Bruno Bouchard
email: bouchard@ceremade.dauphine.fr.

Grégoire Loeper
email: gregoire.loeper@monash.edu

Halil Mete Soner
email: mete.soner@math.ethz.ch.

Chao Zhou

Second order stochastic target problems with generalized market impact

 and exhibit smooth solutions from the classical parabolic equations theory. Up to an additional approximating argument, this allows us to show that the super-hedging price solves the parabolic equation and that a perfect hedging strategy can be constructed when the coefficients are smooth enough. This representation leads to a general dual formulation. We finally provide an asymptotic expansion around a model without impact.

Introduction

Inspired by [START_REF] Abergel | Pricing and hedging contingent claims with liquidity costs and market impact[END_REF][START_REF] Loeper | Option pricing with linear market impact and non-linear Black and Scholes equations[END_REF], the authors in [START_REF] Bouchard | Almost sure hedging with permanent price impact[END_REF][START_REF] Bouchard | Hedging of covered options with linear market impact and gamma constraint[END_REF] considered a financial market with permanent price impact (and possibly a resilience effect), in which the impact function behaves as a linear function (around the origin) in the number of purchased stocks. This class of models is dedicated to the pricing and hedging of derivatives in situations where the notional of the product hedged is such that 1 the delta-hedging is non-negligible compared to the average daily volume traded on the underlying asset. As opposed to [START_REF] Bouchard | Almost sure hedging with permanent price impact[END_REF], the options considered in [START_REF] Bouchard | Hedging of covered options with linear market impact and gamma constraint[END_REF][START_REF] Loeper | Option pricing with linear market impact and non-linear Black and Scholes equations[END_REF] are covered, meaning that the buyer of the option delivers, at the inception, the required initial delta position, and accepts a mix of stocks (at their current market price) and cash as payment for the final claim. This is a common practice which eliminates the cost incurred by the initial and final hedge. In [START_REF] Loeper | Option pricing with linear market impact and non-linear Black and Scholes equations[END_REF], the author considers a Black-Scholes type model, while the model of [START_REF] Bouchard | Hedging of covered options with linear market impact and gamma constraint[END_REF] is a local volatility one. Motivated by these works, we consider in this paper a general abstract model of market impact in which the dynamics of the stocks X, the wealth1 V and the number of stocks Y held in the portfolio follow dynamics of the form

X = x + • t µ(s, X s , γ s , b s )ds + • t σ(s, X s , γ s )dW s Y = y + • t b s ds + • t γ s dX s V = v + • t F (s, X s , γ s )ds + • t Y s dX s
where (y, b, γ) are the controls, and we consider the general super-hedging problem: v(t, x) := inf{v = c + yx : (c, y) ∈ R 2 s.t. G(t, x, v, y) = ∅}, in which G(t, x, v, y) = (b, γ) : V t,x,v,φ T ≥ g(X t,x,φ T ) for φ := (y, b, γ) , and g is the payoff function associated to a European claim. One can easily be convinced, by using formal computations based on the geometric dynamic programming principle of [START_REF] Soner | The dynamic programming equation for second order stochastic target problems[END_REF], see also the discussion just after Remark 3.1, that v should be a super-solution of the fully nonlinear parabolic equation

0 ≤ -∂ t v -F (•, ∂ 2 x v) and (|F | + |σ|)(•, ∂ 2 x v) < ∞. in which F (t, x, z) := 1 2 σ(t, x, z) 2 z -F (t, x, z).
The right-hand side constraint in the previous inequalities is of importance. Indeed (F, σ)(t, x, •) can typically be singular and only finite on an interval of the form (-∞, γ(t, x)), as it is the case in [START_REF] Bouchard | Hedging of covered options with linear market impact and gamma constraint[END_REF]. Under this last assumption, one can actually expect that v is a viscosity solution of

min{-∂ t v -F (•, ∂ 2 x v) , γ -∂ 2 x v} = 0 on [0, T ) × R, (1)
with T -terminal condition given by the smallest function ĝ ≥ g such that ∂ 2 x ĝ ≤ γ(T, •). In [START_REF] Bouchard | Hedging of covered options with linear market impact and gamma constraint[END_REF], the authors impose a strong (uniform) constraint on the controls of the form γ ≤ γ(•, X t,x,φ ) with γ such that F (•, γ) ≤ C for some C > 0, and obtain that v is actually the unique viscosity solution of [START_REF] Abergel | Pricing and hedging contingent claims with liquidity costs and market impact[END_REF] with γ in place of γ, and terminal condition ĝ (defined with γ as well). Their proof of the super-solution property mimicks arguments of [START_REF] Cheridito | The multi-dimensional superreplication problem under gamma constraints[END_REF], and we can follow this approach. As for the sub-solution property, they could not prove the appropriate dynamic programming principle, and the standard direct arguments could not be used. Instead, they employed a regularization argument for viscosity solutions, inspired by [START_REF] Krylov | On the rate of convergence of finite-difference approximations for bellmans equations with variable coefficients[END_REF], together with a verification procedure. In [START_REF] Bouchard | Hedging of covered options with linear market impact and gamma constraint[END_REF], the authors critically use the fact that F is convex. Our setting here is different. First, as in [START_REF] Loeper | Option pricing with linear market impact and non-linear Black and Scholes equations[END_REF], we do not impose a uniform constraint on our strategies. Our controls can take values arbitrarily close to the singularity γ(•, X t,x,φ ) and the equation ( 1) is possibly degenerate. Even for F defined as in [START_REF] Bouchard | Hedging of covered options with linear market impact and gamma constraint[END_REF] our setting is more general in a sense. Second, F is not assumed to be convex. For these reasons, we can not reproduce the smoothing/verification argument of [START_REF] Bouchard | Hedging of covered options with linear market impact and gamma constraint[END_REF] to deduce that v is actually a subsolution.

In this paper, we therefore proceed differently and generalise arguments used in [START_REF] Loeper | Option pricing with linear market impact and non-linear Black and Scholes equations[END_REF] in the context of a Black-Scholes type model. Namely, we directly use the theory of parabolic equations to prove the existence of smooth solutions to (1) whenever ĝ is smooth and satisfies a constraint of the form ∂ 2

x ĝ ≤ γ(T, •)ε, for some ε > 0. Our analysis heavily relies on new a priori estimates, see Proposition 3.9 below, thanks to which one can appeal to the continuity method in a rather classical way, see the proof of Theorem 3.10. We then let ε go to 0 to conclude that v indeed solves (1) in the viscosity solution sense, see Theorem 3.5 below.

We also discuss two important issues that were not considered in [START_REF] Bouchard | Hedging of covered options with linear market impact and gamma constraint[END_REF] but already studied in [START_REF] Loeper | Option pricing with linear market impact and non-linear Black and Scholes equations[END_REF] in a Black-Scholes type model: -The first one concerns the asymptotic expansion of the price around a model without market impact. As in [START_REF] Loeper | Option pricing with linear market impact and non-linear Black and Scholes equations[END_REF], we show that a first order expansion can be established, see Proposition 4.3 below. But, we also prove that one can deduce from it a strategy that matches the terminal face-lifted payoff ĝ at any prescribed level of precision in L ∞ -norm, see Proposition 4.6.

-The second one concerns the existence of a dual formulation. It can be established when F is convex in its last argument, see Theorem 5.2. Applied to the model discussed in [START_REF] Bouchard | Hedging of covered options with linear market impact and gamma constraint[END_REF], see Example 2.1 below, it takes the form

v(t, x) = sup s E ĝ(X t,x,s T ) - T t 1 2 (s s -σ • (t, X t,x,s s )) 2 f (X t,x,s s ) ds = sup s E g(X t,x,s T ) - T t 1 2 (s s -σ • (t, X t,x,s s )) 2 f (X t,x,s s ) ds in which X t,x,s = x + • t s s dW s , σ
• is the volatility surface in a the market without impact and f > 0 is the impact function, the limit case f ≡ 0 corresponding to the absence of impact. It can be interpreted as the formulation of the super-hedging price with volatility uncertainty. The difference being that the formula is penalized by the squared distance of the realized volatility term s to the original local volatility σ • (•, X t,x,s ) associated to the model, weighted by the inverse of the impact function f (X t,x,s ). It can also be seen as a martingale optimal transport problem, see [START_REF] Loeper | Option pricing with linear market impact and non-linear Black and Scholes equations[END_REF]Section 4.1] for details.

To conclude, let us refer to [START_REF] Becherer | Optimal asset liquidation with multiplicative transient price impact[END_REF][START_REF] Becherer | Optimal liquidation under stochastic liquidity[END_REF][START_REF] Becherer | Stability for gains from large investors' strategies in m1/j1 topologies[END_REF][START_REF] Çetin | Liquidity risk and arbitrage pricing theory[END_REF][START_REF] Cheridito | The multi-dimensional superreplication problem under gamma constraints[END_REF][START_REF] Frey | Perfect option hedging for a large trader[END_REF][START_REF] Liu | Option pricing with an illiquid underlying asset market[END_REF][START_REF] Schönbucher | The feedback effects of hedging in illiquid markets[END_REF][START_REF] Sircar | Generalized black-scholes models accounting for increased market volatility from hedging strategies[END_REF][START_REF] Soner | The dynamic programming equation for second order stochastic target problems[END_REF], and the references therein. Also for related works, see [START_REF] Bouchard | Hedging of covered options with linear market impact and gamma constraint[END_REF] for a discussion.

The rest of this paper is organized as follows. The general abstract market model is described in Section 2 and the characterization of v as a solution of a parabolic equation is proved in Section 3. The asymptotic expansion and the dual formulation are provided and discussed in Sections 4 and 5.

General notations. Throughout this paper, Ω is the canonical space of continuous functions on R + starting at 0, P is the Wiener measure, W is the canonical process, and F = (F t ) t≥0 is the augmentation of its raw filtration F • = (F • t ) t≥0 . All random variables are defined on (Ω, F ∞ , P). We denote by |x| the Euclidean norm of x ∈ R n , the integer n ≥ 1 is given by the context. Unless otherwise specified, inequalities involving random variables are taken in the Pa.s. sense. We use the convention x/0 = sign(x) × ∞ with sign(0) = +. We denote by ∂ n

x ϕ the nth-order derivative of a function ϕ with respect to its x-component, whenever it is well-defined. For E, F, G, three subsets of R, We denote by C h,k b (E × F ) the set of continuous functions on E × F which have bounded partial derivatives of order from 1 to h with respect to the first variable and from 1 to k to the second variable. We denote by C h,k,l (E × F × G) the set of continuous functions on E × F × G which have partial derivatives of order from 1 to h with respect to the first variable, from 1 to k to the second variable and from 1 to l to the third variable. We denote by C h b (E × F ) the set of continuous functions on E × F which have bounded partial derivatives of order 1 to h. If in addition its h-th order derivatives are uniformly α-Hölder, with α ∈ (0, 1), we say that it belongs to C h+α b (E × F ). We omit the spaces E, F, G if they are clearly given by the context.

Abstract market impact model

We first describe our abstract market with impact. It generalizes the model studied in [START_REF] Bouchard | Almost sure hedging with permanent price impact[END_REF][START_REF] Bouchard | Hedging of covered options with linear market impact and gamma constraint[END_REF][START_REF] Loeper | Option pricing with linear market impact and non-linear Black and Scholes equations[END_REF]. We use the representation of the hedging strategies described in [START_REF] Bouchard | Hedging of covered options with linear market impact and gamma constraint[END_REF], which is necessary to obtain the supersolution characterization of the super-hedging price of Proposition 3.7 below. How to get to the market evolution [START_REF] Bouchard | Almost sure hedging with permanent price impact[END_REF][START_REF] Bouchard | Hedging of covered options with linear market impact and gamma constraint[END_REF][START_REF] Bouchard | Weak dynamic programming principle for viscosity solutions[END_REF]

E [sup {|ζ s ′ -ζ s |, t ≤ s ≤ s ′ ≤ s + δ ≤ T } |F • t ] ≤ kδ for all 0 ≤ δ ≤ 1 and t ∈ [0, T -δ].
We then define

A • := ∪ k A • k .
Let F : [0, T ] × R 2 → R ∪ {∞} be a continuous map and let D := {F < ∞} be its domain. We assume that there exists a map

(t, x) → γ(t, x) ∈ R ∪ {+∞} such that D = {(t, x, z) ∈ [0, T ] × R × R : z ∈ (-∞, γ(t, x))}, (2)
and that γ is either uniformly continuous, or identically equal to +∞.

We now let µ : D × R → R and σ : D → R be two continuous maps such that, for all ε > 0, µ is Lipschitz, with linear growth in its second variable, on

D ε,ε -1 × R, (4)
σ is Lipschitz, with linear growth in its second variable, on D ε,ε -1 , where

D ε := {(t, x, z) ∈ [0, T ] × R 2 : F (t, x, z) ≤ ε -1 }, (5) D ε,k := D ε ∩ ([0, T ] × R × [-k, k]) for k > 0. Then, given (t, x, v) ∈ [0, T ] × R × R and φ = (y, b, γ) ∈ R × A • , we define (X t,x,φ , Y t,x,φ , V t,x,v,φ ) as the solution on [t, T ] of X = x + • t µ(s, X s , γ s , b s )ds + • t σ(s, X s , γ s )dW s (6)
Y = y + • t b s ds + • t γ s dX s (7)
V = v + • t F (s, X s , γ s )ds + • t Y s dX s (8) satisfying (X t , Y t , V t ) = (x, y, v), whenever (•, X, γ) takes values in D ε,k on [0, T ],
for some ε, k > 0. If this is the case, we say that φ belongs to A ε k . For ease of notations, we set A := ∪ ε,k>0 A ε k . For a payoff function g : R → R the super-hedging price of the covered European claim associated to g is then defined as

v(t, x) := inf{v = c + yx : (c, y) ∈ R 2 s.t. G(t, x, v, y) = ∅}, (9)
in which

G(t, x, v, y) = φ = (y, b, γ) ∈ A : V t,x,v,φ T ≥ g(X t,x,φ T )
whenever this set is non-empty. Note that

v(t, x) = inf ε>0 v ε (t, x) where v ε (t, x) := inf k>0 v ε k (t, x) (10)
in which v ε k is defined as v but in terms of A ε k . In the following, we assume as in [START_REF] Bouchard | Hedging of covered options with linear market impact and gamma constraint[END_REF] that g is lower-semicontinuous, bounded from below, and g + has linear growth. [START_REF] Crandall | User's guide to viscosity solutions of second order partial differential equations[END_REF] Example 2.1 (Example of derivation of the evolution equations). We close this section with an example of formal derivation of the above abstract dynamics. In the spirit of [START_REF] Abergel | Pricing and hedging contingent claims with liquidity costs and market impact[END_REF][START_REF] Loeper | Option pricing with linear market impact and non-linear Black and Scholes equations[END_REF], let us consider a linear market impact model in which an (infinitesimal) order to buy dY t stocks at t leads to a permanent price move of f (t, X t , γ t )dY t , and to an average execution price of X t + f (t, X t , γ t )dY t + f (t, X t , γ t )dY t . Then, following the computations done in [START_REF] Abergel | Pricing and hedging contingent claims with liquidity costs and market impact[END_REF][START_REF] Loeper | Option pricing with linear market impact and non-linear Black and Scholes equations[END_REF], see also the rigorous proof in [START_REF] Bouchard | Almost sure hedging with permanent price impact[END_REF] for details2 , the portfolio value V corresponding to the holding in cash plus the number of stocks in the portfolio evaluated at their current price X is given by3

V = v + • t Y s dX s - • t f (s, X s , γ s )d Y, Y s .
The contribution f (s, X s , γ s )d Y, Y s is the spread between the execution price of the trade and the final price after market impact. It can be either positive or negative. The fact that f and f can depend on γ is discussed in [START_REF] Loeper | Option pricing with linear market impact and non-linear Black and Scholes equations[END_REF].

Let us now assume that X evolves according to dX t = σ • (t, X t )dW t +µ • (t, X t )dt in the absence of trade. Then, arguing again as in [START_REF] Bouchard | Almost sure hedging with permanent price impact[END_REF], we obtain the modified dynamics

dX t = σ • (t, X t )dW t + µ • (t, X t )dt + f (t, X t , γ t )dY t + f ′ (t, X t , γ t )γ t σ • (t, X t ) 2 dt.
Combining this with [START_REF] Bouchard | Hedging of covered options with linear market impact and gamma constraint[END_REF], and formally solving in dX, we obtain that

σ(t, X t , γ t ) = σ • (t, X t ) 1 -f (t, X t , γ t )γ t ,
so that the dynamics of V can be written as

V = v + • t Y s dX s - • t f (s, X s , γ s ) σ • (s, X s )γ s 1 -f (s, X s , γ s )γ s 2 ds.
Note that, as observed in [START_REF] Bouchard | Almost sure hedging with permanent price impact[END_REF], the drift µ • is also affected by the market impact, but that this does not affect the pricing equation. It is therefore not taken into account in our abstract model. The model studied in [START_REF] Bouchard | Almost sure hedging with permanent price impact[END_REF][START_REF] Bouchard | Hedging of covered options with linear market impact and gamma constraint[END_REF] corresponds to f = f (x) (no dependency in t, γ) and f = -f /2. In this particular case, the functions σ and F are given by

σ(t, x, z) = σ•(t,x) 1-f (x)z , γ = 1/f F (t, x, z) = 1 2 σ•(t,x)z 1-f (x)z 2 f (x)I {f (x)z<1} +∞I {f (x)z≥1} .

PDE characterization

The parabolic equation associated to v can be formally derived as follows. Assume that v is smooth and that a perfect hedging strategy φ = (y, b, γ) can be found when starting at t from v = v(t, x) if the stock price is x at t. Then, we expect to have V t,x,v,φ = v(•, X t,x,φ ) which, by Itô's lemma combined with ( 6)-( 8), implies that

F (s, X t,x,φ s , γ s )ds + Y t,x,φ s dX t,x,φ s = (∂ t v + 1 2 σ 2 (•, γ s )∂ 2 x v)(s, X t,x,φ s )ds + ∂ x v(s, X t,x,φ s )dX t,x,φ s for s ∈ [t, T ]
. By identifying the different terms, we obtain

F (s, X t,x,φ s , γ s ) = (∂ t v + 1 2 σ 2 (•, γ s )∂ 2 x v)(s, X t,x,φ s ) and Y t,x,φ s = ∂ x v(s, X t,x,φ s ).
Another application of Itô's lemma to the second equation then leads to

γ s = ∂ 2 x v(s, X t,x,φ s ),
recall [START_REF] Bouchard | Hedging of covered options with linear market impact and gamma constraint[END_REF]. The combination of the above reads

0 = -(∂ t v + F (•, ∂ 2 x v))(s, X t,x,φ s ) and (|F | + |σ|)(•, ∂ 2 x v)(s, X t,x,φ s ) < ∞, in which
F (t, x, z) := 1 2 σ(t, x, z) 2 z -F (t, x, z), for (t, x, z) ∈ D. ( 12
)
Remark 3.1. The model discussed in [START_REF] Bouchard | Hedging of covered options with linear market impact and gamma constraint[END_REF] corresponds to

F (t, x, z) = 1 2 σ 2 • (t, x)z 1 -f (x)z I {f (x)z<1} +∞I {f (x)z≥1} .
As usual, perfect equality can not be ensured because of the gamma constraint induced by the above. We therefore only expect to have

0 ≤ -(∂ t v + F (•, ∂ 2 x v))(s, X t,x,φ s ) and (|F | + |σ|)(•, ∂ 2 x v)(s, X t,x,φ s ) < ∞.
Recalling [START_REF] Barles | Solution de viscosités des équations d'Hamilton Jacobi[END_REF], this leads to the fact that v should be a super-solution of the parabolic equation

min{-∂ t ϕ -F (•, ∂ 2 x ϕ) , γ -∂ 2 x ϕ} = 0 on [0, T ) × R. (13)
By minimality, it should indeed be a solution. Moreover, as usual, the gamma constraint ∂ 2

x ϕ ≤ γ needs to propagate up to the boundary, so that we can only expect that v satisfies the terminal condition

lim (t ′ ,x ′ )→(T,x) ϕ(t ′ , x ′ ) = ĝ(x) for x ∈ R, (14)
where ĝ is the face-lift of g, i.e.

ĝ = inf{h ∈ C 2 (R) : h ≥ g and ∂ 2 x h ≤ γ(T, •)}.
See Remark 3.6 below for ease of comparison with [START_REF] Bouchard | Hedging of covered options with linear market impact and gamma constraint[END_REF].

Remark 3.2. When γ ≡ +∞, the above reads

-∂ t ϕ -F (•, ∂ 2 x ϕ) = 0 on [0, T ) × R and lim (t ′ ,x ′ )→(T,x) ϕ(t ′ , x ′ ) = g(x) for x ∈ R.
In order to prove that v is actually a continuous viscosity solution of the above, we need some additional assumptions. First, we assume that F is smooth enough,

F ∈ C 1 (D) and F ∈ C 1,3,3 b (D ε,ε -1 ), ε ∈ (0, ε • ], (15)
F is uniformly continuous on D ε , ε ∈ (0, ε • ], (16)
where ε • > 0, and that

F (•, 0) = 0, . (17)
For later use, note that the above implies

F (•, 0) = 0. (18)
We also assume that there exists L • , M > 0 such that, on D and for all ε ∈ (0, ε • ],

|∂ t F / F | ≤ L • , and |∂ 2 x F (•, z)| ≤ M |z| for all z ∈ (-∞, 0], (19)
that

∂ z F > 0 on D ε and sup {(t,x,z)∈D ε,ε -1 } (|∂ z F | + |1/∂ z F |) < ∞, (20)
inf

D ε,ε -1 σ > 0. (21)
F is uniformly continuous on D ε ,

and that, for all ε ∈ (0, ε • ], there exists a continuous map γε such that

D ε = {(t, x, z) ∈ [0, T ] × R 2 : z ≤ γε (t, x)} (23)
Remark 3.3. All these conditions are satisfied in the model of [START_REF] Bouchard | Hedging of covered options with linear market impact and gamma constraint[END_REF].

Remark 3.4. As a corollary of ( 17) and ( 22), we have that

sup Dε |F | < ∞, (24)
Finally, we assume that

ĝε := inf{h ≥ g : h ∈ C 2 (R), F (T, •, ∂ 2 x h) < ε -1 } (25)
satisfies ĝε is uniformly continuous, bounded from below and has linear growth (26)

and that there exists k • ≥ 1 such that

[v ε k ] + has linear growth, uniformly in k ≥ k • , (27)
for all 0 < ε ≤ ε • , in which we use the convention 1/0 = ∞ and identify ĝ with ĝ0 .

Under the above conditions, we can state the main result of this section.

Theorem 3.5. The function v is a continuous viscosity solution of (13) such that

lim t ′ ↑T,x ′ →x v(t ′ , x ′ ) = ĝ(x) for all x ∈ R. If moreover there exists α ∈ (0, 1) such that ĝ ∈ C 4+α b , |∂ 2 x ĝ| ≤ ε -1 and (T, •, ∂ 2 x ĝ) ∈ D ε for some ε > 0, then, for each (t, x) ∈ [0, T ) × R, we can find φ ∈ A such that V t,x,v,φ T = ĝ(X t,x,φ T ) with v = v(t, x).
In [START_REF] Bouchard | Hedging of covered options with linear market impact and gamma constraint[END_REF], the authors also provide a viscosity solution characterization of v, but in their case (i) admissible strategies should satisfy γ ≤ γ(•, X t,x,φ ) for some given function

γ < γ (uniformly on [0, T ] × R), (ii) F (•, γ) < ∞, (iii) F (t, x, •) is convex on (-∞, γ(t, x)] for all (t, x) ∈ [0, T ] × R.
None of these assumptions are imposed here, and we also consider the case γ ≡ +∞.

Still, the supersolution property can essentially be proved by mimicking the arguments of [10, Section 5], up to considering a weak formulation of our stochastic target problem. This will only provide a supersolution of (13) that will serve as a lower bound, see Proposition 3.7 for a precise statement. In [START_REF] Bouchard | Hedging of covered options with linear market impact and gamma constraint[END_REF], the subsolution property could not be proved directly as in [START_REF] Cheridito | The multi-dimensional superreplication problem under gamma constraints[END_REF]. The reason is that the feedback effect of the controled state dynamics (X, Y, V ) prevented them to establish the required geometric dynamic programming principle. Instead, they used a smoothing argument in the spirit of [START_REF] Krylov | On the rate of convergence of finite-difference approximations for bellmans equations with variable coefficients[END_REF]. This however requires F to be convex, which, again, is not the case in our generalized setting. We will instead rely on the theory of parabolic equations, which, up to regularization arguments, will allow us to construct smooth subsolutions of ( 13) from which superhedging strategies can be deduced, see Corollary 3.11. As in [START_REF] Bouchard | Hedging of covered options with linear market impact and gamma constraint[END_REF], combining these two results will prove Theorem 3.5.

We conclude this section with a remark on our definition of the face-lift of g. Remark 3.6. In [START_REF] Bouchard | Hedging of covered options with linear market impact and gamma constraint[END_REF], the face-lift is defined as the smallest function above g that is a viscosity supersolution of the equation γ -∂ 2

x ϕ = 0. It is obtained by considering any twice continuously differentiable function Γ such that ∂ 2

x Γ = γ, and then setting ḡ := (g -Γ) conc + Γ, in which the superscript conc means concave envelope, cf. [START_REF] Soner | Superreplication under gamma constraints[END_REF]Lemma 3.1]. This actually corresponds to our definition. The fact that ĝ ≥ ḡ is trivialy deduced from the supersolution property in the definition of ḡ. Let us prove the converse inequality. Fix ε ∈ (0, ε • ], and define ḡε as ḡ but with γε in place of γ. Fix ψ ∈ C ∞ b with compact support, such that ψ(y)dy = 1 and ψ ≥ 0, and define ḡε n (x) := ḡε (y) 1 n ψ(n(yx))dy for n ≥ 1. Since ḡε is the sum of a concave function and a C 2 function, one can consider the measure m ε associated to its second derivative and it satisfies m ε (dy) ≤ (γ(y)ε)dy. Then, ∂ 2

x ḡε n (x) = ḡε (y)n∂ 2 x ψ(n(y -x))dy = 1 n ψ(n(y -x))dm ε (y) ≤ 1 n ψ(n(y -x))(γ(y) -ε)
dy. Now, note that ḡ is continuous and therefore uniformly continuous on compact sets. Then, up to using the approximation from above argument of [7, Lemma 3.2], we can assume that it is uniformly continuous. Since γ is also uniformly continuous, see

(3), one can find κ, ε > 0 such that ḡε,κ n : x ∈ R → ḡε n (x) + κ is C 2 , ∂ 2 x ḡε,κ n ≤ γ and ḡε,κ n ≥ g. By definition, it follows that ḡε,κ n ≥ ĝ. Clearly, (ḡ ε,κ n ) ε,κ>0,n≥1
converges pointwise to ḡ as n → ∞ and (ε, κ) → 0 in a suitable way. This shows that ḡ ≥ ĝ.

Supersolution property of a lower bound and partial comparison

In this section, we produce a supersolution of a version of (13) that is associated to v ε , recall [START_REF] Cheridito | The multi-dimensional superreplication problem under gamma constraints[END_REF], and that is a lower bound for v ε . We also prove a partial comparison result on this version that will be of important use later on. Recall the definition of ĝε in (25).

Proposition 3.7. For each ε ∈ (0, ε • ] small enough, there exists a continuous function v ε ≤ v ε that has linear growth, is bounded from below, is a viscosity super-solution of

min{-∂ t ϕ -F (•, ∂ 2 x ϕ) , ε -1 -F (•, ∂ 2 x ϕ)} = 0 on [0, T ) × R (Eq ε )
and satisfies lim inf

t ′ ↑T,x ′ →x v ε (t ′ , x ′ ) ≥ ĝε (x) for all x ∈ R.
Proof. This follows from exactly the same arguments as in [7, Section 3.1]. We only explain the differences. As in [7, Section 3.2], we first introduce a sequence of weak formulations. On (C(R + )) 5 , let us denote by ( ζ := (γ, b, α, β), W ) the coordinate process and let F• = ( F • s ) s≤T be its raw filtration. We say that a probability measure P belongs to Ãk if W is a P-Brownian motion and if for all 0 ≤ δ ≤ 1 and r ≥ 0 it holds P-a.s. that

γ = γ0 + • 0 βs ds + • 0 αs d Ws for some γ0 ∈ R, (28)
sup R+ | ζ| ≤ k , (29)
and

E P sup | ζs ′ -ζs |, r ≤ s ≤ s ′ ≤ s + δ | F • r ≤ kδ. ( 30
)
For φ := (y, γ, b), y ∈ R, we define ( Xx, φ, Ỹ φ, Ṽ x,v, φ) as in ( 6)-( 7)-( 8) associated to the control (γ, b) with time-t initial condition (x, y, v), and with W in place of W . For t ≤ T and k ≥ 1, we say that P ∈ Gk,ε (t, x, v, y) if

Ṽ x,v, φ T ≥ g( Xx, φ T ), F (•, Xx, φ, γ) ≤ ε -1 and γ ∈ [-k, k] on R + P -a.s.
(31) We finally define

ṽε k (t, x) := inf{v = c + yx : (c, y) ∈ R × [-k, k] s.t. Ãk ∩ Gk,ε (t, x, v, y) = ∅}.
Step 1. We first provide bounds for ṽε k . Note that ṽε k ≤ v ε k , so that (27) implies that [ṽ ε k ] + has linear growth, uniformly in k ≥ k • . Moreover, note that the fact that σ is Lipschitz with linear growth in its second variable, uniformly on D ε,k × R (see (4)), implies that Xt,x, φ is a square integrable martingale under P for any φ := (y, γ, b), and that the same holds for

• t Ỹ t, φ s d Xt,x, φ s . Then, the inequality v + T t F (s, Xt,x, φ s , γs )ds + T t Ỹ t, φ s d Xt,x, φ s ≥ g( Xt,x, φ T
) combined with (24) and [START_REF] Crandall | User's guide to viscosity solutions of second order partial differential equations[END_REF]

implies that v ≥ -sup |g -| -T sup Dε F > -∞. This shows that ṽε k is bounded from below, uniformly in k ≥ k 0 . Step 2. We claim that v ε (t, x) := lim inf (k, t ′ , x ′ ) → (∞, t, x) (t ′ , x ′ ) ∈ [0, T ) × R ṽε k (t ′ , x ′ ), (t, x) ∈ [0, T ] × R,
is a viscosity supersolution of (Eq ε ).

To prove this, it suffices to show that it holds for each ṽε k , with k ≥ k • , and then to apply standard stability results, see e.g. [START_REF] Barles | Solution de viscosités des équations d'Hamilton Jacobi[END_REF]. By the same arguments as in [START_REF] Bouchard | Hedging of covered options with linear market impact and gamma constraint[END_REF]Proposition 3.15], each ṽε

k is lower- semicontinuous. Given a C ∞ b test function ϕ and (t 0 , x 0 ) ∈ [0, T ) × R such that (strict) min [0,T )×R (ṽ ε k -ϕ) = (ṽ ε k -ϕ)(t 0 , x 0 ) = 0,
we first use [START_REF] Soner | The dynamic programming equation for second order stochastic target problems[END_REF] and the arguments of [7, Step 1-2, proof of Theorem 3.16] to obtain that there exists γ0 such that

∂ 2 x ϕ(t 0 , x 0 ) ≤ γ0 and F (t 0 , x 0 , γ0 ) ≤ ε -1 .
Then, the same arguments as in [7, Step 3.a., proof of Theorem 3.16] combined with ( 12) and ( 20) lead to

0 ≤F (t 0 , x 0 , γ0 ) -∂ t ϕ(t 0 , x 0 ) - 1 2 σ 2 (t 0 , x 0 , γ0 ) 2 ∂ 2 x ϕ(t 0 , x 0 ) - 1 2 γ0 -∂ 2 x ϕ(t 0 , x 0 ) σ 2 (t 0 , x 0 , γ0 ) = -∂ t ϕ(t 0 , x 0 ) -F (t 0 , x 0 , γ0 ) ≤ -∂ t ϕ(t 0 , x 0 ) -F (t 0 , x 0 , ∂ 2
x ϕ(t 0 , x 0 )).

Finally, the T -boundary condition is obtained as in [7, Step 3.b., proof of Theorem 3.16], recall our assumption [START_REF] Crandall | User's guide to viscosity solutions of second order partial differential equations[END_REF], as well as Remark 3.6.

We now provide a partial comparison result that will be used later on. Note that a full comparison result could be proved as in [START_REF] Bouchard | Hedging of covered options with linear market impact and gamma constraint[END_REF]Theorem 3.11] when F is convex, by mimicking their arguments. It is however not the case in general. Given the strategy of our proof, it is not required in this paper. In the following, we interpret (Eq ε ) by using the convention 0 -1 = ∞ in the case ε = 0. Proposition 3.8. Let U be an upper semicontinuous viscosity subsolution of (Eq ε ) for ε ∈ [0, ε • ]. Let V be a lower semicontinuous viscosity supersolution of (Eq ε ′ ) for some ε ′ ∈ (ε, ε • ]. Assume that U and V have linear growth and that

U ≤ V on {T } × R, then U ≤ V on [0, T ] × R. Proof. Set Û (t, x) := e ρt U (t, x), V (t, x) := e ρt V (t, x) for some ρ > 0. Then, Û is a subsolution of min ρϕ -∂ t ϕ -e ρ• F (•, e -ρ• ∂ 2 x ϕ), ε -1 -F (•, e -ρ• ∂ 2 x ϕ) = 0 ( 32
)
and V is a supersolution of

min ρϕ -∂ t ϕ -e ρ• F (•, e -ρ• ∂ 2 x ϕ), (ε ′ ) -1 -F (•, e -ρ• ∂ 2 x ϕ) = 0 (33) on [0, T ) × R. Assume that sup [0,T ]×R ( Û -V ) > 0.
Then, there exists η > 0 such that, for all n > 0 and all λ > 0 small enough,

sup (t,x,y)∈[0,T ]×R 2 Û (t, x) -V (t, y) - λ 2 |x| 2 - n 2 |x -y| 2 ≥ η > 0. ( 34
)
Denote by (t n , x n , y n ) the point at which this supremum is achieved. Since V (T, •) ≥ Û (T, •), we have t n < T . Moreover, standard arguments, see e.g., [START_REF] Crandall | User's guide to viscosity solutions of second order partial differential equations[END_REF]Proposition 3.7], lead to

lim n→∞ n|x n -y n | 2 = 0. ( 35
)
We now apply Ishii's lemma to obtain the existence of (a

n , M n , N n ) ∈ R 3 such that (a n , n(x n -y n )+λx n , M n ) ∈ P2,+ Û (t n , x n ) (a n , -n(x n -y n ), N n ) ∈ P2,-V (t n , y n ),
in which P2,+ and P2,denote as usual the closed parabolic super-and subjets, see [START_REF] Crandall | User's guide to viscosity solutions of second order partial differential equations[END_REF], and

M n 0 0 -N n ≤ 3n 1 -1 -1 1 + 3λ + λ 2 n -λ -λ 0 .
In particular, M n ≤ N n + 2λ + λ 2 /n. Since V is a supersolution of (33) and ε < ε ′ , ( 22) and (35) imply that F (t n , x n , e -ρtn M n ) < ε -1 for λ > 0 small enough and n large enough. Hence,

ρ Û (t n , x n ) -a n -e ρtn F (t n , x n , e -ρtn M n ) ≤ 0.
On the other hand, the supersolution property of V combined with ( 16) and ( 20) implies that

0 ≤ρ V (t n , y n ) -a n -e ρtn F (t n , y n , e -ρtn N n ) ≤ρ V (t n , y n ) -a n -e ρtn F (t n , y n , e -ρtn M n ) + e ρtn δ(e -ρtn (2λ + λ 2 /n))
in which δ(z) → 0 as z → 0. Hence,

ρ( Û (t n , x n ) -V (t n , y n )) ≤ e ρtn F (t n , x n , e -ρtn M n ) -F (t n , y n , e -ρtn M n ) + e ρtn δ(e -ρtn (λ + λ 2 /n)).
Recalling (35) and ( 16), we obtain a contradiction to (34) by sending n → ∞ and then λ → 0.

Regularity of solutions to (Eq ε )

To complete the characterization of Proposition 3.7, we now study the regularity of solutions to (Eq ε ). We shall indeed show that (Eq ε ) admits a smooth solution u ε such that (•, ∂ 2 x u) ∈ D ε on [0, T ] × R, for ε > 0 small enough and for a certain class of terminal conditions. A simple verification argument will then show that u ε dominates the super-hedging price v if the terminal data Φ ε associated to u ε dominates ĝ. A lower bound u ε for v can also be constructed by considering a terminal condition Φ ε ≤ ĝ and using our comparison result of Proposition 3.8 combined with Proposition 3.7. Then, letting Φ ε , Φ ε → ĝ in a suitable way will be enough to show that v is actually a solution of (Eq 0 ), i.e. to conclude the proof of Theorem 3.5. The strategy we employ consists in establishing a priori estimates for the second derivative of the solution to (Eq ε ). Once established, the equation becomes uniformly parabolic, and higher regularity follows by standard parabolic regularity (see [START_REF] Lieberman | order parabolic differential equations[END_REF]). Then, the continuity method (see [START_REF] Gilbarg | Elliptic partial differential equations of second order, volume 224 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences[END_REF]) allows us to actually construct the solution to (Eq ε ).

Let us start with uniform estimates for solutions to (Eq ε ) such that (•, •, ∂ 2

x u) ∈ D ε ′ for some ε ′ > 0, in the case where the terminal condition Φ is smooth and satisfies a similar constraint. Proposition 3.9. Let u and Φ be two continuous functions such that

(i) Φ ∈ C 2 (R) with |∂ 2 x Φ| ≤ K Φ for some K Φ > 0, (ii) (T, •, ∂ 2 x Φ) ∈ D εΦ for some ε Φ > 0, (iii) u ∈ C 1,4 ([0, T ) × R)∩C 0,2 ([0, T ] × R) with |∂ 2 x u| ≤ K ′ for some K ′ > 0, (iv) (•, •, ∂ 2 x u) ∈ D ε ′ for some ε ′ > 0. Assume that u solves ∂ t u + F (•, ∂ 2 x u) = 0 on [0, T ) × R, (Eq 0 ) u(T, •) = Φ on R. (36) Then, a. (•, ∂ 2 x u) ∈ D ε on [0, T ] × R, for some ε > 0 that depends only on ε Φ and L • , b. |∂ 2
x u| ≤ K on [0, T ] × R where K depends only on K Φ . c. If Φ is globally Lipschitz, then u is also globally Lipschitz with Lipschitz constant controlled by the one of Φ.

d. u is the unique C 1,2 ([0, T ) × R) ∩ C 0 ([0, T ] × R) solution of (Eq 0 )-(36) such that (•, •, ∂ 2 x u) ∈ D ε ′′ for some ε ′′ > 0. Proof. a. Let V := F (•, ∂ 2 x u). Then, on [0, T ) × R, ∂ t V = ∂ t F (•, ∂ 2 x u) + ∂ z F (•, ∂ 2 x u)∂ t ∂ 2 x u
in which, by (Eq 0 ),

∂ t ∂ 2 x u + ∂ 2 x V = 0. Hence, ∂ t V + ∂ z F (•, ∂ 2 x u)∂ 2 x V = ∂ t F (•, ∂ 2 x u) = ∂ t F (•, ∂ 2 x u) F (•, ∂ 2 x u) V, (37)
recall [START_REF] Schönbucher | The feedback effects of hedging in illiquid markets[END_REF]. For (t, x) ∈ [0, T ] × R, let Xt,x be the solution of

X = x + • t (2∂ z F (•, ∂ 2 x u)(s, Xs )) 1 2 dW s .
By (iv), ( 15) and ( 20), it is well-defined. Combining Itô's Lemma and a standard localizing argument using ( 15) and ( 19), we obtain

V (t, x) = E[V (T, Xt,x T )e -T t (∂t F (•,∂ 2 x u)/ F (•,∂ 2 x u))(s, Xt,x s )ds ]. (38)
By definition of V and the fact that

∂ 2 x u(T, •) = ∂ 2 x Φ by (iii), this shows that (•, ∂ 2 x u) ∈ D ε on [0, T ] × R,
for some ε > 0 that depends only on L • and ε Φ . b. To obtain the bound on ∂ 2

x u, we first differentiate twice (Eq 0 ) with respect to x, recall ( 15) and (iii). Letting Z(t, x) = ∂ 2

x u(t, x), this yields

∂ t Z + 2∂ x ∂ z F ∂ x Z + ∂ z F ∂ 2 x Z + ∂ 2 z F (∂ x Z) 2 = -∂ 2 x F .
We now consider

(t, x) → Z(t, x) := min{0, inf Z(T, •)}e M(T -t) ,
in which M is given in [START_REF] Schönbucher | The feedback effects of hedging in illiquid markets[END_REF]. Then,

∂ t Z + 2∂ x ∂ z F ∂ x Z + ∂ z F ∂ 2 x Z + ∂ 2 z F (∂ x Z) 2 = -M Z ≥ -∂ 2 x F (t, x, Z).
Under the current assumptions, Z is uniformly bounded on [0, T ]×R. Moreover, from assumption (15), ∂ 2

x F is uniformly continuous on D ε,ε -1 , for all ε > 0 small enough, hence, by [START_REF] Loeper | Option pricing with linear market impact and non-linear Black and Scholes equations[END_REF] and [11, Proof of comparison, Theorem 5.1], the comparison principle holds between Z and Z, and yields that Z ≤ Z globally on [0, T ] × R. The upper bound is obtained in the exact same way. c. The assertion about the Lipschitz regularity also follows from the linearised equation satisfied by κ = ∂ x u:

∂ t κ + ∂ z F (•, ∂ 2 x u)∂ 2 x κ + ∂ x F (•, ∂ x κ) = 0, κ(T, •) = ∂ x Φ.
Under the assumptions ( 20), [START_REF] Krylov | On the rate of convergence of finite-difference approximations for bellmans equations with variable coefficients[END_REF], and [START_REF] Loeper | Option pricing with linear market impact and non-linear Black and Scholes equations[END_REF], this implies that

κ(t, x) = E[∂ x Φ( Xt,x T )] where Xt,x = x + • t 2∂ z F (•, ∂ 2 x u) 1 2 (s, Xt,x s )dW s + • t ∂ x F (•, ∂ 2 x u) ∂ 2 x u (s, Xt,x s )ds,
and the result follows. (Note that, since F (•, 0) = 0 and

F ∈ C 1,3,3 b (D ε,ε -1 ), the map z → ∂x F (•,z)
z is bounded and Lipschitz -after extending it to ∂ z ∂ x F (•, 0) at 0.) d. Consider another solution u ′ . Then, b. implies that u and u ′ have at most a quadratic growth. Moreover, a. allows one to consider a uniformly parabolic equation. Then, the fact that u = u ′ follows from standard arguments.

We are now in position to construct a smooth solution to (Eq 0 ).

Theorem 3.10. Let Φ be a continuous map such that |∂ 2

x Φ| ≤ ε -1 and (T, •, ∂ 2

x Φ) ∈ D ε for some ε > 0. Then, there exists a solution u of (Eq 0 )-( 36) that belongs to C([0, T ]×R)∩C 1,4 loc b . Proof. This follows by using the continuity method (cf. [START_REF] Gilbarg | Elliptic partial differential equations of second order, volume 224 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences[END_REF]Chap. 17.2]). We first mollify Φ into a function Φ n so that ∂ 5

([0, T )×R), such that |∂ 2 x u| ≤ (ε Φ,L• ) -1 and (•, ∂ 2 x u) ∈ D εΦ,L • on [0, T ] × R, for some ε Φ,L• > 0 that
x Φ n is bounded, and at the same time F so that F (•, •, z) ∈ C ∞ ([0, T ] × R) locally uniformly with respect to z. This is possible, since γ and F are uniformly continuous (recall ( 3) and ( 16)), by taking a compactly supported smoothing kernel ψ ∈ C ∞ (R) and considering

Φ n = 1 n R Φ(y)ψ(n(y -•))dy, Fn = 1 n 2 [0,T ]×R F (s, y, •)ψ(n(s -•))ψ(n(y -•))dsdy.
For later use, note that Fn (T,

•, ∂ 2 x Φ n ) ≤ 2ε -1 , for n large enough. Set G n (ϕ, θ) := [∂ t ϕ + Fn (•, ∂ 2 x ϕ)]I [0,T ) + I {T } (ϕ -θΦ n ) for ϕ ∈ C 1,4 b ,
and let E n ⊂ [0, 1] be the set of real number θ ∈ [0, 1] for which a C 1,4 b solution u n θ to G n (u n θ , θ) = 0 exists such that it satisfies the condition (iii)-(iv) of Proposition 3.9. By [START_REF] Loeper | Option pricing with linear market impact and non-linear Black and Scholes equations[END_REF], u 0 ≡ 0 solves G n (u 0 , 0) = 0 so that 0 ∈ E n . Hence, E n is non empty. Moreover, for every θ ∈ E n , the linearised operator associated to G n is

(ũ, θ) ∈ C 1,2 ×E n → L n (ũ, θ) := [∂ t ũ+∂ z Fn (t, x, ∂ 2 x u)∂ 2 x ũ]I [0,T ) +I {T } (ũ-θΦ n ).
It is uniformly parabolic (recall ( 20)) with coefficients in C ∞ . For θ fixed, the equation L n (ũ, θ) = 0 is therefore a linear, uniformly parabolic equation, with smooth coefficients. The terminal data is smooth, has linear growth and bounded derivatives of order 1 up to 5. Standard parabolic regularity theory (see [START_REF] Friedman | Partial Differential Equations of Parabolic Type[END_REF]) yields that the linearised equation with respect to u is solvable in C 1,4 b . By the implicit function theorem, see e.g. [START_REF] Gilbarg | Elliptic partial differential equations of second order, volume 224 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences[END_REF]Theorem 17.6], E n is open in [0, 1]. By the a priori estimates of Proposition 3.9, E n is also closed. Therefore, E n = [0, 1] and u n 1 is well defined. Since ( Fn ) n≥1 is uniformly parabolic, uniformly in n,

and (Φ n ) n≥1 is bounded in C 4+α b uniformly in n, then (u n 1 ) n≥1 is C 1,4
b uniformly in n. It remains to send n → ∞ and to appeal again to the a priori estimates of Proposition 3.9 to deduce the required result.

Full chacterization of the super-hedging price and perfect hedging in the smooth case

We are now about to conclude the proof of Theorem 3.5. Let û be the function constructed in Theorem 3.10 for Φ = ĝ, assuming that ĝ satisfies the required constraints. We first establish that û permits to apply a perfect hedging strategy of the face-lifted payoff whenever it is smooth enough, and that it coincides with the super-hedging price.

Corollary 3.11. Assume that there exists α ∈ (0, 1)

such that ĝ ∈ C 4+α b , that |∂ 2 x ĝ| ≤ ε -1 and (T, •, ∂ 2 x ĝ) ∈ D ε for some ε > 0.
Let û be the function constructed in Theorem 3.10 for Φ = ĝ. Then, v = û and, for each

(t, x) ∈ [0, T ] × R, we can find φ ∈ A such that V t,x,v,φ T = ĝ(X t,x,φ T ).
Proof. It follows from Theorem 3.10, Itô's lemma and ( 12) that û induces an exact replication strategy:

ĝ(X t,x,φ T ) =û(t, x) + T t ∂ t û + 1 2 σ(•, ∂ 2 x û) 2 ∂ 2 x û (s, X t, x,φ s )ds +
T t ∂ x û(s, X t,x,φ s )dX t,x,φ s =û(t, x) + T t F (s, X t,x,φ s , γ s )ds + T t Y t,x,φ s dX t,x,φ s in which φ = (y, b, γ) with y = ∂ x û(t, x), b = ([∂ t + 1 2 σ(•, γ) 2 ∂ 2 x ]∂ x û)(•, X t,x,φ • ), γ = ∂ 2 x û(•, X t,x,φ • ).
Hence, û ≥ v. Moreover, û is a viscosity subsolution of (Eq ε ′ ) for all ε ′ ≥ 0 small enough. Since ĝ is globally Lipschitz, û is also globally Lipschitz (Theorem 3.10), and therefore has linear growth. By Proposition 3.7, v ε ≥ v ε that is a supersolution of (Eq ε ) and satisfies lim inf

t ′ ↑T,x ′ →x v ε (t ′ , x ′ ) ≥ ĝε (x) ≥ ĝ(x) = û(T, x) for all x ∈ R. Then, Proposition 3.8 implies that v ε ≥ û. Taking the inf over ε > 0 leads to v ≥ û.
We can now conclude the proof of Theorem 3.5.

Proof of Theorem 3.5.

For ε > 0, let Φ ε , Φ ε ∈ C 2 be such that, for Ψ ∈ {Φ ε , Φ ε }, Ψ ∈ C 5 b (R), |∂ 2 x Ψ| ≤ ε -1 , (T, •, ∂ 2 x Ψ) ∈ D ε , and
Φ ε ≤ ĝ ≤ Φ ε , Φ ε -Φ ε ≤ δ(ε),
in which lim ε→0 δ(ε) = 0. Such functions can be constructed as in Remark 3.6, and we can further assume that Φ ε (resp. Φ ε ) is non-increasing (resp. nondecreasing) with respect to ε. Let u ε and u ε be the (smooth) solutions to (Eq 0 ) associated to Φ ε and Φ ε respectively, as in Theorem 3.10. By applying Corollay 3.11 to Φ ε in place of ĝ, we deduce that u ε is the super-hedging price of Φ ε ≥ ĝ so that u ε ≥ v. Similarly u ε ≤ v, and therefore u ε ≤ v ≤ u ε . By the comparison principle, we also have

0 ≤ u ε -u ε ≤ sup{Φ ε -Φ ε } ≤ δ(ε).
It follows that v is the uniform limit of a sequence of continuous functions, and is therefore continuous. Each of the functions u ε solves (13), recall [START_REF] Barles | Solution de viscosités des équations d'Hamilton Jacobi[END_REF]. Standard stability results, see e.g. [START_REF] Barles | Solution de viscosités des équations d'Hamilton Jacobi[END_REF], imply that v is a viscosity solution to ( 13)-( 14). The other assertions in Theorem 3.5 are immediate consequences of Corollary 3.11.

Asymptotic analysis

We now consider the case where the impact of the γ process in the dynamics of (X, V ) is small. Our aim is to obtain an asymptotic expansion around an impact free model. More precisely, we consider the dynamics

X ǫ,t,x,φ = x + • t µ(s, X ǫ,t,x,φ s , ǫγ s , ǫb s )ds + • t σ(s, X ǫ,t,x,φ s , ǫγ s )dW s V ǫ,t,x,v,φ = v + • t ǫ -1 F (s, X ǫ,t,x,φ s , ǫγ s )ds + • t Y ǫ,t,x,φ s dX ǫ,t,x,φ s , ǫ > 0,
and denote by v ǫ the corresponding super-hedging price. We place ourself in the context of Corollary 3.11 for the coefficients µ(•, ǫ•, ǫ•), σ(•, ǫ•) and ǫ -1 F (•, ǫ•). In particular, we assume that ĝ ∈ C 2 is such that ǫ -1 F (T, •, ǫ∂ 2

x ĝ) is bounded on R, for ǫ > 0 small enough. In the following, we use the notation

( F0 , ∂ n z F0 ) := ( F (•, 0), ∂ n z F (•, 0)), for n = 1, 2.
Remark 4.1. Note that the model of [START_REF] Bouchard | Hedging of covered options with linear market impact and gamma constraint[END_REF] corresponds to

σ(t, x, ǫz) = σ • (t, x) 1 -ǫf (x)z , ǫ -1 F (t, x, ǫz) = 1 2 σ • (t, x)z 1 -ǫf (x)z 2 ǫf (x).
Our scaling therefore amounts to consider a small impact function x → ǫf (x).

In order to interpret the result of Proposition 4.3 below, also observe that

(2∂ z F0 (t, x)) 1 2 = σ • (t, x) and ∂ 2 z F0 (t, x) = σ 2 • (t, x)f (x).
Our expansion is performed around the solution v 0 of

∂ t v 0 + ∂ z F0 ∂ 2 x v 0 = 0 on [0, T ) × R and v 0 (T, •) = ĝ on R. ( 39
)
Remark 4.2. Let the conditions of Corollary 3.11 hold and assume that F ∈ C 1,3,1 loc (D) with

|∂ x ∂ z F0 | + |∂ 2 x ∂ z F0 | uniformly bounded. (40)
Then, v 0 is the unique solution in

C 1,2 b ([0, T ] × R) ∩ C 1,3 ([0, T ) × R]
) of (39). This follows from [START_REF] Sircar | Generalized black-scholes models accounting for increased market volatility from hedging strategies[END_REF] and standard estimates.

The following expansion requires some additional regularity on ĝ that will in general not be satisfied in applications. However, one can reduce to it up to a slight approximation argument.

F ǫ := ǫ -1 F (•, ǫ•) in place of F , uniformly in ǫ ∈ (0, ǫ • ], for some ǫ • > 0. Assume further that F ∈ C 1,2,3 loc (D), that (40) and sup Dǫ |∂ 2 z F0 | + |∂ 3 z F0 | + |∂ x ∂ 2 z F0 | + |∂ 2 x ∂ 2 z F0 | < ∞ (41)
hold. Then, there exists some o(ε), which does not depend on x, such that

v ǫ (0, x) =v 0 (0, x) + ǫ 2 E T 0 [∂ 2 z F0 (∂ 2 x v 0 ) 2 ](s, X0 s )ds + o(ǫ) =v 0 (0, x) + ǫ 2 E ∂ x ĝ(T, X0 T ) ỸT + o(ǫ)
where, for z ∈ R, Xz is the solution on [0, T ] of

Xz = x + • t (2∂ z F (•, z∂ 2 x v 0 (•))) 1 2 (s, Xz s )dW s , (42)
and

Ỹ := ∂ z Xz | z=0 , solves Ỹ = 1 √ 2 • t ∂ x ∂ z F0 (s, X0 s ) Ỹs + ∂ 2 z F0 ∂ 2 x v 0 (s, X0 s ) ∂ z F0 (s, X0 s ) dW s .
Proof. By Corollary 3.11, each v ǫ associated to ǫ ∈ (0, ǫ • ] solves

∂ t v ǫ + ǫ -1 F (•, ǫ∂ 2 x v ǫ ) = 0.
Moreover, it follows from our assumptions and Corollary 3.11 that (•, v ǫ ) ∈ D ǫ for all ǫ ∈ (0, ǫ • ]. Then, the fact that F (•, 0) = 0 implies that

∂ t v ǫ + ∂ z F0 ∂ 2 x v ǫ + 1 2 ǫ∂ 2 z F0 (∂ 2 x v ǫ ) 2 = O(ǫ 2 ),
O(ǫ) =∂ t ∆v ǫ + ∂ z F0 ∂ 2 x ∆v ǫ + 1 2 ∂ 2 z F0 (∂ 2 x v 0 ) 2 + 1 2 ǫ 2 ∂ 2 z F0 (∂ 2 x ∆v ǫ ) 2 + ǫ∂ 2 z F0 ∂ 2 x ∆v ǫ ∂ 2 x v 0 ,
in which O(ǫ) is uniform on [0, T ) × R. By Theorem 3.10, Remark 4.2, and the same arguments as in this remark,

(∂ 2 x ∆v ǫ , ∂ 2 z F0 , ∂ 2 x v 0 ) 0<ǫ≤ǫ• is locally bounded. Since ∆v ǫ (T, •) = 0, it follows that ∆v ǫ (0, x) = E 1 2 T 0 [∂ 2 z F0 (∂ 2 x v 0 ) 2 ](s, X0 s )ds + O(ǫ).
Hence, ∆v := lim ǫ→0 ∆v ǫ is given by

∆v(0, x) = E 1 2 T 0 [∂ 2 z F0 (∂ 2 x v 0 ) 2 ](s, X0 s )ds . (43) Moreover, ∂ x v 0 satisfies ∂ t (∂ x v 0 ) + ∂ x ∂ z F0 ∂ 2 x v 0 + ∂ z F0 ∂ 2 x (∂ x v 0 ) = 0, (44)
recall Remark 4.2. Applying Itô's lemma to ∂ x v 0 (t, X0 t ) Ỹt , we obtain

d(∂ x v 0 (t, X0 t ) Ỹt ) = ∂ t ∂ x v 0 (t, X0 t ) Ỹt dt + ∂ 2 x v 0 (t, X0 t ) Ỹt d X0 t + ∂ x v 0 (t, X0 t )d Ỹt + ∂ 2 x v 0 (t, X0 t )d Ỹ , X0 t + 1 2 ∂ 2 x (∂ x v 0 (t, X0 t )) Ỹt d X0 t = ∂ t ∂ x v 0 (t, X0 t ) + ∂ 2 x v 0 (t, X0 t )∂ x ∂ z F0 (t, X0 t ) + ∂ 2 x (∂ x v 0 (t, X0 t ))∂ z F0 (t, X0 t ) Ỹt dt + ∂ 2 z F0 (t, X0 t )(∂ 2 x v 0 (t, X0 t )) 2 dt + ∂ 2 x v 0 (t, X0 t ) Ỹt d X0 t + ∂ x v 0 (t, X0 t )d Ỹt = ∂ 2 z F0 (t, X0 t )(∂ 2 x v 0 (t, X0 t )) 2 dt + ∂ 2 x v 0 (t, X0 t ) Ỹt d X0 t + ∂ x v 0 (t, X0 t )
d Ỹt where we use (44) to get the last equality. Therefore, taking expectation on both sides, we have

E ∂ x v 0 (T, X0 T ) ỸT = E T 0 [∂ 2 z F0 (∂ 2 x v 0 ) 2 ](s, X0 s )ds , which leads to ∆v(0, x) = 1 2 E ∂ x v 0 (T, X0 T ) ỸT = 1 2 E ∂ x ĝ(T, X0 T ) ỸT .
Remark 4.4. For later use, note that the above proof implies that ∆v defined in (43) satisfies

∂ t ∆v + ∂ z F0 ∂ 2 x ∆v + 1 2 ∂ 2 z F0 (∂ 2 x v 0 ) 2 = 0 on [0, T ) × R.
Remark 4.5. A more tractable formulation can be obtained in the particular case where (∂ z F0 , ∂ 2 z F0 ) = (λ 1 , λ 2 ) is constant and ∂ x ∂ z F0 = 0. This is the case in the model of [START_REF] Bouchard | Hedging of covered options with linear market impact and gamma constraint[END_REF], see Example 2.1, whenever σ • and f are constant, see e.g. Remark 4.1. Then,

∂ x v 0 (•, X0 ) = ∂ x v 0 (0, x) + • 0 √ 2λ 1 ∂ 2 x v 0 (s, X0 s )dW s by (44), so that ǫ 2 E T 0 [∂ 2 z F0 (∂ 2 x v 0 ) 2 ](s, X0 s )ds = ǫλ 2 4λ 1 E T 0 [ 2λ 1 ∂ 2 x v 0 (s, X0 s )] 2 ds = ǫλ 2 4λ 1 E (∂ x ĝ( X0 T ) -∂ x v 0 (0, x)) 2 = ǫλ 2 4λ 1 E (∂ x ĝ( X0 T ) -E[∂ x ĝ( X0 T )]) 2 = ǫλ 2 4λ 1 Var ∂ x ĝ( X0 T )
and the computation of the gamma ∂ 2 x v 0 is not required. Such a formulation does not seem available in general.

The expansion of Proposition 4.3 leads to a natural approximate hedging strategy. The result is stated in terms of the function ∆v introduced in the proof of Proposition 4.3, see (43). Proposition 4.6. Assume that the conditions of Proposition 4.3 hold and that

(i) ∂ 2 z F0 ∈ C 1,2 b ([0, T ] × R) ∩ C 0,4 b ([0, T ] × R), (ii) (t, x, z) ∈ [0, T ]×R×R → 1 2ǫ σ 2 (t,
x, ǫz) is bounded and uniformly Lipschitz in its two last components, uniformly in ǫ ∈ (0, ǫ 0 ].

Then, there exists a constant C > 0 such that, for each ǫ ∈ (0, ǫ 0 ] and x ∈ R,

|V ǫ,0,x,v ǫ ,φ ǫ T -ĝ(X ǫ,0,x,φ ǫ T )| ≤ Cǫ 2 in which v ǫ := v 0 (0, x) + ǫ∆v(0, x) and φ ǫ = (y ǫ , b ǫ , γ ǫ ) ∈ A with y ǫ = ∂ x (v 0 + ǫ∆v)(0, x), b ǫ = ∂ t + 1 2ǫ σ 2 (•, ǫ∂ 2 x (v 0 + ǫ∆v))∂ 2 x ∂ x (v 0 + ǫ∆v)(•, X ǫ,0,x,φ ǫ ), γ ǫ = ∂ 2 x (v 0 + ǫ∆v)(•, X ǫ,0,x,φ ǫ ).
Proof. For ease of notations, we write σ ǫ for ǫ -1 2 σ(•, ǫ•). We let Y ǫ = ∂ x (v 0 + ǫ∆v)(•, X ǫ,0,x,φ ǫ ), and only write X ǫ for X ǫ,0,x,φ ǫ in the following. Note that (42), (43), (i) and [START_REF] Sircar | Generalized black-scholes models accounting for increased market volatility from hedging strategies[END_REF]

imply that ∆v ∈ C 1,2 b ([0, T ] × R) ∩ C 0,4 b ([0, T ] × R).
Then, the dynamics are well-defined thanks to Remark 4.2, and φ ǫ ∈ A. Set

(X ǫ T ) -v ǫ - T 0 Y ǫ t dX ǫ t - T 0 F ǫ (t, X ǫ t , γ ǫ t )dt =v 0 (T, X ǫ T ) + ǫ∆v(T, X ǫ T ) -v 0 (0, x) -ǫ∆v(0, x) - T 0 Y ǫ t dX ǫ t - T 0 F ǫ (•, ∂ 2 x (v 0 + ǫ∆v))(t, X ǫ t )dt = T 0 Fǫ (•, ∂ 2 x (v 0 + ǫ∆v)) -∂ z F0 ∂ 2 x (v 0 + ǫ∆v) - ǫ 2 ∂ 2 z F0 (∂ 2 x v 0 ) 2 (t, X ǫ t )dt.
Recalling that ( 15) is assumed to hold for Fǫ , uniformly in ǫ ∈ (0, ǫ • ], that ∂ 2 x v 0 and ∂ 2

x ∆v are bounded, as well as ( 18), a second order Taylor expansion implies

Fǫ (•, ∂ 2 x (v 0 + ǫ∆v)) -∂ z F0 ∂ 2 x (v 0 + ǫ∆v) - ǫ 2 ∂ 2 z F0 (∂ 2 x v 0 ) 2 = O(ǫ 2 ), in which O(ǫ 2 ) is uniform on [0, T ] × R.

Dual representation formula in the convex case

In this last section, we assume that z ∈ R → F (t, x, z) is convex and bounded from below, (45) lim

z→γ(t,x) ∂ z F (t, x, z) = ∞ for all (t, x) ∈ [0, T ] × R. (46)
Note that the second assumption is automatically satisfied if γ < ∞, since in this case lim z→γ(t,x) F (t, x, z) = ∞. Both are satisfied is the model studied in [START_REF] Bouchard | Hedging of covered options with linear market impact and gamma constraint[END_REF], see Remark 3.1.

Whenever γ < ∞, let us now use the extension F (•, z) := ∞ for z ∈ [γ, ∞) and define the Fenchel-Moreau transform

F * (•, v) := sup z∈R 1 2 vz -F (•, z) , v ∈ R.
The conditions (45) and (46) ensure that F * (t, x, •) is finite on R + and takes the value +∞ on R -\ {0}. The function F being lower-semicontinuous on R + , convex and proper in its last argument, it follows that

F (•, z) = sup s∈R+ 1 2 s 2 z -F * (•, s 2 ) . ( 47
) F * (•, 2∂ z F (•, z)) = ∂ z F (•, z)z -F (•, z), for z < γ. ( 48
) Remark 5.1. It follows from (47) that a function V is a viscosity supersolution (resp. subsolution) on [0, T ) × R of min{-∂ t ϕ -F (•, ∂ 2 x ϕ) , γ -∂ 2 x ϕ} = 0 if and only if it is a viscosity supersolution (resp. subsolution) on [0, T ) × R of inf s∈R+ F * (•, s 2 ) -∂ t ϕ - 1 2 s 2 ∂ 2 x ϕ = 0. (49)
This suggests, in the spirit of [START_REF] Soner | Dual formulation of second order target problems[END_REF], that v admits a dual formulation in terms of an optimal control problem.

Theorem 5.2. Assume that (45) and (46) hold. Let S denote the collection of non-negative bounded predictable processes. Then, for all (t, x)

∈ [0, T ) × R, v(t, x) = sup s∈S E ĝ(X t,x,s T ) - T t F * (s, X t,x,s s , s 2 s )ds (50) = sup s∈S E g(X t,x,s T ) - T t F * (s, X t,x,s s , s 2 s )ds in which X t,x,s = x + • t s s dW s , s ∈ S.
If moreover the conditions of Corollary 3.11 hold, then the optimum is achieved by the Markovian control ŝt,x := (2∂ z F (•, ∂ 2 x v)(•, X t,x,ŝt,x ))

1 2 .
Remark 5.3. The model studied in [START_REF] Bouchard | Hedging of covered options with linear market impact and gamma constraint[END_REF] corresponds to

F * (t, x, s 2 ) = 1 2 (s -σ • (t, x)) 2 f (x)
, for s ≥ 0.

See Remark 3.1. The result of Theorem 5.2 above can then be formally interpreted as follows. The larger the impact function f , the more the optimal control can deviate from the volatility associated to the model without market impact. When f tends to 0, the optimal control needs to converge to the volatility of the impact free model σ • , and one recovers the usual pricing rule at the limit.

Proof of Theorem 5.2. 1. We first prove the first equality in (50) in the case where the conditions of Corollary 3.11 hold. Let v denote the right-hand side of (50). Recalling from Remark 5.1, Corollary 3.11 and Theorem 3.10 that v is a smooth supersolution of (49), we deduce that v ≥ v by a simple verification argument. Let now X be the solution of

X = x + • t (2∂ z F (•, ∂ 2 x v)(s, Xs )) 1 2 dW s .
It is well defined, recall Corollary 3.11, Theorem 3.10, ( 20) and ( 15), and corresponds to X t,x,ŝ with ŝ := (2∂ z F (•, ∂ 2 x v)(•, X))

1 2 , which is bounded. Moreover, (48) implies that v(t, x) = E ĝ( XT ) -T t F * (s, Xs , ŝ2 s )ds , which shows that v ≤ v since ŝ is bounded.

2. We now extend the first equality in (50) to the general case. Let {Φ ε , Φ ε } be as in the proof of Theorem 3.5 at the end of Section 3, and let u ε and u ε be the (smooth) solutions to (Eq 0 ) associated to Φ ε and Φ ε respectively, as in Theorem 3.5. Then Φ ε ≤ ĝ ≤ Φ ε , u ε ≤ v ≤ u ε and (u εu ε , Φ ε -Φ ε ) ε>0 converges uniformly to 0 as ε → 0. Define v ε and v ε as v but with Φ ε and Φ ε in place of ĝ. Then, v ε ≤ v ≤ v ε and (v εv ε ) ε>0 converges uniformly to 0 as ε → 0. Since, by 1., (v ε , v ε ) = (u ε , u ε ), the required result follows. In view of 2., we know that ṽ is bounded from above by v. Since F * (•, 0) + and g -are bounded, see ( 45) and [START_REF] Crandall | User's guide to viscosity solutions of second order partial differential equations[END_REF], it is also bounded from below, by a constant. Then, it follows from [START_REF] Bouchard | Weak dynamic programming principle for viscosity solutions[END_REF] that the lower-semicontinuous enveloppe ṽ * of ṽ is a viscosity supersolution of (49) such that ṽ * (T, •) ≥ g, recall [START_REF] Crandall | User's guide to viscosity solutions of second order partial differential equations[END_REF]. It is in particular a supersolution of γ -∂ 2 x ϕ ≥ 0 on [0, T ) × R, by Remark 5.1. Then, the same arguments as in [7, Step 3.b., proof of Theorem 3.16] imply that ṽ * (T, •) ≥ ĝ. By [START_REF] Bouchard | Weak dynamic programming principle for viscosity solutions[END_REF] again, we also have that ṽ(t, x) ≥ E ṽ * (T, X t,x,s T ) -T t F * (s, X t,x,s s , s 2 s )ds , for any s ∈ S.

Hence, ṽ(t, x) ≥ sup s∈S E ĝ(X t,x,s T ) -T t F * (s, X t,x,s s , s 2 s )ds .

We conclude this section with a result showing that any optimal control control ŝ should be such that ĝ(X t,x,ŝ T ) = g(X t,x,ŝ T ). Then, (X t,x,s n T ) n≥1 is tight, and any limiting law ν associated to a subsequence satisfies ν(ĝ > g) = 0.

Proof. We only write X n for X t,x,s n and let

J n := E g(X n T ) - T t F * (

  only depends on Φ and L • . If Φ is globally Lipschitz, then u is also globally Lipschitz with Lipschitz constant controlled by the one of Φ. If moreover there exists α ∈ (0, 1) such that Φ ∈ C 4+α b then u ∈ C 1,4

Proposition 4 . 3 .

 43 Assume that the conditions of Corollary 3.11 hold with

in which the O(ǫ 2 )

 2 is uniform since |∂ 3 z F0 | is uniformly bounded on D ε by assumption. Let ∆v ǫ := (v ǫv 0 )/ǫ. By the above, (39) and Remark 4.2, it solves

F

  ǫ := F (•, ǫ •)/ǫ. By applying Itô's Lemma, using Remark 4.2, Remark 4.4 and the definition of Fǫ together with[START_REF] Loeper | Option pricing with linear market impact and non-linear Black and Scholes equations[END_REF], we obtain ĝ

3 .F

 3 It remains to prove the second equality in (50). Defineṽ(t, x) * (s, X t,x,s s , s 2 s )ds , (t, x) ∈ [0, T ) × R.

Proposition 5 . 4 .F

 54 Let the condition of Theorem 5.2 hold and assume that F (•, κ) is uniformly bounded on [0, T ] × R for some κ > 0. Fix (t, x) ∈ [0, T ) × R and let (s n ) n≥1 be such that v(t, x) = lim n↑∞ E g(X t,x,s n * (s, X t,x,s n s , (s n s ) 2 )ds .

  is explained briefly in Example 2.1.

	F-adapted processes (b, γ) such that
	•	•
	γ = γ 0 +	β s ds +	α s dW s
	0	0
	where (α, β) is continuous, F-adapted, and ζ := (b, γ, α, β) is essentially bounded
	by k and such that
	More precisely, given k ≥ 1, we denote by A • k the collection of continuous and

  s, X Let ν n be the law associated to X n T . The above shows that (ν n ) n≥1 is tight. Let us consider a subsequence (ν n k ) k≥1 that converges to some law ν. If ν(ĝ > g) > 0, then one can find δ > 0 such that E[ĝ(X n k

								n s , (s n s ) 2 )ds ,
	n ≥ 1. Then, (11) and (45) imply that one can find C > 0 such that
	-C ≤ E[C +	κ 4	|X n T | 2 -	t	T	κ 2	(s n s ) 2 ds + T sup F (•, κ)]
	≤ E[C -	t	T	κ 4	(s n

s ) 2 ds + T sup F (•, κ)]. Hence, sup n≥1 E[ T t (s n s ) 2 ds] < ∞. T )] ≥ E[g(X n k T )] + δ for all k ≥ 1 large enough, which would imply that lim k→∞ E ĝ(X n k T ) -T t F * (s, X n k s , (s n k s ) 2 )ds ≥ lim k→∞ J n k + δ,

a contradiction to Theorem 5.2.

More precisely: the value of the cash plus the number of stocks in the portofolio times the current value of the stocks.

The continuous time version is obtained by considering the limit dynamics of a discrete time trading model, as the speed of trading goes to infinity.

Obviously, this is only a theoretical value, the liquidation value of the portfolio being different.

ANR Grant CAESARS (ANR-15-CE05-0024), Initiative de Recherche "Méthodes non-linéaires pour la gestion des risques financiers" sponsored by AXA Research Fund. † Monash University, School of Mathematical Sciences & Centre for Quantitative Finance and Investment Strategies (CQFIS), Partially supported by the ETH Foundation, Swiss Finance Institute and the Swiss National Foundation through SNF 200020-172815. Research supported by Singapore MOE AcRF Grants R-146-000-219-112 and R-146-000-255-114.


D Bourcier

J P Féraud

D Ode

D Colson

F Puel

Computational Filterability and dewatering determination based on shape and size distribution of solid suspensions

Keywords: Deliquoring, Pressure Filtration, Modeling, Solid-liquid-separation, Suspension 1. Introduction

Dewatering after filtration is an important issue, as it determines the transferability and the final cost of dewatering for the product. Any optimization in the dewatering step will improve the process. This paper shows the effect of particle properties such as shape factor and size distribution on dewatering efficiency. An experimental plan was established to show the influence of the working pressures (filtration and dewatering pressures) on residual moisture and dewatering time. This plan was repeated on four calcium carbonate particle shapes: needles, plates, spheres, and cubes. The first data from these experiments showed that the volume shape factor of the particle has no effect on cake behavior during dewatering. The same-shaped iso-response curves of residual moisture as a function of the filtration pressure and the dewatering pressure were obtained for each particle shape. It was also shown that raising the dewatering pressure can in fact lead to a wetter cake if this pressure is higher than the cake filtration pressure. This fact points out a counterproductive phenomenon that the industrial operator has to avoid. The experimental considerations were compared to a model for dewatering based on work by Tarleton and Wakeman combined with a specially-developed filtration model. This combination enabled the computation of cake resistance and compressibility, as well as cake dewatering behavior, using easily measurable parameters. It can serve the industrial process in the choice of the appropriate liquid-solid separation device, and provide information as to the pressure conditions necessary to obtain the wettest cake possible for a time-fixed dewatering or to see if a wet target can be reached and in how much time.

The deliquoring helps to cut down the costs of drying and improves the transferability of the cake between the different steps of the process. Knowing the optimal deliquoring pressure will also improve the productivity of the process and increase the profits of an industrial application.

The residual moisture after filtration has to be linked with particle properties such as Particle size Distribution or shape factor.

This paper aims at studying the variation of the behavior of the cake during deliquoring as a function of the shape factor. A model for dewatering based on work by Tarleton and Wakeman (2007) combined with a specially-developed filtration model will also be presented. This combination enabled the computation of cake resistance and compressibility, as well as cake dewatering behavior, using easily measurable parameters.

Simulation procedure

The simulation of deliquoring is based on work by Tarleton and Wakeman (2007). The first step is to consider Darcy's law combined with Ergun's equation (Equation 1):

𝑘 = 1 𝛼𝜌 (1 -𝜀) = 𝜀 𝐷 180(1 -𝜀) (1)
Thanks to equation 1, the effective particle diameter can be calculated:

𝐷 = 180(1 -𝜀) 𝛼𝜌 𝜀 = 13.4 (1 -𝜀) 𝛼𝜌 𝜀 (2)
Wakeman proposes a correlation to calculate the irreducible saturation thanks to this effective diameter and a calculation of the capillary number (Equation 3 and 4):

𝐶𝑎 = 𝜀 𝐷 (𝜌 𝑔𝐿 + ∆𝑃) (1 -𝜀) 𝐿𝜎 (3) 𝑆 ∝ = 0.155(1 + 0.031𝐶𝑎 . ) (4)
If the objective is to calculate the residual moisture obtained in a cake after a fixed time and pressure of deliquoring, the dimensionless time and pressure are calculated:

𝜏 = 𝑘𝑝 𝜇𝜀(1 -𝑆 ) 𝐿 𝑡 (5) 𝑝 * = ∆𝑃 𝑝 (6)
The product 𝜏𝑝 * can be linked with the reduced saturation thanks to an experimental curve. The reduced saturation is obtained and the saturation S calculated. This saturation gives access to the residual moisture in the cake:

𝑆 = 𝑆 -𝑆 1 -𝑆 (7) 𝑆 = (1 -𝜀) 𝜀 𝜌 𝜌 𝐻 1 -𝐻 (8)
If the residual moisture is the target and the deliquoring time needed is wanted, the calculation can be done by calculating Sr, having the product 𝜏𝑝 * thanks to the curve and calculating the deliquoring time td.

Experimental considerations

A study of the influence of the particle shape on cake deliquoring is proposed thanks to multiple synthesis calcium carbonate. Four shapes were obtained: spheres, cubes, platelets and needles. The values of cake residual moisture as a function of filtration and deliquoring pressure were measured. For instance, Figure 2 presents the iso-responses curves for calcium carbonate spheres. The table 1 summarizes the results for each shape. For each shape, a linear polynomial regression is proposed:

𝐻 = 𝑎 + 𝑎 𝑋 + 𝑎 𝑋 (9)
where X1 stands for the centered variable for deliquoring pressure and X2 for the centered variable for filtration pressure.

The table 2 gives the measured coefficients a1 and a2 for each shape: An analysis of coefficients shows that the influence of deliquoring pressure is more important for shapes close to the sphere and that the influence of filtration pressure is more important for shapes far from the sphere.

Comparison experiments/modeling

The model presented in section 2 and the experiments presented in section 3 were compared in figure 3. The surface represents the difference between the modeled results and the experiments. The correlation is good and better at low pressure.

Conclusion

The correlation between the model and the experiments is good. The effect of the shape on deliquoring has to be taken into account. This comparison gives the industrial a great tool to adapt the filtration and deliquoring pressures to the product needed, in order to improve the productivity and the dryness of the solid at the end of the separation process. The optimal pressures can be found with the Filtration Performance Test by FlowerSEP.

Figure 1 :

 1 Figure 1: Correlation between reduced saturation and the product of the time and pressure adimensional

Figure 2 :

 2 Figure 2: Isoresponses curves of residual moisture as a function of deliquoring and filtration pressure

Figure 3 :

 3 Figure 3: Comparison modelling-experiments for calcium carbonate spheres

Table 1 :

 1 Experimental results of residual moisture measurements

	Shape	Deliquoring Pressure	Filtration Pressure 1 3	5
		1	23.3	24.0	24.6
	Spheres	3	18.8	19.8	20.3
		5	9.0	9.7	10.1
		1	21.0	25.7	31.4
	Cubes	3	14.8	19.3	25.1
		5	9.0	14.5	18.4
			34.3	38.2	43.0
	Platelets	3	32.8	36.8	40.2
		5	30.2	34.7	39.1
		1	12.0	21.6	32.0
	Needles	3	8.4	18.1	27.9
		5	6.2	15.6	24.3

Table 2 :

 2 Polynomial coefficients for each shape

	Shape	Shape factor	a0	a1	a2	a1+a2
	Spheres	0.92	17.74	-3.6	0.33	-3.27
	Cubes	0.81	19.91	-3.02	2.51	-0.51
	Platelets	0.3	36.59	-0.96	2.08	1.12
	Needles	0.17	18.46	-1.63	4.80	3.17