on the associated stochastic target problem, and therefore does not appeal to any form of dual formulation, see Theorem 2.3 below. It turns out to be sufficient to provide a PDE characterization for the associated value function. This approach led to a series of papers providing a direct way to characterize super-hedging prices, see e.g. [START_REF] Bouchard | Explicit Solution of the Multivariate Super-Replication Problem under Transaction Costs[END_REF], [START_REF] Cheridito | The multi-dimensional super-replication problem under Gamma constraints[END_REF], [START_REF] Soner | The problem of super-replication under constraints[END_REF] and [START_REF] Touzi | Direct characterization of the value of super-replication under stochastic volatility and portfolio constraints[END_REF]. Up to the recent work of Bouchard, Elie and Touzi [START_REF] Bouchard | Stochastic target problems with controlled loss[END_REF], this approach was however limited to super-hedging problems which in turn typically lead to high prices which are not reasonable in practice, see e.g. [START_REF] Cvitanić | A closed-form solution to the problem of superreplication under transaction costs[END_REF] and [START_REF] Cvitanić | Super-replication in stochastic volatility models with portfolio constraints[END_REF]. Apart from technical improvements, the main result of Bouchard, Elie and Touzi [START_REF] Bouchard | Stochastic target problems with controlled loss[END_REF] is that pricing problems under risk constraints of the form: find the minimal Y φ (0) such that there exists a control φ ∈ A satisfying E ℓ(Y φ (T )g(X φ (T ))) ≥ p, for some "loss function" ℓ and a threshold p, can actually be treated via the stochastic target approach of Soner and Touzi [START_REF] Soner | Super-replication under Gamma constraint[END_REF] and [START_REF] Soner | Stochastic target problems, dynamic programming and viscosity solutions[END_REF]. For ℓ of the form ℓ(r) = 1 r≥0 and p ∈ (0, 1), one retrieves the quantile hedging problem of Follmer and Leukert [START_REF] Föllmer | Quantile Hedging[END_REF]. When ℓ stands for a utility function and p := sup{E ℓ(Y φ (T )) : Y φ (0) = y 0 , φ ∈ A}, this corresponds to a utility indifference pricing problem. More generally, one can treat risk constraints of the form E Ψ(X φ (T ), Y φ (T )) ≥ p, for a general class of "risk functions" -Ψ. The success ratio hedging problem of Follmer and Leukert [START_REF] Föllmer | Quantile Hedging[END_REF] enters into this framework. Finally, American type constraints can be introduced, see Bouchard and Vu [START_REF] Bouchard | The American version of the geometric dynamic programming principle, Application to the pricing of american options under constraints[END_REF]. This provides a general framework for a direct characterization of risk based prices of contingent contracts. In Bouchard, Elie and Touzi [START_REF] Bouchard | Stochastic target problems with controlled loss[END_REF], the authors restrict to dynamics given by Brownian SDEs in which only the drift and the volatility coefficients are controlled. In this paper, we show how their results can be extended to the case where the dynamics are controlled by processes with bounded variations and state constraints have to be satisfied. This extension is mainly motivated by the pricing of a VWAP-type1 book liquidation contract, however the domain of application is vast, in particular it perfectly suits to partial hedging problems under proportional transaction costs, see Example 2.3 below. We therefore first consider a general abstract formulation that could be used in many different practical situations/models. It is presented in Section 2 together with examples of application. The associated general PDE characterization is provided in Section 3. The pricing problem of a VWAP-type book liquidation contract is fully discussed in Section 4. The proofs of our abstract results are collected in Section 5.

Notations: We denote by x i the i-th component of a vector x ∈ R d , which will always be viewed as a column vector, with transposed vector x ⊤ , and Euclidean norm |x|. The element e i ∈ R d is the i-th unit vector: e j i = 1 i=j , i, j ≤ d. The set M d is the collection of ddimensional square matrices M with coordinates M ij , and norm |M | defined by viewing M as an element of R d×d . We denote by S d the subset of elements of M d that are symmetric. For a subset O of R d , we denote by Ō its closure, by int(O) its interior, by ∂O its boundary, and by dist(x, O) the Euclidean distance from x to O with the convention dist(x, ∅) = ∞. We denote by B r (x) the open ball of radius r > 0 centered at x ∈ R d . If B = [s, t] × O for s ≤ t and O ⊂ R d , we write ∂ p B := ([s, t) × ∂O) ∪ ({t} × Ō) for its parabolic boundary. Given a smooth function ϕ : (t, x 1 , . . . , x k ) ∈ R + × R kd → R, we denote by ∂ t ϕ its derivative with respect to its first variable, we write Dϕ and D 2 ϕ for the Jacobian and Hessian matrix with respect to (x 1 , . . . , x k ), and D x i ϕ and D 2

x i ϕ the Jacobian and Hessian matrix with respect to x i , i ≥ 1. Any inequality or inclusion involving random variables has to be taken in the a.s. sense. For a process L with bounded variations, we write |L| • to denote its total variation.

2 Abstract formulation and dynamic programming 2.1 The general singular stochastic target problem with state constraints

We first describe the abstract model. We refer to Section 2.2 for examples of typical dynamics in finance, and to Section 4 for a full discussion of its application to the pricing of VWAP-type book liquidation contracts.

From now on, we let (Ω, F, P) be a probability space supporting a d-dimensional Brownian motion W , d ≥ 1, F := (F t ) t≥0 denote the right-continuous completed filtration generated by W , and T > 0 be a finite time horizon.

The abstract stochastic target problem is defined as follows.

Our set of controls is U × L, where U stands for the set of all progressively measurable process ν in L 2 ([0, T ] × Ω) taking values in a given closed subset U of R d , and L denotes the set of continuous R d -valued adapted processes L which are non-decreasing (component by component) and such that

E |L| 2 T < ∞. For t ∈ [0, T ], z := (x, y) ∈ R d × R and φ := (ν, L) ∈ A := U × L, the controlled process Z φ t,z := (X φ t,x , Y φ t,x,y ) is defined as the R d × R-valued unique strong solution of the stochastic differential equation X φ t,x (s) = x + s t µ X (X φ t,x (r), ν r )dr + β X (X φ t,x (r))dL r + s t σ X (X φ t,x (r), ν r )dW r Y φ t,x,y (s) = y + s t µ Y (Z φ t,x,y (r), ν r )dr + β Y (Z φ t,x,y (r)) ⊤ dL r + s t σ Y (Z φ t,x,y (r), ν r ) ⊤ dW r , (2.1) where (µ X , σ X ) : (x, u) ∈ R d × U → R d × M d , (µ Y , σ Y ) : (z, u) ∈ R d+1 × U → R × R d , β X ∈ C 2 (R d , M d ) and β Y ∈ C 2 (R d+1 , R d
) are assumed to be Lipschitz continuous. Given a family of non-empty Borel subsets (O(t)) t≤T of R d+1 , the stochastic target problem consists in characterizing the value function

(t, x) ∈ [0, T ] × R d → v(t, x) := inf {y ∈ R : (x, y) ∈ V (t)} , (2.2)
where the set valued map V is defined as

t ∈ [0, T ] → V (t) := {z ∈ R d+1 : A t,z = ∅} , (2.3)
and

A t,z := {φ ∈ A : Z φ t,z (s) ∈ O(s) for all s ∈ [t, T ] P -a.s.} . (2.4)
In order to fully characterize the set valued map V in terms of the value function v, we shall assume all over this paper the following:

Standing Assumption 1: For all (t, x) ∈ [0, T ] × R d : (x, y) ∈ O(t) and y ′ ≥ y ⇒ (x, y ′ ) ∈ O(t).
Remark 2.1. It follows from Standing Assumption 1 and standard comparison arguments for stochastic differential equations that A t,x,y ′ ⊂ A t,x,y for y ′ ≥ y. In particular, (x, y ′ ) ∈ V (t) whenever (x, y) ∈ V (t) and y ′ ≥ y, so that V (t) can be (at least when the infimum in the

definition of v is achieved) identified to {(x, y) ∈ R d × R : y ≥ v(t, x)}.
In order to give a sense to the following discussions, we also assume that:

Standing Assumption 2: v is locally bounded on DY where

D Y := {(t, x) ∈ [0, T ) × R d : ∃ y ∈ R s.t. (x, y) ∈ O(t)}.
Remark 2.2. Obviously, the fact that all the relevant quantities take values in R d is used to save notations. One could without difficulty restrict to the case where some components of X only take positive values, which is typically the case for prices of stocks or bonds. By putting to 0 part of the coefficients, one can also retrieve situations where W , L and X do not have the same effective dimension. One could similarly add a time dependence in the coefficients, e.g. by considering the first component of X as a time parameter.

Examples of application

Before to go further in the general treatment, let us immediately discuss some typical examples of application that motivate this work (see also Section 4 for an application to optimal book liquidation that will be studied in details).

Example 2.1. Let us first consider the case where (β X , β Y ) = 0,

µ X (x, u) = diag[x]µ , σ X (x, u) = diag[x]σ and µ Y (x, y, u) = u ⊤ diag[x]µ , σ Y (x, y, u) = u ⊤ diag[x]σ
where diag[x] stands for the diagonal matrix with x i as the i-th diagonal element, µ ∈ R d and σ ∈ M d . The dynamics (2.1) then read, for s ∈ [t, T ]:

X t,x (s) = x + s t diag [X t,x (r)] µdr + s t diag [X t,x (r)] σdW r Y ν t,x,y (s) = y + s t ν ⊤ r dX t,x, (r) ,
where we only write X for X ν,L and Y ν for Y ν,L because X is not affected by the control and Y depends on (ν, L) only through ν.

Restricting to initial condition x ∈ (0, ∞) d , this corresponds to the d-dimensional Black and Scholes model: X i models the dynamics of a financial asset, the risk free interest rate is 0, ν i t stands for the number of units of X i held in a financial portfolio at time t, and Y is the associated wealth process starting from the initial endowment y.

If we now take O of the form:

O(t) = (0, ∞) d × R1 t<T + 1 t=T (x, y) ∈ (0, ∞) d × R : y ≥ g(x) , t ≤ T ,
for some measurable map g : R d → R, the value function can be written as

v(t, x) := inf y ∈ R : Y ν t,x,y (T ) ≥ g (X t,x (
T )) for some ν ∈ U . This corresponds to the usual definition of the super-hedging price of an European option of payoff function g. For O of the form

O(t) = (x, y) ∈ (0, ∞) d × R : y ≥ g(x) , t ≤ T ,
this corresponds to the super-hedging price of an American option.

Example 2.2. Let us now consider a two-dimensional model d = 2 with the following parameters

µ 1 X (x, u) = x 1 µ , µ 2 X (t, u) = x 2 µ , σ 11 X (x, u) = x 1 σ , σ 21 X (x, u) = x 2 σ and β 21 X (x) = -1 , β 22 X (x) = 1 , β Y (x, y) = (1 -λ, -1 -λ)
, where µ ∈ R, σ > 0 and λ ∈ (0, 1), and the other parameters are equal to 0. The dynamics (2.1) then read, for s ∈ [t, T ]:

X 1 t,x (s) = x 1 + s t X 1 t,x (r)µdr + s t X 1 t,x (r)σdW 1 r X 2,L t,x (s) = x 2 + s t X 2,L t,x (r) X 1 t,x (r) dX 1 t,x (r) - s t dL 1 r + s t dL 2 r Y L t,y (s) = y + s t (1 -λ)dL 1 r - s t (1 + λ)dL 2 r .
This corresponds to the one-dimensional model with proportional transaction costs studied in [START_REF] Cvitanić | A closed-form solution to the problem of superreplication under transaction costs[END_REF]. More precisely, there is only one risky asset, X 1 , with a Black and Scholes type dynamics.

When buying or selling this risky asset, the investor pays a proportional transaction costs λ ∈ (0, 1). The process L := L 2 -L 1 stands for the cumulative net amount of money invested in the risky asset from time 0, i.e. L 2 r (resp. L 1 r ) is the cumulated value of bought (resp. sold) shares of X 1 . Each time a buying or selling operation d L is done, the investor pays, in money, a proportional transaction cost λd| L|. The wealth process is described by the two dimensional process (Y, X 2 ) where Y models the evolution of the cash account, and X 2 corresponds to the value of the part of the portfolio invested in X 1 , when taking into account the transaction costs.

For O defined as

O(t) = (0, ∞) × R 2 1 t<T + 1 t=T (x, y) ∈ (0, ∞) × R 2 : Λ(y, x) ≥ g(x) , t ≤ T , where Λ(y, x) := y + x 1 x 2 -λ|x 2 |x 1
provides the value in cash of a terminal position (y, x 2 ) if the value of the stock is x 1 , one retrieves the notion of super-hedging price of an European option with cash delivery, in the financial market model with proportional transaction costs. Obviously, one can consider similarly markets with more than one risky asset, see e.g. [START_REF] Bouchard | Explicit Solution of the Multivariate Super-Replication Problem under Transaction Costs[END_REF].

Example 2.3. As shown in [START_REF] Cvitanić | A closed-form solution to the problem of superreplication under transaction costs[END_REF] and [START_REF] Bouchard | Explicit Solution of the Multivariate Super-Replication Problem under Transaction Costs[END_REF], the super-hedging criteria is much too strict in markets with proportional transaction, as it leads to degenerate strategies of buy-and-hold type which do not reflect the market behavior. It follows that it should be relaxed by using, for instance, quantile or expected loss approaches as studied in [START_REF] Föllmer | Quantile Hedging[END_REF], [START_REF] Föllmer | Efficient hedging : cost versus shortfall risk[END_REF], for frictionless markets. The loss function pricing approach consists in choosing a non-decreasing (typically concave) function ℓ : R → R and defining the price at time t of an European option of payoff (say with cash delivery) g X 1 t,x (T ) as v(t, x 1 , 0; p) with v(t, x; p)

:= inf y ∈ R : E ℓ Λ(Y L t,y (T ), X L t,x (T )) -g X 1 t,x (T ) ≥ p for some L ∈ L ,
where p is a given threshold in R, and Λ is defined as in the previous example.

Assuming that Ξ L t,x,y (T ) := ℓ Λ(Y L t,y (T ), X L t,x (T ))g X 1 t,x (T ) ∈ L 2 for all initial conditions and control L, the arguments of Proposition 3.1 in [START_REF] Bouchard | Stochastic target problems with controlled loss[END_REF] then show that v(t, x; p) = inf y ∈ R : Ξ L t,x,y (T ) ≥ X 3,ν t,p (T ) for some (L, ν) ∈ L × U ,

with U = R 2 and X 3,ν t,p := p + • t ν 1 s dW 1 s .
Indeed, if Ξ L t,x,y (T ) ≥ X 3,ν t,p (T ) then taking expectation leads to E[Ξ L t,x,y (T )] ≥ p, while, if

p 0 := E[Ξ L t,
x,y (T )] ≥ p, then the martingale representation theorem implies that we can find ν ∈ U such that Ξ L t,x,y (T ) = X 3,ν t,p 0 (T ) ≥ X 3,ν t,p (T ). Hence, this last example enters into our general framework with the dynamics given in Example 2.2 and an additional controlled process X 3,ν defined as above.

Example 2.4. Influence of the trading strategies can be incorporated in the previous example without much difficulties. It suffices to consider more general models in which the dynamics of X 1 depends on L. It can for instance take the form

X 1,L t,x (s) = x 1 + s t X 1,L t,x (r)µdr + s t X 1,L t,x (r)σdW 1 r - s t X 1,L t,x (r)β -dL 1 r + s t X 1,L t,x (r)β + dL 2 r
with β -, β + ≥ 0. In this case, a buying order drives the price up, while a selling order pushes the price down. Note that constraints on the liquidation value of the portfolio (X 2,L , Y L ) could also be incorporated by playing with the definition of O. For instance,

O(t) = (x, y) ∈ (0, ∞) × R 2 : Λ(y, x) ≥ -c 1 t<T + (x, y) ∈ (0, ∞) × R 2 : Λ(y, x) ≥ g(x) 1 t=T , t ≤ T ,
means that the liquidation value of the portfolio should never be less than -c.

Dynamic programming

We now come back to the abstract problem (2.2).

In order to provide a PDE characterization of the value function v, we shall appeal to the geometric dynamic programming principle introduced in [START_REF] Soner | Stochastic target problems, dynamic programming and viscosity solutions[END_REF] and [START_REF] Soner | Dynamic programming for stochastic target problems and geometric flows[END_REF] in the case O(t) = R d+1 for t < T , and extended in [START_REF] Bouchard | The American version of the geometric dynamic programming principle, Application to the pricing of american options under constraints[END_REF] This heuristical reasoning can be made rigorous under the following right-continuity assumption:

Standing Assumption 3. [Right-continuity of the target] For all sequence (t n , z n ) n of [0, T ] × R d+1 such that (t n , z n ) → (t, z), we have t n ≥ t n+1 and z n ∈ O(t n ) ∀ n ≥ 1 =⇒ z ∈ O(t) .
In the statement below, we denote by T [t,T ] the set of stopping times with values in [t, T ], for t ≤ T , and use the notation

O τ,θ V := O(τ ) 1 τ ≤θ + V (θ) 1 τ >θ for θ, τ ∈ T [0,T ] .
Theorem 2.3 (Geometric Dynamic Programming Principle). For all t ≤ T ,

V (t) = z ∈ R d+1 : ∃ φ ∈ A s.t. Z φ t,z (θ ∧ τ ) ∈ O τ,θ V for all θ, τ ∈ T [t,T ] .
Proof. Note that the formulation is slightly different from Theorem 2.1 in [START_REF] Bouchard | The American version of the geometric dynamic programming principle, Application to the pricing of american options under constraints[END_REF], however it should be clear that their result can be stated in the above form, see the proofs of Lemma 2.1 and 2.2 in [START_REF] Bouchard | The American version of the geometric dynamic programming principle, Application to the pricing of american options under constraints[END_REF]. It thus suffices to check that the conditions A1-A2 and Z1-Z5 of Section 2.1 in [START_REF] Bouchard | The American version of the geometric dynamic programming principle, Application to the pricing of american options under constraints[END_REF] hold. Clearly, A1 holds. Also note that A is a separable metric space so that A2 holds by Lemma 2.1 and the discussion in Section 2.5 in [START_REF] Soner | Dynamic programming for stochastic target problems and geometric flows[END_REF]. The condition Z1 is satisfied with the additional convention Z φ t,z (s) = 0 for s < t. The verification of Z2 is standard in our Brownian diffusion framework, up to passing to the canonical space, see e.g. Section 3.2 of [START_REF] Bouchard | The American version of the geometric dynamic programming principle, Application to the pricing of american options under constraints[END_REF]. The flow and causality property Z3 and Z4 follow from the uniqueness of the solution to (2.1) for any φ ∈ A and any initial condition. As for the last condition Z5, one easily deduces from the Lipschitz continuity of the coefficients in (2.1) that, for all s ≥ t,

|Z φ t,z (s) -Z φ ′ t ′ ,z ′ (s)| converges to 0 in L 2 (Ω) when (t ′ , z ′ ) → (t, z), ν ′ → ν in L 2 (Ω × [0, T ]), and |L ′ -L| T → 0 in L 2 (Ω)
, where we used the identification φ = (ν, L) and φ ′ = (ν ′ , L ′ ). ✷

Under our Standing Assumption 1, recall Remark 2.1, Theorem 2.3 translates in terms of the value function v as follows:

Corollary 2.4. Fix (t, x, y) ∈ [0, T ] × R d+1 . (GDP1): If y > v(t, x), then there exists φ ∈ A such that, for all θ ∈ T [t,T ] , Y φ t,x,y (θ) ≥ v(θ, X φ t,x (θ)) and Z φ t,x,y (s ∧ θ) ∈ O(s ∧ θ) for all s ∈ [t, T ] . (GDP2): If y < v(t, x), then P Y φ t,x,y (θ) > v(θ, X φ t,x (θ)) and Z φ t,x,y (s ∧ θ) ∈ O(s ∧ θ) ∀ s ∈ [t, T ] < 1 ∀ (φ, θ) ∈ A × T [t,T ] .

PDE characterization in the abstract model

Our main result is a direct PDE characterization of the risk constraint based pricing function v.

Formal derivation

Before to state our main result rigorously, let us first explain formally how it can be deduced from Corollary 2.4.

Interior of the domain

In the case where O(t) = R d+1 for all t < T and β X = β Y = 0, it is shown in [START_REF] Bouchard | Stochastic target problems with controlled loss[END_REF] and [START_REF] Soner | Dynamic programming for stochastic target problems and geometric flows[END_REF] that v should (in a suitable sense) solve on [0,

T ) × R d sup F u x, v(t, x), ∂ t v(t, x), Dv(t, x), D 2 v(t, x) , u ∈ N 0 (x, v(t, x), Dv(t, x)) = 0 (3.1)
where, for Θ = (x, y, r, p,

Q) ∈ R d × R × R × R d × S d , u ∈ U , and ε ≥ 0, F u (Θ) := µ Y (x, y, u) -r -p ⊤ µ X (x, u) - 1 2 Tr σ X σ ⊤ X (x, u)Q , N ε (Θ) := {u ∈ U : |N u (x, y, p)| ≤ ε} with N u (x, y, p) := σ Y (x, y, u) -σ X (x, u) ⊤ p . (3.2)
The reasoning behind the above result is the following. If y = v(t, x), if the infimum in the definition of v is achieved, and if v is smooth, then Theorem 2.3 implies that there exists ν ∈ U such that, with φ = (ν, 0), dY φ t,x,y (t) ≥ dv(t, X φ t,x (t)). Formally, this implies that ν t should thus be such that σ

Y (X φ t,x (t), Y φ t,x,y (t), ν t ) = σ X (X φ t,x (t), ν t ) ⊤ Dv(t, X φ t,x (t)) and µ Y (X φ t,x (t), Y φ t,x,y (t), ν t ) ≥ L νt X v(t, X φ t,x (t))
where, for a smooth function ϕ and u ∈ U ,

L u X ϕ := ∂ t ϕ + Dϕ ⊤ µ X (•, u) + 1 2 Tr σ X σ ⊤ X (•, u)D 2 ϕ . Since (X φ t,x (t), Y φ t,x,y (t)) = (x, y) = (x, v(t, x)
), this imposes that the left-hand side of (3.1) is non-negative. On the other hand, the "optimality" of v should lead to equality in (3.1).

In our situation where β X , β Y = 0, one can also use the bounded variation process L in the dynamics (2.1) to insure that dY φ t,x,y (t) ≥ dv(t, X φ t,x (t)). It suffices to find a direction

ℓ ∈ ∆ + := [0, ∞) d ∩ B 1 (0) such that G ℓ (x, y, Dv(t, x)) > 0, where G ℓ (x, y, p) := β Y (x, y) ⊤ -p ⊤ β X (x) ℓ ,
and to "push in this direction". This corresponds to reflecting the process (s, X(s), Y (s)) s on the boundary of the set

{(t ′ , x ′ , y ′ ) : y ′ ≥ v(t ′ , x ′ )}. Assuming v smooth enough, it is possible only if such a ℓ exists.
It thus follows that v should satisfy either (3.1) or G ℓ (x, v(t, x), Dv(t, x)) > 0 for some ℓ ∈ ∆ + , i.e., at least,

H 0 (x, v(t, x), ∂ t v(t, x), Dv(t, x), D 2 v(t, x)) ≥ 0 (3.3) where, for Θ = (x, y, r, p, Q) ∈ R d × R × R × R d × S d , and ε ≥ 0, H ε (Θ) := max {F ε (Θ) , G(Θ)} with F ε (Θ) := sup {F u (Θ), u ∈ N ε (Θ)} , G(Θ) := max β Y (x, y) ⊤ -p ⊤ β X (x) ℓ, ℓ ∈ ∆ + .

Space boundary

We also have to take care of the state constraint (X, Y ) ∈ O. To this purpose, we shall assume that the set

D := {(t, x, y) ∈ [0, T ] × R d+1 : (x, y) ∈ O(t)} . (3.4)
is smooth enough:

Standing Assumption 4: There exists a locally

C 1,2 function δ on [0, T ) × R d+1 such that δ > 0 in intD, δ = 0 on ∂ 0 D := ∂D ∩ ([0, T ) × R d+1 ), and δ < 0 on ([0, T ) × R d+1 ) \ D.
For (t, x) such that (t, x, y) = (t, x, v(t, x)) ∈ ∂ 0 D, we can then follow the same reasoning as above, taking into account the fact that now, the control φ = (ν, L) should be such that, at the same time, dδ(t, X φ t,x (t), Y φ t,x,y (t)) ≥ 0 and dY φ t,x,y (t) ≥ dv(t, X φ t,x (t)). As above, this can be achieved either through the drift parts, once the Brownian parts are cancelled, or through the bounded variation process, in the case where a suitable inward direction is available. This leads to

H in 0 (x, v(t, x), ∂ t v(t, x), Dv(t, x), D 2 v(t, x)) ≥ 0 (3.5)
where, for Θ = (t, x, y, r, p, Q)

∈ [0, T ] × R d × R × R × R d × S d , u ∈ U , and ε ≥ 0, H in ε (Θ) := max F in ε (Θ) , G in (Θ)
with

F in ε (Θ) := sup u∈N in ε (Θ) min{F u (Θ) , L u Z δ(t, x, y)} G in (Θ) := max ℓ∈∆ + min β Y (x, y) ⊤ -p ⊤ β X (t, x) ℓ , Dδ(t, x, y) ⊤ β Z (x, y)ℓ .
and

N in ε (Θ) := u ∈ N ε (Θ) : |Dδ ⊤ σ Z (x, y, u)| ≤ ε , L u Z ϕ := ∂ t ϕ + Dϕ ⊤ µ Z (•, u) + 1 2 Tr σ Z σ ⊤ Z (•, u)D 2 ϕ µ Z := (µ ⊤ X , µ Y ) ⊤ , σ Z := [σ ⊤ X σ Y ] ⊤ , β Z := [β ⊤ X β Y ] ⊤
for ϕ smooth.

Terminal condition

In order to fully characterize the value function v, it remains to define appropriate boundary conditions. We first note that (x, v(T -, x)) ∈ O(T ) can be expressed as

v(T -, x) ≥ w(x) := inf {y ∈ R : (x, y) ∈ O(T )} . It follows that v(T -, •) should formally satisfy v(T -, •) ≥ w.
On the other hand, the fact that v satisfies (3.3) imposes a constraint on v and its gradient through N and G: N 0 (x, v(t, x), Dv(t, x)) = 0 or G(t, x, v(t, x), Dv(t, x)) ≥ 0. As usual it should propagate up to the boundary. In order to take care of this constraint, we follow [START_REF] Bouchard | Stochastic target problems with controlled loss[END_REF] and introduce the set valued map

N(x, y, p) := r ∈ R d : r = N u (x, y, p) for some u ∈ U , (3.6)
together with the signed distance function from its complement set N c to the origin:

R := dist (0, N c ) -dist (0, N) . (3.7) Then, 0 ∈ int (N(x, y, p)) iff R(x, y, p) > 0 . (3.8)
With these notations, the terminal condition formally reads:

min {v(T -, x) -w(x) , M (x, v(T, x), Dv(T, x))} = 0 , (3.9)
where

M (x, v(T, x), Dv(T, x)) := max{R(x, v(T -, x), Dv(T -, x)) , G(x, v(T -, x), Dv(T -, x))} .
However, the above expression does not incorporate the part of the state constraint that may be imposed on (t, x). In order to take care of this, we shall make the following assumption.

Standing Assumption 5: The function δ admits a locally C 1,2 extension on [0, T ] × R d+1 .

Under the above additional condition, we shall show that v(T -, •) indeed satisfies the constrained boundary condition

min v(T -, x) -w(x) , M in (T, x, v(T -, x), Dv(T -, x)) = 0 , (3.10) when (T, x, v(T -, x)) ∈ ∂D T := {T } ×   0<r<T 0<ε≤r ∂O(T -ε)   , (3.11)
where

M in := max{R in , G in } with R in defined as R with N in in place of N and N in (t, x, y, p) := {r ∈ R d : r = N u (x, y, p) and Dδ(t, x, y) ⊤ σ Z (x, y, u) = 0 for some u ∈ U } .
For later use, we set

intD T := ({T } × O(T )) \ ∂D T .
(3.12)

Main results

As in [START_REF] Bouchard | Stochastic target problems with controlled loss[END_REF], the operators F and F in are in general neither upper-semicontinuous nor lowersemicontinuous and need to be relaxed, i.e. we have to consider their semi-relaxed upperand lower-semicontinuous envelopes.

F * (Θ) := lim sup εց0,Θ ′ →Θ F ε (Θ ′ ), F * (Θ) := lim inf εց0,Θ ′ →Θ F ε (Θ ′ ), H * := max{F * , G}, H * := max{F * , G}
and we define similarly

F in * , F in * , H in * , H in * from F in , H in as well as R * , R * , R in * , R in * , M in * , M in * , M * , M * .
For ease of notations, we shall simply write H * ϕ for H * (•, ϕ, ∂ t ϕ, Dϕ, D 2 ϕ), and use similar notations for all the above defined operators. We shall also write w * and w * for the lower-and upper-semicontinuous envelopes of w.

Remark 3.1. (i) It follows from the convention sup ∅ = -∞ that F ε (Θ) = -∞ whenever N ε (x, y, q) = ∅. (ii) Since F ε is non-decreasing in ε ≥ 0, we have H * (Θ) = lim inf Θ ′ →Θ H 0 (Θ ′ ).
In particular, F * (Θ) > -∞ implies that there exists a neighborhood of Θ on which N 0 = ∅. (iii) The same reasoning holds for F in .

Since the value function v may not be continuous, we also introduce the corresponding semicontinuous envelopes:

v * (t, x) := lim inf (t ′ , x ′ ) → (t, x) (t ′ , x ′ ) ∈ D Y v(t ′ , x ′ ) , v * (t, x) := lim sup (t ′ , x ′ ) → (t, x) (t ′ , x ′ ) ∈ D Y v(t ′ , x ′ ) , (t, x) ∈ DY , (3.13)
where D Y is defined as in Standing Assumption 2 and DY denotes its closure. Before to state our main results, we need to introduce the following continuity assumption, compare with Assumption 2.1 in [START_REF] Bouchard | Stochastic target problems with controlled loss[END_REF], which will be used to prove the subsolution property.

Assumption 3.2. Let (t 0 , z 0 , q 0 ) be an element of D × R d .

(i) If N 0 = ∅ on a neighborhood B of (z 0 , q 0 ), then for every ε > 0 and u 0 ∈ N 0 (z 0 , q 0 ) there exists a locally Lipschitz map ν defined on a neighborhood of B ′ of (z 0 , q 0 ) such that |ν(z 0 , q 0 )u 0 | ≤ ε and ν ∈ N 0 on B ′ .

(ii) If N in 0 = ∅ on a neighborhood B of (t 0 , z 0 , q 0 ), then for every ε > 0 and u 0 ∈ N in 0 (t 0 , z 0 , q 0 ) there exists a locally Lipschitz map ν defined on a neighborhood of B ′ of (t 0 , z 0 , q 0 ) such that |ν(t 0 , z 0 , q 0 )u 0 | ≤ ε and ν ∈ N in 0 on B ′ . Under the above assumption, we shall show that v is a discontinuous viscosity solution of (3.3)-(3.5)-(3.9)-(3.10) in the following sense.

Theorem 3.3. v * is a viscosity super-solution on DY of H * ϕ ≥ 0 on DY ∩ ([0, T ) × R d ) min (ϕ -w * )1 {F * ϕ<∞,Gϕ<0} , M * ϕ ≥ 0 on DY ∩ ({T } × R d ) . (3.14) If Assumption 3.2 holds, then v * is a viscosity sub-solution on DY ∩ ([0, T ] × R d ) of        H * ϕ ≤ 0 if (•, ϕ) ∈ intD H in * ϕ ≤ 0 if (•, ϕ) ∈ ∂D on DY ∩ ([0, T ) × R d ) min{ϕ -w * , M * ϕ} ≤ 0 if (•, ϕ) ∈ intD T min{ϕ -w * , M in * ϕ} ≤ 0 if (•, ϕ) ∈ ∂D T on DY ∩ ({T } × R d ) . (3.15)
Note that, as usual, the state constraints appear only on the subsolution property, see e.g. [START_REF] Soner | Optimal control with state-space constraint I[END_REF] and [START_REF] Soner | Optimal control with state-space constraint II[END_REF]. The proof of this result is reported in Section 5.

Application in optimal book liquidation

In this section, we study an application of our general model to the pricing of a book liquidation contract under a VWAP (volume weighted average price) constraint.

For sake of simplicity, we shall restrict to the case where W is a one-dimensional Brownian motion although d = 2, which amounts to set part of the coefficients equal to 0. We shall also consider time-dependent coefficients, which corresponds to adding a component interpreted as time in the process X and can always be done by suitably choosing the drift parameter.

Moreover, the dynamics will be only controlled by a real valued non-decreasing process L.

We shall therefore only write X L and Y L , and now consider L as the set of continuous realvalued non-decreasing adapted processes L satisfying E L 2 T < ∞. Still, similar arguments as those used in Example 2.3 will lead to the introduction of an additional control in U, see Proposition 4.2 below.

Description of the model

The optimal book liquidation problem is the following. A financial agent asks a broker to sell on the market a total of K > 0 stocks on a time interval [0, T ]. The broker takes the engagement that he will obtain a mean selling price which corresponds to (at least) γ ∈ (0, 1) times the mean price of the market, i.e. the observed selling prices weighted by the volume of the corresponding transactions initiated by all the traders that are acting on the market on [0, T ]. Such contracts are referred to as VWAP guaranteed. The financial agent pays to the broker a premium y at time 0.

The cumulated number of stocks sold by the broker on the market since time 0 is described by a continuous real-valued non-decreasing adapted process L. Given L ∈ L, the dynamic of the broker's portfolio Y L is given by dY L (t) = X L,1 (t)dL t , Y L (0) = 0 where X L,1 represents the stock's selling price dynamics and is assumed to solve

dX L,1 (t) = X L,1 (t)µ(t, X L,1 (t))dt + X L,1 (t)σ(t, X L,1 (t))dW t -X L,1 (t)β(t, X L,1 (t))dL t where µ, σ, β : [0, T ] × R → R are continuous functions satisfying x ∈ [0, ∞) → x (µ(t, x), σ(t, x), β(t, x)) is uniformly Lipschitz, uniformly in t ∈ [0, T ] (4.1) and β : [0, T ] × R → R + is C 2 in space, uniformly in time.
Note that we allow the trading strategy of the broker to have an impact on the price dynamics if β = 0. For sake of simplicity, we model the intensity of all the transactions on the market by a deterministic non-negative integrable process ϑ, so that Θ(t) := t 0 ϑ(s)ds denotes the cumulated number of stocks sold on the market since time 0. Then, the mean price of selling orders in the market, denoted by X L,2 , has the dynamics

dX L,2 (t) = X L,1 (t)ϑ(t)dt , X L,2 (0) = 0 .
In order to accept the contract, a highly risk adverse broker should ask for an initial premium y such that

y + Y L (T )/K -γX L,2 (T )/Θ(T ) K ≥ 0 for some L ∈ L s.t. L T -L 0 = K,
i.e. which a.s. compensates the loss made if the mean selling price of the market is not matched. In practice, it is clear that the above problem does not make sense and needs to be relaxed. We shall therefore consider problems of the form Find the minimal y s.t., for some L ∈ L with L 0 = 0 ,

L T = K and E ℓ y + Y L (T )/K -γX L,2 (T )/Θ(T ) K ≥ p ,
for p ∈ R and ℓ : R → R non-decreasing.

Moreover, practitioners typically impose bounds on the cumulated number of sold stocks X L,3 := L -L 0 . We shall therefore restrict to strategies L ∈ L such that

X L,3 (s) ∈ [Λ(s), Λ(s)] for all s ≤ T ,
where Λ and Λ are assumed here to be C 1 deterministic functions such that Λ < Λ on [0, T ) .

Value function and problem reduction

In order to define the associated value function, we now extend the above dynamics to arbitrary initial conditions. Given L ∈ L, we set X L := (X L,1 , X L,2 , X L,3 ). We write Z L t,x,y = (X L t,x , Y L t,x,y ) the corresponding processes satisfying the initial condition Z L t,x,y (t) = (x, y). In the following, we restrict to initial conditions y ≥ 0 and x = (x 1 , x 2 , x 3 ) ∈ (0, ∞) × [0, ∞) 2 to be consistent with the fact that the above quantities should be non-negative and that the process X L,1 takes positive values if X L,1 (0) > 0. In order to simplify our analysis, we make the following assumption: Λ(T ) = Λ(T ) = K and DΛ, DΛ ∈ (0, M ] on [0, T ] for some M > 0.

(4.

2)

The first condition allows us to impose the constraint X 3,L (T ) = K via the simpler one X 3,L ∈ [Λ, Λ], while the assumption on the right-hand side will be used in the proof of Proposition 4.7 below in order to provide boundary conditions which will turn easier to handle.

In view of Remark 4.1 and the left-hand side of (4.2), the value function associated to the above stochastic target problem can then be written as

v(t, x, p) := inf{y ≥ 0 : ∃L ∈ L s.t. X 3,L t,x ∈ [Λ, Λ] and E Ψ(Z L t,x,y (T )) ≥ p} ,
with Ψ(x, y) = ℓ(yγx 2 ) and γ > 0.

In order to convert the above problem into a stochastic target problem in the form of the one studied in the previous sections, we use the key argument of [START_REF] Bouchard | Stochastic target problems with controlled loss[END_REF] as explained in Example 2.3.

In the following we set A := U × L, where U denotes the set of all progressively measurable process ν in L 2 ([0, T ] × Ω) taking values in R.

Proposition 4.2. Assume that ℓ has polynomial growth. Then, for all (t, x, p)

∈ [0, T ] × (0, ∞) × [0, ∞) 2 × R, v(t, x, p) := inf{y ≥ 0 : A t,x,y,p = ∅} ,
where A t,x,y,p denotes the set of processes (ν, L) ∈ A such that (Z L t,x,y , P ν t,p ) ∈ V on [t, T ] with

V := (x, y, p) ∈ (0, ∞) × [0, ∞) 2 × R 2 : x 3 ∈ [Λ, Λ] 1 [0,T ) + (x, y, p) ∈ (0, ∞) × [0, ∞) 2 × R 2 : x 3 = K and ℓ(y -γx 2 ) ≥ p 1 {T } , and
P ν t,p := p + • t ν s dW s .
Proof. If ℓ has polynomial growth, then it is clear that ℓ(Y L t,x,y (T ) -γX 2,L t,x (T )) ∈ L 2 for all L ∈ L such that X 3,L t,x (T ) ≤ K. It then suffices to reproduce the arguments used in Example 2.3 or in the proof of Proposition 3.1 in [START_REF] Bouchard | Stochastic target problems with controlled loss[END_REF]. ✷

In the following, we set

D Y := {(t, x, p) ∈ [0, T ) × (0, ∞) × [0, ∞) 2 × R : x 3 ∈ [Λ(t), Λ(t)]} ,
which is the natural domain on which our problem is stated. It satisfies the Standing Assumption 2 under the additional condition (4.4) below, see Proposition 4.5 below. In the following, v * and v * are defined as in (3.13) for D Y as above.

Additional assumptions and a-priori estimates

In the context of the above problem, the sets (N ε ) ε reads

N ε ϕ = u ∈ R : |uD p ϕ + x 1 σD x 1 ϕ| ≤ ε and
F * ϕ = F * ϕ = F 0 ϕ if D p ϕ = 0 , (4.3)
where

F 0 ϕ := -L X ϕ - (x 1 σ) 2 2 |D x 1 ϕ/D p ϕ| 2 D 2 p ϕ -2(D x 1 ϕ/D p ϕ)D 2 (x 1 ,p) ϕ with L X ϕ := ∂ t ϕ + x 1 µD x 1 ϕ + x 1 ϑD x 2 ϕ + 1 2 (x 1 σ) 2 D 2 x 1 ϕ .
In order to provide a PDE characterization in terms of the continuous operator F 0 rather than in terms of F * and F * , we need to ensure that D p ϕ = 0 for any test function for v * or v * . Moreover, the proof of a comparison principle will require a control of the ratio D x 1 ϕ/D p ϕ which appears in F 0 ϕ. In order to control the last term, we shall assume from now on that:

ℓ admits right-and left-derivatives, there exists ǫ > 0 s.t. ǫ ≤ D -ℓ , D + ℓ ≤ ǫ -1 , (4.4) and lim

r→∞ D + ℓ(r) = lim r→∞ D -ℓ(r) =: Dℓ(∞) , (4.5)
where D + and D -denote the right-and left-derivatives respectively.

The above conditions indeed induce the following controls on v, in which we use the notation e 1 := (1, 0, 0).

Proposition 4.3. For all (t, x, p) ∈ DY and h ∈ (-(x 1 ∧ 1), 1)

v(t, x, p) ≥ max{v(t, x, p -ǫ -1 |h|) + |h| , v(t, x + he 1 , p -C(x)|h|)} ,
where

C : [0, ∞) 3 → R + is a continuous map. (4.6)
Proof. a. We start with the first inequality v(t, x, p) ≥ v(t, x, pǫ -1 |h|) + |h|. Fix y > v(t, x, p). Then, there exists φ = (ν, L)

∈ A such that (Z L t,x,y , P ν t,p ) ∈ V on [t, T ]. Since, by (4.4), ℓ(r -|h|) ≥ ℓ(r) -ǫ -1 |h|, we have E ℓ Y L t,x,y (T ) -|h| -γX L,2 t,x (T ) ≥ p -ǫ -1 |h| .
Since Y L t,x,y -|h| = Y L t,x,y-|h| and X L does not depend on the initial value of Y L , this implies the required result by arbitrariness of y > v(t, x, p) and the definition of the value function v. b. Before to prove the second inequality, let us observe that standard computations based on Burkholder-Davis-Gundy's inequality, Gronwall's Lemma and the Lipschitz continuity assumption on our coefficients imply that there exists a continuous map

C : [0, ∞) 3 → R + such that, for all h ∈ [-1, 1] and L ∈ L such that |L| T ≤ K, E Ψ Z L t,x+he 1 ,y (T ) -Ψ Z L t,x,y (T ) ≤ C(x)|h| .
c. We now turn to the second inequality v(t, x, p)

≥ v(t, x+he 1 , p-C(x)|h|). Fix y > v(t, x, p) and consider φ = (ν, L) ∈ A such that (Z L t,x,y , P ν t,p ) ∈ V on [t, T ]. It follows from b. above that E Ψ Z L t,x+he 1 ,y (T ) ≥ E Ψ Z L t,x,y (T ) -C(x)|h| ≥ p -C(x)|h| .
As above, the required result then follows from the arbitrariness y > v(t, x, p). ✷

The immediate consequence of the above estimates is a control on D x 1 ϕ/D p ϕ for test functions of v * or v * . We now provide additional estimates that will be used later on to establish a comparison principle on the PDE associated to v. We first show that the conditions (4.2)-(4.4) allows us to deduce a classical growth condition on v. Proof. Define L := max{x 3 , Λ}, which belongs to L by (4.2). Then, it follows from (4.2) again that X L,3 t,x (T ) = K and X L,3 t,x ∈ [Λ, Λ] on [t, T ]. Moreover, standard estimates imply

E |X L t,x (T )| 2 1 2 ≤ η(1 + |x|)
for some η > 0 which does not depend on (t, x). In particular, for y > 0, the above inequality combined with our assumption (4.4) leads to

E ℓ Y L t,x,y (T ) -γX L,2 t,x (T ) ≥ E ℓ y -γX L,2 t,x (T ) ≥ ǫ (y -γη(1 + |x|)) + ℓ(0) .
By choosing y equal to the right-hand side of (4.9), we obtain E ℓ Y L t,x,y (T ) -γX L,2 t,x (T ) ≥ p. The required result follows from the definition of v. ✷

We finally provide suitable boundary conditions for v.

Proposition 4.6. Fix (t, x, p) ∈ DY . Then, v * (t, 0, x 2 , x 3 , p) = v * (t, 0, x 2 , x 3 , p) = Ψ -1 (0, x 2 , x 3 , p) , (4.10)

where Ψ -1 (x, p) := inf{y ≥ 0 : Ψ(x, y) ≥ p} .

Moreover, for all sequence

(t n , x n , p n ) n ⊂ D Y such that (t n , x n ) → (t, x) ∈ [0, T ] × [0, ∞) 3 , lim n→∞ v * (t n , x n , p n ) = lim n→∞ v * (t n , x n , p n ) = 0 if p n → -∞ , (4.11) lim n→∞ v * (t n , x n , p n )/p n = lim n→∞ v * (t n , x n , p n )/p n = 1/Dℓ(∞) if p n → ∞ . (4.12)
Proof. a. We start with the first assertion. Let (t n , x n , p n ) n be a sequence in D Y that converges to (t, 0, x 2 , x 3 , p) and fix y > Ψ -1 (x, p) with x = (0, x 2 , x 3 ). Then, the Lipschitz continuity of the coefficients implies that X 0 tn,xn (T ) → (0, x 2 , x 3 ) Pa.s. and in L q for any q ≥ 2. Since Ψ is Lipschitz continuous, it follows that lim n→∞ E Ψ(Z 0 tn,xn,y (T )) = Ψ(x, y) > p. Hence, E Ψ(Z 0 tn,xn,y ) ≥ p n for n large enough, and therefore v(t n , x n , p n ) ≤ y. The arbitrariness of y thus implies that lim sup n→∞ v(t n , x n , p n ) ≤ Ψ -1 (x, p). We next deduce from the Lipschitz continuity of the coefficients again that, for any L n ∈ L such that L n T ≤ K and (y n ) n≥ such that E Ψ(Z L n tn,xn,yn (T )) ≥ p n , we have Z L n tn,xn,yn (T ) → (0, x 2 , x 3 , y) Pa.s. and in L q for any q ≥ 2, whenever y n → y. Hence, lim n→∞ E Ψ(Z L n tn,xn,y (T )) = Ψ(x, y) ≥ p so that y ≥ Ψ -1 (x, p). Taking y n := v(t n , x n , p n ) + 1/n with (t n , x n , p n ) n such that v(t n , x n , p n ) → v * (t, 0, x 2 , x 3 , p) then shows that v * (t, 0, x 2 , x 3 , p) ≥ Ψ -1 (x, p). b. We now turn to the second assertion. It follows from the following easy observation. Fix (t, x) ∈ [0, T )×(0, ∞)×[0, ∞) 2 such that x 3 ∈ [Λ(t), Λ(t)]. Then, for L defined by L = Λ+x 3 -Λ(t), one obtains X L,3 t,x ∈ [Λ, Λ] on [t, T ], recall (4.2), and p(t, x) := E Ψ Z L t,x,0 (T ) > -∞. It follows that v(t, x, p) = 0 for p ≤ p(t, x), where the function p is clearly locally bounded. c. We finally prove the last assertion. Since p n → ∞ and, for any strategy L n ∈ L such that L n T ≤ K, Z L n tn,xn,0 (T ) is uniformly bounded in any L q , q ≥ 2, one has y n → ∞ whenever E Ψ(Z L n tn,xn,yn (T )) ≥ p n for all n. Using (4.5), one deduces that, for all ε > 0, ∃ r ε ∈ R such that

Ψ(Z L n tn,xn,yn (T )) ≤ ℓ(r ε )+(Dℓ(∞)+ε) y n + Y L n tn,xn,0 (T ) -γX 2,L n tn,xn (
T )r ε for n large P-a.s.

It follows that 1 ≤ lim sup n→∞ E Ψ(Z L n tn,xn,yn (T )) /p n ≤ (Dℓ(∞) + ε) lim sup n→∞ y n /p n .
This implies that lim inf n→∞ v * (t n , x n , p n )/p n ≥ 1/ (Dℓ(∞) + ε). Choosing ε arbitrarily small leads to the required result. On the other hand, for y n := p n (Dℓ(∞)ε) -1 with ε ∈ (0, Dℓ(∞)), we have, by similar arguments,

Ψ(Z 0 tn,xn,yn (T ))/p n → Dℓ(∞)(Dℓ(∞) -ε) -1 > 1 so that lim inf n→∞ E Ψ(Z 0 tn,xn,yn (T )) /p n > 1 and therefore v(t n , x n , p n ) ≤ p n (Dℓ(∞)-ε) -1 for n large enough. This implies that lim sup n→∞ v * (t n , x n , p n )/p n ≤ (Dℓ(∞) -ε) -1
, which yields the required result by arbitrariness of ε > 0. ✷

PDE characterization

We can now provide the main results of this section. We first report the PDE characterization of v.

Proposition 4.7. The functions v * is a viscosity supersolution on

D Y of max F 0 ϕ , x 1 + x 1 βD x 1 ϕ -D x 3 ϕ = 0 . (4.13) The function v * is a subsolution on D Y of min ϕ , max F 0 ϕ , x 1 + x 1 βD x 1 ϕ -D x 3 ϕ = 0 if Λ < x 3 < Λ min ϕ , x 1 + βD x 1 ϕ -D x 3 ϕ = 0 if Λ = x 3 min {ϕ , F 0 ϕ} = 0 if x 3 = Λ . (4.14) Moreover, v * (T, x, p) = v * (T, x, p) = Ψ -1 (x, p) for all (x, p) ∈ [0, ∞) 2 × {K} × R . (4.15)
Proof. a. We first discuss the PDE characterization. In view of Theorem 3.3, we already know that v * is a supersolution on

D Y of max F * ϕ , x 1 + x 1 βD x 1 ϕ -D x 3 ϕ = 0 and that v * is a subsolution on D Y ∩ {v * > 0} of max F * ϕ , x 1 + x 1 βD x 1 ϕ -D x 3 ϕ = 0 if Λ < x 3 < Λ max min{F * ϕ , -DΛ} , min{x 1 + x 1 βD x 1 ϕ -D x 3 ϕ , 1} = 0 if Λ = x 3 max min{F * ϕ , DΛ} , min{x 1 + x 1 βD x 1 ϕ -D x 3 ϕ , -1} = 0 if x 3 = Λ .
Note that the boundary x 2 = 0 does not play any role here since the process X 2,L is nondecreasing. Since, by Proposition 4.3, v is strictly increasing in its p variable, any test function ϕ such that (t 0 , x 0 , p 0 ) achieves a local maximum (resp. minimum) of v *ϕ (resp. v *ϕ) must then satisfy D p ϕ(t 0 , x 0 , p 0 ) > 0. It follows that F * and F * can be replaced by F 0 , see (4.3). In order to simplify the subsolution property for t < T , it then suffices to use the fact that > 0 and DΛ > 0 by assumption (4.2). b. It remains to prove the boundary condition at T . b.1. We first discuss the supersolution property at T . Let (t n , x n , p n ) n≥1 be a sequence in D Y , with t n < T for all n, such that (t n , x n , p n ) → (T, x 0 , p 0 ) ∈ DY and

v(t n , x n , p n ) → v * (T, x 0 , p 0 ). Set y n := v(t n , x n , p n ) + 1/n and let L n be such that E [Ψ(Z n (T ))] ≥ p n where Z n = (X n , Y n ) := (X L n tn,xn , Y L n tn,xn,yn ). Since X n,2 (T ) = K, we have L n (T ) -L n (t n ) = K -x 3 n . Since L n is non-decreasing, this shows that sup tn≤t≤T L n (t) -L n (t n ) → 0 in L ∞ . This implies that Z n (T ) → z 0 := (x 0 , y 0 ) in any L q , q ≥ 2. It then follows from the dominated convergence theorem that Ψ(z 0 ) -p 0 = lim n→∞ E [Ψ(Z n (T ))] -p n ≥ 0 .
This shows that v * (T, x 0 , p 0 ) ≥ Ψ -1 (x 0 , p 0 ). b.2. We finally prove the subsolution property. Let (T, x 0 , p 0 ) ∈ DY and ϕ be a smooth function such that (T, x 0 , p 0 ) achieves a strict local maximum of v *ϕ such that (v *ϕ)(T, x 0 , p 0 ) = 0 and ϕ(T, x 0 , p 0 ) > Ψ -1 (x 0 , p 0 ) . (4.16)

Let (t n , x n , p n ) n≥1 be a sequence in D Y , with t n < T for all n, which converges to (T, x 0 , p 0 ) and such that v(t n , x n , p n ) → v * (T, x 0 , p 0 ). Set

y n := v(t n , x n , p n ) -1/n.
|α n t | ≤ sup n≥1 sup t∈[tn,T ] DΛ(t) ∨ DΛ(t) ≤ M . (4.18)
Since ϕ is smooth, we can also define the control

ν n := -σ(•, X L n ,1 tn,xn )(D x 1 ϕ/D p ϕ)(•, X L n ,1
tn,xn ), recall from the above discussion that we must have D p ϕ > 0 on a neighborhood of (T, x 0 , p 0 ). For ease of notations, we write Z n = (X n , Y n ) := (X L n tn,xn , Y L n tn,xn,yn ) and P n := P ν n tn,pn . Let φ be defined by φ(t, x, p) := ϕ(t, x, p) + √ Tt + ι -√ ι for some ι > 0. It follows from the identity v(T, •) = Ψ -1 and (4.16) that max

({T }× Bε(x0,p0))∪([T -ε,T ]×∂Bε(x 0 ,p 0 )) (v * -φ) := -ζ < 0 ,
for ε > 0 small enough. Moreover, for ε, ι > 0 small enough, one has

inf |α|≤M x 1 α + F 0 φ + β(x 1 )αD x 1 φ -αD x 3 φ ≥ 0 on [T -ε, T ] × ∂B ε (x 0 , p 0 ) , (4.19)
since ∂ t φ → -∞ as t → T and ι → 0. We next define the stopping times

θ o n := inf {s ≥ t n : (s, X n (s), P n (s)) / ∈ [T -ε, T ] × B ε (x 0 , p 0 )} , θ n := inf {s ≥ t n : |Y n (s) -ϕ(s, X n (s), P n (s))| ≥ ε} ∧ θ o n .
Using (4.17)-(4.18)-(4.19), the same arguments as in the proof of Proposition 5.2 below show that

Y n (θ n ) -v (θ n , X n (θ n ), P n (θ n )) > 0 for n large enough. Recalling that y n = v(t n , x n , p n ) -n -1 < v(t n , x n , p n ),
this is in contradiction with (GDP2) of Corollary 2.4. ✷

Comparison principle and uniqueness

In order to complete the characterization of Proposition 4.7, it remains to provide a comparison theorem for (4.13)-(

Note that the term D x 1 ϕ/D p ϕ which appears in the definition of F 0 can be shown to be bounded because viscosity super-and subsolution of (4. In what follows, we shall therefore reduce to a bounded domain by adding the following condition:

∃ x1 > 0 s.t. µ(•, x1 ) = σ(•, x1 ) = 0 . (4.20)
This implies that x1 is an absorbing point for X L,1 . In particular, it remains bounded as well as X L,2 which is bounded by Θ(T )x 1 . In particular, we can then restrict to the bounded domain

DY := D Y ∩ [0, T ) × (0, 2x 1 ) × [0, 2Θ(T )x 1 ) × [0, K] × R .
For x 1 > x1 , the value function v can be easily computed explicitly, since the problem becomes deterministic, and is continuous:

v(t, x, p) = Ψ -1 x 1 , γ x 2 + T t x 1 ϑ(s)ds -x 1 (K -x 3 ), x 3 , p for x 1 > x1 . (4.21)
Note that it is not a real limitation for practical applications, since x1 can be arbitrary large.

Proposition 4.8. Assume that (4.20) holds. Let U (resp. V ) be a non-negative lowersemicontinuous supersolution of (4.13) (resp. upper-semicontinuous subsolution of (4.14)) on DY , such that U and V are continuous in x 3 . Assume that

U (t, x, p) ≥ V (t, x, p) if t = T or x 1 ∈ {0, 2x 1 }, (4.22)
and that there exists c + > 0 and c -∈ R such that lim sup

(t ′ ,x ′ ,p ′ )→(t,x,∞) V (t ′ , x ′ , p ′ )/p ′ ≤ c + ≤ lim inf (t ′ ,y ′ ,p ′ )→(t,y,∞) U (t ′ , y ′ , p ′ )/p ′ , (4.23)
lim sup Proof. Since ℓ is non-decreasing, so is v, in the p-variable. It thus suffices to show that lim sup |h|→0 v(t, x, p + |h|) ≤ v(t, x, p). To see this fix y > v(t, x, p) and L ∈ L such that Z L t,x,y ∈ V and E Ψ(Z L t,x,y (T )) > p, which is possible since ℓ is strictly increasing and y > v(t, x, p). It follows that y ≥ v(t, x, p + |h|) for h small enough. Sending |h| → 0 and then y → v(t, x, p) leads to the required result. We now turn to the continuity with respect to x 3 . Fix h ∈ R such that |h| ≤ min{x 3 -Λ(t), Λ(t)-x 3 }. Denote e 3 := (0, 0, 1), and let L ∈ L and y ≥ 0 be such that

(t ′ ,x ′ ,p ′ )→(t,x,-∞) V (t ′ , x ′ , p ′ ) ≤ c -≤ lim inf (t ′ ,y ′ ,p ′ )→(t,y,-∞) U (t ′ , y ′ , p ′ ) (4.24) ∀ (t, x) ∈ [0, T ) × [0, ∞) 3 . If
Z L t,x,y ∈ V . Then, L T -L t ≤ K and sup t≤s≤T |Z L t,x,0 | ≤ c for some c > 0. Set L h := 1 [t,T ] (L -L t + x 3 + h) ∧ Λ ∨ Λ. Then, Z L h t,x+he 3 ,y ∈ V and sup t≤s≤T |Z L t,x,0 -Z L h t,x+he 3 ,0 | ≤ c|h| where c > 0 does not de- pend on (t, x, p). This implies that v(t, x + he 3 , p) ≤ v(t, x, p) + c|h|. Similarly, v(t, x, p) ≤ v(t, x + he 3 , p) + c|h|. ✷
We conclude with the proof of Proposition 4.8.

Proof of Proposition 4.8. We assume that U is a viscosity supersolution of (4.7) which is continuous in p. The case where V is a viscosity subsolution of (4.8) which is continuous in p is treated similarly. As usual, we argue by contradiction and assume that there exists (t 0 , x 0 , p 0 ) ∈ DY such that (V -U )(t 0 , x 0 , p 0 ) =: 4η 0 > 0 .

Given κ > 0 and ζ > 1, we define Ũζ and Ṽ by Ũζ (t, x, p) := e κ(t+x 3 ) U (t, x, pζ) and Ṽ (t, x, p) := e κ(t+x 3 ) V (t, x, p). Since U is continuous in its p-variable, one has sup

DY ( Ṽ -Ũζ ) =: 2η ≥ 2η 0 > 0 , (4.25)
for κ > 0 and ζ > 1 small enough.

1. Note that η < ∞ and that 2η = sup

DY ( Ṽ -Ũζ ) = ( Ṽ -Ũζ )( tζ , xζ , pζ ) (4.26)
for some ẑζ := ( tζ , xζ , pζ ) in the closure of DY . Indeed, if (t k , x k , p k ) k≥1 is a maximizing sequence, then (t k , x k ) k≥1 is bounded and therefore converges along a subsequence. If p k → -∞, we obtain a contradiction by appealing to (4.24). If

p k → ∞, then lim sup k→∞ V (t k , x k , p k )/p k ≤ c + < c + ζ ≤ ζ lim sup k→∞ U (t k , x k , p k ζ)/(p k ζ)
, which also leads to a contradiction. The fact that the supremum is achieved then follows from the upper-semicontinuity of Ṽ -Ũζ . From now on, we assume that

x3 ζ = Λ( tζ ) (4.27)
for all ζ > 1 small enough. The case where x3

ζ = Λ( tζ ) (resp. x3 ζ ∈ (Λ( tζ ), Λ( tζ ))
) can be treated similarly by replacing |x 3y 3 -δ| 2 in the definition of Λ n below by |y 3x 3 -δ| 2 (resp. 0).

2.

For n ≥ 1, we now set Θ n : (t, x, y, p, q) → Ṽ (t, x, p) -Ũζ (t, y, q) -Λ n (t, x, y, p, q) where Λ n (t, x, y, p, q

) := n 2 |p -q| 2 + |x 1 -y 1 | 2 + |x 3 -y 3 -δ| 2 + 1 2 |t -tζ | 2 + |x 3 -x3 ζ | 4
for some δ > 0. Note that, by continuity of V in x 3 and (4.26),

lim δ→0 inf n≥1 Θ n ( tζ , xζ + δe 3 , xζ , pζ , pζ ) = 2η
, with e 3 := (0, 0, 1). It follows that sup

D2 Y Θ n ≥ η for all n ≥ 1 , (4.28)
for δ > 0 small enough, where

D2 Y := {(t, x, y, p, q) ∈ [0, T ] × [0, ∞) 6 × R 2 : ((t, x, p), (t, y, q)) ∈ DY × DY , x 2 = y 2 } .
Moreover, the same arguments as in step 1. above show that there exists zn := (t n , x n , y n , p n , q n ) in the closure of DY satisfying

Θ n (z n ) = sup D2 Y Θ n ≥ η for all n ≥ 1 . ( 4
| 2 + n|p n -q n | 2 + n|x 3 n -y 3 n -δ| 2 = 0 lim n→∞ ( Ṽ -Ũζ )(z n ) = ( Ṽ -Ũζ )(z ζ,δ ) and lim δ→0 (t ζ,δ , x 3 ζ,δ ) = ( tζ , x3 ζ ) , lim δ→0 ( Ṽ -Ũζ )(z ζ,δ ) = ( Ṽ -Ũζ )(ẑ ζ ) = 2η > η 0 . (4.31)
Note that, combined with (4.27), this implies that

x 1 ζ,δ = y 1 ζ,δ , x 2 ζ,δ = y 2 ζ,δ , p ζ,δ = q ζ,δ and Λ(t ζ,δ ) < y 3 ζ,δ + δ = x 3 ζ,δ < Λ(t ζ,δ ) , (4.32)
for δ > 0 small enough. c. We can also not have x 1

ζ,δ ∈ {0, 2x 1 } for all δ > 0 small enough. To see this assume the contrary and note that the fact that U is a supersolution of (4.7) implies that it is non-decreasing in p. First assume that q ζ,δ ≥ 0 for all ζ > 1 small enough. Since ζ > 1, it then follows from the upper-semicontinuity of V, -U and from (4.32) that 4. Now observe that, by assumption, Ũζ and Ṽ are super and subsolutions on DY of max κϕ + F0 ϕ , κϕ + x 1 + x 1 βD x 1 ϕ -D x 3 ϕ ≥ 0 (4.33) and max κϕ + F0 ϕ , κϕ

V (t ζ,δ , x ζ,δ , p ζ,δ ) -U (t ζ,δ , y ζ,δ , ζq ζ,δ ) ≤ (V -U )(t ζ,δ , x ζ,δ , p ζ,δ ) + O(δ). Recalling
+ x 1 + x 1 βD x 1 ϕ -D x 3 ϕ ≤ 0 if Λ < x 3 < Λ and Ṽ > 0 (4.34)
respectively, with

F0 ϕ := -∂ t ϕ -x 1 µD x 1 ϕ -x 1 ϑ(t)D x 2 ϕ - (x 1 σ) 2 2 D 2 x 1 ϕ + |D x 1 ϕ/D p ϕ| 2 D 2 p ϕ -2(D x 1 ϕ/D p ϕ)D 2 (x 1 ,p) ϕ ,
and that Ũζ is a viscosity supersolution of

min ζ -1 D p ϕ -ǫ , D x 1 ϕ -C(x)ζ -1 D p ϕ , -D x 1 ϕ + C(x)ζ -1 D p ϕ = 0 . (4.35)
Let P+ Ṽ and P-Ũζ denote the super-and subjets of Ṽ and Ũζ , with (t, x 2 , x 3 ) taken as a first order term. It then follows from Ishii's Lemma, see e.g. [START_REF] Crandall | User's guide to viscosity solutions of second order partial differential equations[END_REF], that we can find 2-dimensional symmetric matrices (X n , Y n ) ∈ S 2 such that

(D t Λ n , (D x Λ n , D p Λ n ), X n )(t n , x n , y n , p n , q n ) ∈ P+ Ṽ (t n , x n , p n ) (-D t Λ n , -(D y Λ n , D q Λ n ), Y n )(t n , x n , y n , p n , q n ) ∈ P- Ũζ (t n , y n , q n ),
where the super-and subjets are defined with (t, x 2 , x 3 ) viewed as a first order term, that satisfy

X n 0 0 -Y n ≤ 3n I -I -I I (4.36)
where I denotes the 2-dimensional identity matrix.

We now study two cases: Case 1. We first assume that

κ Ũζ (t n , y n , q n ) + y 1 n -y 1 n β(t n , y 1 n )D y 1 Λ n (z n ) + D y 3 Λ n (z n ) ≥ 0 ,
along a subsequence. Then, by (4.29), (4.30), (4.32), (4.34), step 3. and the Lipschitz of x → xβ(t, x),

-κη ≥ κ( Ũζ (t n , y n , q n ) -Ṽ (t n , x n , p n )) ≥ -(y 1 n -x 1 n ) + y 1 n β(t n , y 1 n )D y 1 Λ n (z n ) + x 1 n β(t n , x 1 n )D x 1 Λ n (z n ) -D y 3 Λ n (z n ) -D x 3 Λ n (z n ) ≥ O(n|x 1 n -y 1 n | 2 + |x 1 n -y 1 n |) + O(|x 3 n -x3 ζ |) .
It then follows from (4.30)-(4.31) that -κη ≥ 0 which leads to a contradiction since η > 0. Case 2. We now assume that κ Ũζ (t n , y n , q n ) + y

1 n -y 1 n β(t n , y 1 n )D y 1 Λ n (z n ) + D y 3 Λ n (z n ) < 0 ,
along a subsequence. Then, using (4.30), (4.32), (4.33), (4.34), step 3. again, we deduce that

0 ≤ κ Ũζ + D t Λ n + y 1 n µ(t n , y 1 n )D y 1 Λ n + y 1 n ϑ(t n )D y 2 Λ n (z n ) - (y 1 n σ(t n , y 1 n )) 2 2 Y 11 n + |A n | 2 Y 22 n -2A n Y 12 n (4.37) and 0 ≥ κ Ṽ -D t Λ n -x 1 n µ(t n , x 1 n )D x 1 Λ n -x 1 n ϑ(t n )D x 2 Λ n (z n ) - (x 1 n σ(t n , x 1 n )) 2 2 X 11 n + |A n | 2 X 22 n -2A n X 12 n , (4.38)
where

A n := -D y 1 Λ n (z n ) -D q Λ n (z n ) = D x 1 Λ n (z n ) D p Λ n (z n ) .
Moreover, the continuity of C, recall (4.6), and the viscosity supersolution properties of Ũζ in (4.35), together with (4.30), imply that

|A n | ≤ ζ -1 C(y ζ,δ ) + 1 (4.39)
for n large. We now use (4.36), the Lipschitz continuity of the coefficients and (4.39) to obtain

κ( Ṽ -Ũζ )(z n ) ≤ n|x 1 n µ(t n , x 1 n ) -y 1 n µ(t n , y 1 n )||x 1 n -y 1 n | + O(|ẑ ζ -(t n , x n , p n )|) + O (1 + |A n | 2 )|x 1 n σ(t n , x 1 n ) -y 1 n σ(t n , y 1 n )| 2 ≤ O(n|x 1 n -y 1 n | 2 ) + O(|ẑ ζ -(t n , x n , p n )|) .
Recalling (4.30)-(4.31) and sending n → ∞ and then δ → 0 then leads to a contradiction since η > 0. ✷

Proof of the viscosity property in the abstract model

We now provide the proof of Theorem 3.3. It is divided in several subsections. We first consider the case (t 0 , x 0 ) ∈ DY with t 0 < T . The proof follows from almost exactly the same arguments as in [START_REF] Bouchard | Stochastic target problems with controlled loss[END_REF]. The only difference comes from the part of the control with bounded variations, however it is easily handled. We provide it for completeness.

Proposition 5.1. Let (t 0 , x 0 ) ∈ DY , with t 0 < T , and let ϕ be a smooth function such that

(strict) min [0,T ]×R d (v * -ϕ) = (v * -ϕ)(t 0 , x 0 ) = 0 . (5.1)
Then, H * ϕ(t 0 , x 0 ) ≥ 0.

Proof. We assume to the contrary that

H * ϕ(t 0 , x 0 ) ≤ -2η (5.2)
for some η > 0, and work towards a contradiction. It follows from (5.2) and the definition of H * that we may find ε > 0 such that

max ξ∈∆ + β Y (x, y) ⊤ -Dϕ(t, x) ⊤ β X (x) ξ ≤ -η µ Y (x, y, u) -L u ϕ(t, x) ≤ -η ∀ u ∈ N ε (x, y, Dϕ(t, x)) (5.3) ∀ (t, x, y) ∈ [0, T ] × R d+1 s.t. (t, x) ∈ B ε (t 0 , x 0 ) ∩ DY , |y -ϕ(t, x)| ≤ ε .
For later use, observe that, by (5.1) and the definition of ϕ,

ζ := min ∂pBε(t 0 ,x 0 ) (v * -ϕ) > 0 , (5.4)
where ∂ p B ε (t 0 , x 0 ) denotes the parabolic boundary of B ε (t 0 , x 0 ). Let (t n , x n ) n≥1 be a sequence in D Y which converges to (t 0 , x 0 ) and such that v(t n , x n ) → v * (t 0 , x 0 ). Set y n = v(t n , x n ) + n -1 and observe that

γ n := y n -ϕ(t n , x n ) → 0 . (5.5)
For each n ≥ 1, we have y n > v(t n , x n ). It thus follows from (GDP1) of Corollary 2.4, that there exists some

φ n = (ν n , L n ) ∈ A such that Y n (t ∧ θ n ) ≥ v(t ∧ θ n , X n (t ∧ θ n )) for t ≥ t n , (5.6)
where

Z n := (X n , Y n ) := X ν n tn,xn , Y ν n tn,xn,yn and θ n := θ o n ∧ θ 1 n , with θ o n := s ≥ t n : (s, X φ n tn,xn (s)) / ∈ B ε (t 0 , x 0 ) , θ 1 n := s ≥ t n : |Y φ n tn,xn,yn (s) -ϕ(s, X φ n tn,xn (s))| ≥ ε . Let us define A n := s ∈ [t n , θ n ] : µ Y (Z n (s), ν n s ) -L ν n s ϕ (s, X n (s)) > -η , (5.7)
and observe that (5.3) implies that the process

ψ n s := N ν n (Z n s , Dϕ(s, X n s )) satisfies |ψ n s | > ε for s ∈ A n , (5.8) recall (3.2)
. Since L n ∈ L is continuous, so is the path of Z n . Using (5.6), the definition of ζ in (5.4) and the definition of θ n , thus leads to

Y n (t ∧ θ n ) ≥ ϕ (t ∧ θ n , X n (t ∧ θ n )) + ζ1 {θ o n =θn} + ε1 {θ o n >θn} 1 {t=θn} ≥ ϕ (t ∧ θ n , X n (t ∧ θ n )) + (ζ ∧ ε) 1 {t=θn} , t ≥ t n .
Since ϕ is smooth, it then follows from Itô's Lemma, (5.3), (5.5) and the definition of ψ n that

-(ζ ∧ ε) 1 {t<θn} ≤ K n t + t∧θn tn b n s 1 A c n (s)ds + t∧θn tn β ⊤ Y (Z n (s)) -Dϕ(s, X n (s)) ⊤ β X (X n (s)) dL n s ≤ K n t , (5.9)
where

K n t := γ n -(ζ ∧ ε) + t∧θn tn b n s 1 An (s)ds + t∧θn tn ψ n s dW s , and
b n s := µ Y (Z n (s), ν n s ) -L ν n s ϕ (s, X n (s)) .
Let M n be the exponential local martingale defined by M n tn = 1 and, for s ≥ t n ,

dM n s = -M n s b n s |ψ n s | -2 (ψ n s ) ⊤ 1 An (s)dW s ,
which is well defined by (5.8), the Lipschitz continuity of the coefficients and our definition of the set of admissible controls U. By Itô's formula and (5.9), we see that M n K n is a local martingale which is bounded from below by the submartingale -(ζ ∧ ε) M n . Then, M n K n is a supermartingale, and it follows from (5.9) that (5.5), which leads to a contradiction. ✷

0 = E -(ζ ∧ ε) 1 {θn<θn} ≤ E M n θn K n θn ≤ γ n -(ζ ∧ ε) < 0 , for n large enough, recall

Subsolution property on [0, T )

We first consider the case where (t 0 , x 0 , v * (t 0 , x 0 )) ∈ intD. The first part of the proof is similar to those provided in [START_REF] Bouchard | Stochastic target problems with controlled loss[END_REF]. The novelty comes from the second part where we play with the part of the control with bounded variations to obtain a contradiction.

Proposition 5.2. Let (t 0 , x 0 ) ∈ DY , with t 0 < T , and ϕ be a smooth function such that

(strict) max DY (v * -ϕ) = (v * -ϕ)(t 0 , x 0 ) = 0 . (5.10)
Assume that (t 0 , x 0 , v * (t 0 , x 0 )) ∈ intD. Then, H * ϕ(t 0 , x 0 ) ≤ 0.

Proof. We assume to the contrary that (t 0 , x 0 , v * (t 0 , x 0 )) ∈ intD and H * ϕ(t 0 , x 0 ) ≥ 2η (5.11) for some η > 0, and work towards a contradiction. For later use note that (5.11) implies that, for ε > 0 small enough,

(t, x, y) ∈ [0, T ] × R d+1 s.t. (t, x) ∈ B ε (t 0 , x 0 ), |y -ϕ(t, x)| ≤ ε ⊂ intD .
(5.12) Also observe that, by (5.10) and the definition of ϕ,

-ζ := max ∂pBε(t 0 ,x 0 ) (v * -ϕ) < 0 . (5.13)
Moreover, we can find a sequence (t n , x n ) n≥1 in D Y which converges to (t 0 , x 0 ) and such that v(t n , x n ) → v * (t 0 , x 0 ). Set y n = v(t n , x n )n -1 and observe that

γ n := y n -ϕ(t n , x n ) → 0 . (5.14)
We now consider two cases. First case. We first assume that

F * ϕ(t 0 , x 0 ) ≥ 2η . (5.15)
Then it follows from Assumption 3.2 and Remark 3.1 that may find ε > 0 such that

µ Y (•, ν) -L ν ϕ > η (5.16) ∀ (t, x, y) ∈ [0, T ] × R d+1 s.t. (t, x) ∈ B ε (t 0 , x 0 ), |y -ϕ(t, x)| ≤ ε ,
where ν is a locally Lipschitz map satisfying ν(x, y, Dϕ(t, x)) ∈ N 0 (x, y, Dϕ(t, x)) if (t, x) ∈ B ε (t 0 , x 0 ) and |yϕ(t, x)| ≤ ε . (5.17)

We now fix n large enough so that (t n , x n ) ∈ B ε (t 0 , x 0 ) and let Z n := (X n , Y n ) denote the solution of (2.1) associated to the Markovian control φn and the initial condition Z n (t n ) = (x n , y n ), where φn = (ν n , Ln ) := (ν(•, X n , Y n ), 0) .

We next define the stopping times

θ o n := inf {s ≥ t n : (s, X n (s)) / ∈ B ε (t 0 , x 0 )} , θ n := inf {s ≥ t n : |Y n (s) -ϕ(s, X n (s))| ≥ ε} ∧ θ o n .
where γ(t, x, y) ⊤ := (0, β X (x), β ⊤ Y (x, y)) l. Given u ∈ U , we thus deduce from Theorem 4.8 of [START_REF] Dupuis | SDEs with oblique reflection on nonsmooth domains[END_REF] and the assumption made on our coefficients, that there exists an adapted process Z n = (X n , Y n ) and a continuous real-valued adapted non-decreasing process M n satisfying X n (s) = x n + s tn µ X (X n (r), u)dr + Observe that Z n coincides with the solution of (2.1) for the control (u, lM n ). In view of (5.14) and (5.21), we have Y n (θ n )ϕ(θ n , X n (θ n )) ≥ -2|γ n | > -ε for n large enough. Following the arguments after (5.18) above then leads to the required contradiction. ✷

We now turn to the case where (t 0 , x 0 , v * (t 0 , x 0 )) ∈ ∂D. Assume that (t 0 , x 0 , v * (t 0 , x 0 )) ∈ ∂D. Then, H in * ϕ(t 0 , x 0 ) ≤ 0.

Proof. The fact that (t 0 , x 0 , v * (t 0 , x 0 )) ∈ ∂D and H in * ϕ(t 0 , x 0 ) ≥ 2η (5.23)

for some η > 0, leads to a contradiction to (GDP2) of Corollary 2.4 follows exactly from the same arguments as in the proof of Proposition 5.2. We therefore only sketch the case where F in * ϕ(t 0 , x 0 ) ≥ 2η. In this case, it follows from the definition of F in * and (ii) of Assumption 3.2, see also Remark 3.1, that we may find ε > 0 such that min{µ Y (•, ν) -L ν ϕ , L ν Z δ} > η (5.24) ∀ (t, x, y) ∈ [0, T ] × R d+1 s.t. (t, x) ∈ B ε (t 0 , x 0 ), |yϕ(t, x)| ≤ ε , where ν is a locally Lipschitz map satisfying ν(t, x, y) ∈ N in 0 (t, x, y, Dϕ(t, x)) for (t, x) ∈ B ε (t 0 , x 0 ) and |yϕ(t, x)| ≤ ε .

(5.25) Let (t n , x n ) n≥1 be a sequence in D Y which converges to (t 0 , x 0 ) and such that v(t n , x n ) → v * (t 0 , x 0 ). Set y n = v(t n , x n )n -1 and observe that Recalling that y n = v(t n , x n )n -1 < v(t n , x n ), this is in contradiction with (GDP2) of Corollary 2.4. ✷

Viscosity solution property at T

In this part we follow standard arguments which consist in propagating the boundary condition backward on some small time [Tε, T ] so as to be in position to repeat the arguments used to derive the viscosity solution property on [0, T ). The arguments being standard, see e.g. [START_REF] Bouchard | Stochastic target problems with controlled loss[END_REF] or [START_REF] Soner | Dynamic programming for stochastic target problems and geometric flows[END_REF], we only sketch them.

We begin with the supersolution property. Then, M * ϕ(T, x 0 ) ≥ 0. If moreover F * ϕ(T, x 0 ) < ∞ and Gϕ(T, x 0 ) < 0, then ϕ(T, x 0 )w * (x 0 ) ≥ 0.

Proof. The fact that M * ϕ(T, x 0 ) ≥ 0 is deduced from Proposition 5.1 and the uppersemicontinuity of M * by standard arguments, see e.g. the proof of Lemma 5.2 in [START_REF] Soner | Stochastic target problems, dynamic programming and viscosity solutions[END_REF]. We now prove the second assertion. Assume that F * ϕ(T, x 0 ) < ∞ , Gϕ(T, x 0 ) < 0 and ϕ(T, x 0 ) = v * (T, x 0 ) < w * (x 0 ) , and let us work towards a contradiction. Since v(T, .) = w by the definition of the problem, there is a constant η > 0 such that ϕv(T, •) ≤ ϕw * ≤ -η on B ε (x 0 ) for some ε > 0. Since x 0 is a strict minimizer, 2ζ := min x∈∂Bε(x 0 ) (v * (T, x)ϕ(T, x)) > 0 and it follows that there exists r > 0 such that v(t, By following the arguments in the proof of Proposition 5.1, the latter inequalities lead to a contradiction of (GDP1) of Corollary 2.4.

✷

We now turn to the subsolution property. As in the previous section, we first consider the case where (t 0 , x 0 , v * (t 0 , x 0 )) ∈ intD T .

Proposition 5.5. Let (t 0 , x 0 ) ∈ DY , with t 0 = T , and ϕ be a smooth function such that (strict) max DY (v *ϕ) = (v *ϕ)(t 0 , x 0 ) = 0 .

(5.29)

Assume that (T, x 0 , v * (T, x 0 )) ∈ intD T . Then, min {ϕ(T, x 0 )w * (x 0 ) , M * ϕ(T, x 0 )} ≤ 0.

Remark 4 . 1 .

 41 Up to an obvious change of variables, the initial premium y can be incorporated in the initial condition Y (0) of Y . Similarly, the constant γK/Θ(T ) can be simply written γ > 0 up to a change of variable. It follows that the above problem could be alternatively written as Find the minimal Y L (0) s.t., for some L ∈ L with L 0 = 0 , X L,3 T = K, X L,3 (s) ∈ [Λ(s), Λ(s)] for all s ≤ T , and E ℓ Y L (T ) -γX L,2 (T ) ≥ p , with γ > 0.

Corollary 4 . 4 .

 44 The function v * is a viscosity supersolution ofmin {D p ϕǫ , (D x 1 ϕ -C(x)D p ϕ)1 x 1 >0 , -D x 1 ϕ + C(x)D p ϕ} = 0 on DY (4.7)and v * is a viscosity subsolution ofmax {-D p ϕ + ǫ , (D x 1 ϕ -C(x)D p ϕ)1 x 1 >0 , -D x 1 ϕ + C(x)D p ϕ} = 0 on DY . (4.8)

Proposition 4 . 5 .

 45 There exists η > 0 such that0 ≤ v(t, x, p) ≤ ǫ -1 |pℓ(0)| + γη(1 + |x|) for all (t, x, p) ∈ DY .(4.9)

  13)-(4.14) have to satisfy D p ϕ ≥ ǫ and |D x 1 ϕ/D p ϕ| ≤ C in the viscosity sense. Still obtaining a general comparison theorem for the above PDE in an unbounded domain remains an open question.

  3. a. Clearly, we cannot have Ṽ (t ζ,δ , x ζ,δ , p ζ,δ ) = 0 since Ũζ ≥ 0 by assumption. b. We can neither have t ζ,δ = T since this would imply x 3 ζ,δ = y 3 ζ,δ by definition of DY and (4.2), a contradiction to (4.32).

  (4.22) and the definition of ( Ṽ , Ũζ ), we obtainṼ (t ζ,δ , x ζ,δ , p ζ,δ ) -Ũζ (t ζ,δ , y ζ,δ , q ζ,δ ) ≤ O(δ) < η 0 for δ small enough whenever x 1 ζ,δ ∈ {0, 2x 1 }, a contradiction to(4.31). Now assume that p ζ,δ < 0 for all ζ > 1 small enough. Then (4.24), the fact that (t ζ,δ , x ζ,δ ) takes values in a compact set and the definition of η ≥ η 0 > 0 imply that |p ζ,δ | ≤ ξ for some ξ > 0 which does not depend on ζ or δ. In particular, as ζ → 1, (t ζ,δ , x ζ,δ , y ζ,δ , p ζ,δ , q ζ,δ ) ζ>1 converges to some (t 1,δ , x 1,δ , y 1,δ , p 1,δ , q 1,δ ) such that x1 1,δ ∈ {0, 2x 1 }, (4.32) holds at the limit ζ = 1, and lim sup ζ→1 ( Ṽ (t ζ,δ , x ζ,δ , p ζ,δ ) -Ũζ (t ζ,δ , y ζ,δ , q ζ,δ )) ≤ Ṽ (t 1,δ , x 1,δ , p 1,δ ) -Ũ1 (t 1,δ , x 1,δδe 3 , p 1,δ ) ≤ O(δ) by upper-semicontinuity of V , -U , and(4.22). This shows that Ṽ (t ζ,δ , x ζ,δ , p ζ,δ ) -Ũζ (t ζ,δ , y ζ,δ , q ζ,δ ) < η 0 < 2η for ζ sufficiently close to 1 and δ > 0 small enough, a contradiction to (4.31).

5. 1

 1 Viscosity solution property on [0, T ) 5.1.1 Supersolution property on [0, T )

  X n (r)) l dM n r + s tn σ X (X n (r), u)dW r Y n (s) = y n + s tn µ Y (Z n (r), u)dr + s tn β Y (Z n (r)) ⊤ l dM n r + s tn σ Y (Z n (r), u) ⊤ dW r , Y n (s ∧ θ n ) ≥ ϕ(s ∧ θ n , X n (s ∧ θ n )) -2|γ n | for all s ≥ t n ,(5.21)whereθ o n := inf {s ≥ t n : (s, X n (s)) / ∈ B ε (t 0 , x 0 )} , θ n := inf {s ≥ t n : |Y n (s)ϕ(s, X n (s))| ≥ ε} ∧ θ o n .

Proposition 5 . 3 .

 53 Let (t 0 , x 0 ) ∈ DY , with t 0 < T , and ϕ be a smooth function such that(strict) max DY (v *ϕ) = (v *ϕ)(t 0 , x 0 ) = 0 .(5.22)

γ

  n := y nϕ(t n , x n ) → 0 .(5.26) Let Z n := (X n , Y n ) denote the solution of (2.1) associated to the Markovian control (ν n , 0) and the initial conditionZ n (t n ) = (x n , y n ), where νn = ν(•, X n , Y n ) .We next define the stopping timesθ o n := inf {s ≥ t n : (s, X n (s)) / ∈ B ε (t 0 , x 0 )} , θ n := inf {s ≥ t n : |Y n (s)ϕ(s, X n (s))| ≥ ε} ∧ θ o n .The same arguments as in the proof of Proposition 5.2 show thatY n (θ n )v (θ n , X n (θ n )) > 0for n large enough. Moreover, it follows from (5.24), (5.25) and Itô's Lemma that δ(•, X n , Y n ) ≥ 0 on [t n , θ n ] .

Proposition 5 . 4 .

 54 Fix (T, x 0 ) ∈ DY , and let ϕ be a smooth function such that(strict) min [0,T ]×R d (v *ϕ) = (v *ϕ)(T, x 0 ) = 0 .(5.27)

  x)ϕ(t, x) ≥ ζ > 0 for all (t, x) ∈ [Tr, T ] × ∂B ε (x 0 ). Hence,v(t, x)ϕ(t, x) ≥ ζ ∧ η > 0 for (t, x) ∈ ([Tr, T ] × ∂B ε (x 0 )) ∪ ({T } × B ε (x 0 )) . (5.28)Since F * ϕ(T, x 0 ) < ∞ and Gϕ(T, x 0 ) < 0, we can assume, after possibly changing ε > 0, thatGϕ(t, x) ≤ 0 and µ Y (x, y, u) -L u X ϕ(t, x) ≤ C for all u ∈ N ε (x, y, Dϕ(t, x)) and (t, x, y) ∈ [0, T ] × R d+1 s.t. (t, x) ∈ B ε (T, x 0 ) and |yϕ(x)| ≤ ε for some constant C > 0. Let φ(t, x) := ϕ(x) -√ Tt + ε + √ ε.Then, for sufficiently small ε > 0, we havev(t, x)φ(t, x) ≥ 1 2 (ζ ∧ η) > 0 for (t, x) ∈ ([Tε, T ] × ∂B ε (x 0 )) ∪ ({T } × B ε (x 0 )) G φ(t, x) ≤ 0 and µ Y (x, y, u) -L u X φ(t, x) ≤ 0 for all u ∈ N ε (x, y, D φ(t, x)) (t, x, y) ∈ [Tε, T ] × R d+1 s.t.x ∈ B ε (x 0 ) and |yφ(t, x)| ≤ ε.

  either U is a viscosity supersolution of (4.7) on DY which is continuous in p, or that V is a viscosity subsolution of (4.8) on DY which is continuous in p, then U ≥ V on DY .Before to provide the proof of the above result, we state the following immediate corollary which shows that the characterization of v in Proposition 4.7 is indeed sharp.

Corollary 4.9. Assume that (4.20) holds. Then, v is continuous on the closure of DY . Moreover, it is the unique non-negative viscosity solution of (4.13) in the class of continuous functions that are either supersolutions of (4.7) or subsolutions of (4.8), and satisfy the boundary conditions (4.10)-(4.11)-(4.12)-(4.15)-(4.21). This follows from Proposition 4.6, Corollary 4.4, Proposition 4.7, Proposition 4.8 and the following continuity result. Proposition 4.10. The function v is continuous in its p and x 3 variables, and, therefore, so are v * and v * .

  .29) It then follows from standard arguments, combined with the ones used in step 1. above, see e.g.[START_REF] Crandall | User's guide to viscosity solutions of second order partial differential equations[END_REF], that zn → zζ,δ := (t ζ,δ , x ζ,δ , y ζ,δ , p ζ,δ , q ζ,δ ) in the closure of D2

	where
	lim n→∞	n|x 1 n -y 1 n
		Y as n → ∞	(4.30)

VWAP means Volume Weighted Average Price, see Section 4 for a detailed presentation.

Note that, by definition of φn and (5.16), Y nϕ(•, X n ) is non-decreasing on [t n , θ n ], so that

(5.18)

for n large enough, recall (5.14). Since ϕ ≥ v * ≥ v, it follows that

In view of (5.18), this leads to

for n large enough. Recalling (5.12) and the fact that

Second case. If (5.15) does not hold, then it follows from (5.11) that we can find l ∈ ∆ + such that, for ε > 0 small enough,

(5.20) Proof. Assume to the contrary that min {ϕ(T, x 0 )w * (x 0 ) , M * ϕ(T, x 0 )} =: 2η > 0 .

Let φ be defined by φ(t, x) := ϕ(t, x) + √ Tt + ε -√ ε for ε > 0 small. Clearly, (T, x 0 ) achieves a strict maximum of v *φ, and it follows from the identity v(T, •) = w(x 0 ), the fact that ϕ(T, x 0 )w * (x 0 ) > 0 and (5.29) that max

Also observe that, the fact that M * ϕ(T, x 0 ) > 0, means that max{R * φ(T, x 0 ) , G φ(T, x 0 )} > 0 .

Since ∂ t φ → -∞ as t → T and ε → 0, we can find ε > 0 small enough such that

Moreover, the assumption (T, x 0 , φ(T, x 0 )) = (T, x 0 , v * (T, x 0 )) ∈ intD T , recall (3.11)-(3.12), implies that, for ε > 0 small enough,

Following line by line the arguments of Proposition 5.2 then leads to a contradiction to (GDP2) of Corollary 2.4. ✷

We finally consider the case (T, x 0 , v * (T, x 0 )) ∈ ∂D T .

Proposition 5.6. Let (t 0 , x 0 ) ∈ DY , with t 0 = T , and ϕ be a smooth function such that

.30)

Assume that (T, x 0 , v * (T, x 0 )) ∈ ∂D T . Then, min ϕ(T, x 0 )w * (x 0 ) , M in * ϕ(T, x 0 ) ≤ 0.

Proof. The result follows from an obvious combination of the arguments used in the proofs of Proposition 5.5 and Proposition 5.3 above. ✷