B Bouchard

G Loeper

Y Zou

Hedging of covered options with linear market impact and gamma constraint

Keywords: Hedging, Price impact, Stochastic target AMS 2010 Subject Classification: 91G20, 93E20, 49L20

Within a financial model with linear price impact, we study the problem of hedging a covered European option under gamma constraint. Using stochastic target and partial differential equation smoothing techniques, we prove that the super-replication price is the viscosity solution of a fully non-linear parabolic equation. As a by-product, we show how ε-optimal strategies can be constructed. Finally, a numerical resolution scheme is proposed.

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 Market Impact and Non-Linear Black and Scholes PDEs[END_REF], the authors in [START_REF] Bouchard | Almost-sure hedging with permanent price impact[END_REF] considered a financial market with permanent price impact, in which the impact function behaves as a linear function (around the origin) in the number of bought 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 the delta-hedging is non-negligible compared to the average daily volume traded on the underlying asset. Hence, the delta-hedging strategy will have an impact on the price dynamics, and will also incur liquidity costs. The linear impact models studied in [START_REF] Abergel | Pricing and hedging contingent claims with liquidity costs and market impact[END_REF][START_REF] Bouchard | Almost-sure hedging with permanent price impact[END_REF][START_REF] Loeper | Option Pricing with Market Impact and Non-Linear Black and Scholes PDEs[END_REF] incorporate both the effects into the pricing and hedging of the derivative, while maintaining the completeness of the market (up to a certain extent) and eventually leading to exact replication strategies. As in perfect market models, this approach can provide an approximation of the real market conditions that can be used by practitioners to help them in the design of a suitable hedge in a systematic way, without having to rely on an ad hoc risk criterion.

In [START_REF] Bouchard | Almost-sure hedging with permanent price impact[END_REF], the authors considered the hedging of a cash-settled European option: at inception the option seller has to build the initial delta-hedge, and conversely at maturity the hedge must be liquidated to settle the final claim in cash. It is shown therein that the price function of the optimal super-replicating strategy no longer solves a linear parabolic equation, as in the classical case, rather a quasi-linear one. Moreover, the hedging strategy consists in following a modified delta-hedging rule where the delta is computed at the "unperturbed" value of the underlying, i.e., the one the underlying would have if the trader's position were liquidated immediately.

The approach and the results obtained in [START_REF] Bouchard | Almost-sure hedging with permanent price impact[END_REF] thus differ substantially from [START_REF] Abergel | Pricing and hedging contingent claims with liquidity costs and market impact[END_REF][START_REF] Loeper | Option Pricing with Market Impact and Non-Linear Black and Scholes PDEs[END_REF]. Indeed, while in [START_REF] Abergel | Pricing and hedging contingent claims with liquidity costs and market impact[END_REF][START_REF] Loeper | Option Pricing with Market Impact and Non-Linear Black and Scholes PDEs[END_REF] the impact model considered is the same, the control problem is different in the sense that it is applied to the hedging of covered options. This refers to situations where the buyer of the option delivers at inception the required initial delta position, and accepts a mix of stocks (at their current market price) and cash as payment of the final claim, which eliminates the cost incurred by the initial and final hedge. Quite surprisingly, this is not a genuine approximation of the problem studied in [START_REF] Bouchard | Almost-sure hedging with permanent price impact[END_REF]. The question of the initial and final hedge is fundamental, to the point that the structure of the pricing question is completely different: in [START_REF] Bouchard | Almost-sure hedging with permanent price impact[END_REF] the equation is quasi-linear, while it is fully non-linear in [START_REF] Abergel | Pricing and hedging contingent claims with liquidity costs and market impact[END_REF][START_REF] Loeper | Option Pricing with Market Impact and Non-Linear Black and Scholes PDEs[END_REF].

As opposed to [START_REF] Bouchard | Almost-sure hedging with permanent price impact[END_REF], authors in [START_REF] Abergel | Pricing and hedging contingent claims with liquidity costs and market impact[END_REF][START_REF] Loeper | Option Pricing with Market Impact and Non-Linear Black and Scholes PDEs[END_REF] use a verification argument to build an exact replication strategy. Due to the special form of the non-linearity, the equation is ill-posed when the solution does not satisfy a gamma-type constraint. The aim of the current paper is to provide a direct characterization via stochastic target techniques, and to incorporate right from the beginning a gamma constraint on the hedging strategy.

The super-solution property can be proved by (essentially) following the arguments of [START_REF] Cheridito | The multi-dimensional superreplication problem under gamma constraints[END_REF]. The sub-solution characterization is much more difficult to obtain. Actually, we could not prove the required geometric dynamic programming principle, unlike [START_REF] Cheridito | The multi-dimensional superreplication problem under gamma constraints[END_REF], because of the strong interaction between the hedging strategy and the underlying price process due to the market impact. Instead, we use the smoothing technique developed in [START_REF] Bouchard | Stochastic target games and dynamic programming via regularized viscosity solutions[END_REF]. We construct a sequence of smooth super-solutions which, by a verification argument, provide upper-bounds for the super-hedging price. As they converge to a solution of the targeted pricing equation, a comparison principle argument implies that their limit is the super-hedging price. As a by-product, this construction provides explicit ε-optimal hedging strategies. We also provide a comparison principle and a numerical resolution scheme. For simplicity, we first consider a model that has only permanent price impact, we explain in Section 3 why adding a resilience effect does not affect our analysis. Note that this is because the resilience effect we consider here has no quadratic variation, as opposed to [START_REF] Abergel | Pricing and hedging contingent claims with liquidity costs and market impact[END_REF], in which the resilience can destroy the parabolicity of the equation, and renders the exact replication non optimal.

We close this introduction by pointing out some related references. [START_REF] Çetin | Liquidity risk and arbitrage pricing theory[END_REF] incorporates liquidity costs but no price impact, the price curve is not affected by the trading strategy. It can be modified by adding restrictions on admissible strategies as in [START_REF] Çetin | Option hedging for small investors under liquidity costs[END_REF] and [START_REF] Soner | The dynamic programming equation for second order stochastic target problems[END_REF]. This leads to a modified pricing equation, which exhibits a quadratic term in the second order derivative of the solution, and renders the pricing equation fully non-linear, and even not unconditionally parabolic. Other articles focus on the derivation of the price dynamics through clearing condition, see e.g., [START_REF] Frey | Perfect option hedging for a large trader[END_REF], [START_REF] Sircar | Generalized black-scholes models accounting for increased market volatility from hedging strategies[END_REF], [START_REF] Schönbucher | The feedback effects of hedging in illiquid markets[END_REF] in which the supply and demand curves arise from "reference" and "program" traders (i.e., option hedgers); leading to a modified price dynamics, but without taking into account the liquidity costs, see also [START_REF] Liu | Option pricing with an illiquid underlying asset market[END_REF]. Finally, the series of papers [START_REF] Soner | Superreplication under gamma constraints[END_REF], [START_REF] Cheridito | The multi-dimensional superreplication problem under gamma constraints[END_REF], [START_REF] Soner | The dynamic programming equation for second order stochastic target problems[END_REF] addresses the liquidity issue indirectly by imposing bounds on the "gamma" of admissible trading strategies, no liquidity cost or price impact are modeled explicitly.

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 P -a.s. sense. We use the convention x/0 = sign(x) × ∞ with sign(0) = +.

Model and hedging problem

This section is dedicated to the derivation of the dynamics and the description of the gamma constraint. We also explain in detail how the pricing equation can be obtained and state our main result.

Impact rule and discrete time trading dynamics

We consider the framework studied in [START_REF] Bouchard | Almost-sure hedging with permanent price impact[END_REF]. Namely, the impact of a strategy on the price process is modeled by an impact function f : the price variation due to buying a (infinitesimal) number δ ∈ R of shares is δf (x), given that the price of the asset is x before the trade. The cost of buying the additional δ units is

δx + 1 2 δ 2 f (x) = δ δ 0 1 δ (x + ιf (x))dι, in which δ 0 1 δ (x + ιf (x))dι
can be interpreted as the average cost for each additional unit.

Between two trading instances τ 1 , τ 2 with τ 1 ≤ τ 2 , the dynamics of the stock is given by the strong solution of the stochastic differential equation

dX t = µ(X t )dt + σ(X t )dW t .
Throughout this paper, we assume that f ∈ C 2 b and inf f > 0, (µ, σ) is Lipschitz and bounded, inf σ > 0.

(1.1)

As in [START_REF] Bouchard | Almost-sure hedging with permanent price impact[END_REF], the number of shares the trader would like to hold is given by a continuous Itô process Y of the form

Y = Y 0 + • 0 b s ds + • 0 a s dW s . (1.2) We say 1 that (a, b) belongs to A • k if (a, b) is continuous, F-adapted, a = a 0 + • 0 β s ds + • 0 α s dW s
where (α, β) is continuous, F-adapted, and ζ := (a, b, α, β) is essentially bounded by k and such that

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 .
To derive the continuous time dynamics, we first consider a discrete time setting and then pass to the limit. In the discrete time setting, the position is re-balanced only at times t n i := iT /n, i = 0, . . . , n, n ≥ 1. In other words, the trader keeps the position Y t n i in stocks over each time interval [t n i , t n i+1 ). Hence, his position in stocks at t is

Y n t := n-1 i=0 Y t n i 1 {t n i ≤t<t n i+1 } + Y T 1 {t=T } , (1.3)
and the number of shares purchased at

t n i+1 is δ n t n i+1 := Y t n i+1 -Y t n i .
Given our impact rule, the corresponding dynamics for the stock price process is

X n = X 0 + • 0 µ(X n s )ds + • 0 σ(X n s )dW s + n i=1 1 [t n i ,T ] δ n t n i f (X n t n i -), (1.4)
in which X 0 is a constant. The portfolio process is described as the sum V n of the amount of cash held and the potential wealth Y n X n associated to the position in stocks:

V n = cash position + Y n X n .
It does not correspond to the liquidation value of the portfolio, except when Y n = 0. This is due to the fact that the liquidation of Y n stocks does not generate a gain equal to Y n X n , because of the price impact. However, one can infer the exact composition in cash and stocks of the portfolio from the knowledge of the couple

(V n , Y n ).
Throughout this paper, we assume that the risk-free interest rate is zero (for ease of notations). Then,

V n = V 0 + • 0 Y n s-dX n s + n i=1 1 [t n i ,T ] 1 2 (δ n t n i ) 2 f (X n t n i -). (1.5)
This wealth equation is derived as in [START_REF] Bouchard | Almost-sure hedging with permanent price impact[END_REF] following elementary calculations. The last term of the right-hand side comes from the fact that, at time t n i , δ n t n i shares are bought at the average execution price X n

t n i -+ 1 2 δ n t n i f (X n t n i -)
, and the stock's price ends at X n

t n i -+ δ n t n i f (X n t n i -)
, whence the additional profit term. However, one can check that a profitable round trip trade can not be built, see [START_REF] Bouchard | Almost-sure hedging with permanent price impact[END_REF]Remark 3].

Remark 1.1. Note that in this work we restrict ourselves to a permanent price impact, no resilience effect is modeled. We shall explain in Section 3 below why taking resilience into account does not affect our analysis. See in particular Proposition 3.1.

Continuous time trading dynamics

The continuous time trading dynamics is obtained by passing to the limit n → ∞, i.e., by considering strategies with increasing frequency of rebalancement.

Proposition 1.2. [4, Proposition 1] Let Z := (X, Y, V ) where Y is defined as in (1.2) for some (a, b) ∈ A • , and (X, V ) solves X = X 0 + • 0 σ(X s )dW s + • 0 f (X s )dY s + • 0 (µ(X s ) + a s (σf )(X s ))ds = X 0 + • 0 σ as X (X s )dW s + • 0 µ as,bs X (X s )ds (1.6) with σ as X := (σ + a s f ) , µ as,bs X := (µ+b s f + a s σf ), and
V = V 0 + • 0 Y s dX s + 1 2 • 0 a 2 s f (X s )ds.
(1.7)

Let Z n := (X n , Y n , V n ) be defined as in (1.4)-(1.3)-(1.5).
Then, there exists a constant C > 0 such that

sup [0,T ] E |Z n -Z| 2 ≤ Cn -1
for all n ≥ 1.

For the rest of the paper, we shall therefore consider (1.7)-(1.6) for the dynamics of the portfolio and price processes.

Remark 1.3. As explained in [START_REF] Bouchard | Almost-sure hedging with permanent price impact[END_REF], the previous analysis could be extended to a non-linear impact rule in the size of the order. To this end, we note that the continuous time trading dynamics described above would be the same for a more general impact rule δ → F (x, δ) whenever it satisfies F (x, 0)= ∂ 2 δδ F (x, 0) = 0 and ∂ δ F (x, 0) = f (x). For our analysis, we only need to consider the value and the slope of the impact function at the origin.

Hedging equation and gamma constraint

Given φ = (y, a, b) ∈ R × A • and (t, x, v) ∈ [0, T ] × R × R, we now write (X t,x,φ , Y t,φ , V t,x,v,φ ) for the solution of (1.6)-(1.2)-(1.7) associated to the control (a, b) with time-t initial condition (x, y, v).

In this paper, we consider covered options, in the sense that the trader is given at the initial time t the number of shares Y t = y required to launch his hedging strategy and can pay the option's payoff at T in cash and stocks (evaluated at their time-T value). Therefore, he does not exert any immediate impact at time t nor T due to the initial building or final liquidation of his position in stocks. Recalling that V stands for the sum of the position in cash and the number of held shares multiplied by their price, the super-hedging price at time t of the option with payoff g(X t,x,φ T ) is defined as

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) is the set of elements (a, b) ∈ A • such that φ := (y, a, b) satisfies V t,x,v,φ T ≥ g(X t,x,φ T ).
In order to understand what the associated partial differential equation is, let us first rewrite the dynamics of Y in terms of X:

dY t,φ t = γ at Y (X t,x,φ t )dX t,x,φ t + µ at,bt Y (X t,x,φ t )dt with γ a Y := a σ + f a and µ a,b Y := b -γ a Y µ a,b X . (1.8)
Assuming that the hedging strategy consists in tracking the super-hedging price, as in classical complete market models, then one should have V t,x,v,φ = v(•, X t,x,φ ). If v is smooth, recalling (1.6)-(1.7) and applying Itô's lemma twice implies

Y t,φ = ∂ x v(•, X t,x,φ ) , γ a Y (X t,x,φ ) = ∂ 2 xx v(•, X t,x,φ ), (1.9)
and

1 2 a 2 f (X t,x,φ ) = ∂ t v(•, X t,x,φ ) + 1 2 (σ a X ) 2 (X t,x,φ )∂ 2 xx v(•, X t,x,φ ). (1.10)
Then, the right-hand side of (1.9) combined with the definition of γ a Y leads to

a = σ∂ 2 xx v(•, X t,x,φ ) 1 -f ∂ 2 xx v(•, X t,x,φ ) , σ a X = σ 1 -f ∂ 2 xx v(•, X t,x,φ )
, and (1.10) simplifies to

-∂ t v - 1 2 σ 2 (1 -f ∂ 2 xx v) ∂ 2 xx v (•, X t,x,φ ) = 0 on [t, T ). (1.11)
This is precisely the pricing equation obtained in [START_REF] Abergel | Pricing and hedging contingent claims with liquidity costs and market impact[END_REF][START_REF] Loeper | Option Pricing with Market Impact and Non-Linear Black and Scholes PDEs[END_REF]. Equation (1.11) needs to be considered with some precautions due to the singularity at f ∂ 2 xx v = 1. Hence, one needs to enforce that 1 -f ∂ 2 xx v does not change sign. We choose to restrict the solutions to satisfy 1 -f ∂ 2 xx v > 0, since having the opposite inequality would imply that a does not have the same sign as ∂ 2 xx v, so that, having sold a convex payoff, one would sell when the stock goes up and buy when it goes down, a very counter-intuitive fact.

In the following, we impose that the constraint

-k ≤γ a Y (X t,x,φ ) ≤ γ(X t,x,φ ) , on [t, T ] P -a.e., (1.12)
should hold for some k ≥ 0, in which γ is a bounded continuous map satisfying

ι ≤ γ ≤ 1/f -ι, for some ι > 0. (1.13)
We denote by A k,γ (t, x) the collection of elements (a, b) ∈ A • k such that (1.12) holds and define A γ (t, x) := ∪ k≥0 A k,γ (t, x).

Then, the equation (1.11) has to be modified to take this gamma constraint into account, leading naturally to

F [v γ ] := min -∂ t v γ - 1 2 σ 2 1 -f ∂ 2 xx v γ ∂ 2 xx v γ , γ -∂ 2 xx v γ = 0 on [0, T ) × R, (1.14) in which v γ is defined as v but with G γ (t, x, v, y) := G(t, x, v, y) ∩ A γ (t, x) in place of G(t, x, v, y). More precisely, v γ (t, x) := inf{v = c + yx : (c, y) ∈ R 2 s.t. G γ (t, x, v, y) = ∅}.
(1.15)

As for the T -boundary condition, we know that v γ (T, •) = g by definition. However, as usual, the constraint on the gamma in (1.14) should propagate up to the boundary and g has to be replaced by its face-lifted version ĝ, defined as the smallest function above g that is a viscosity super-solution of the equation From now on, we assume that ĝ is uniformly continuous, g is lower-semicontinuous, g -is bounded and g + has linear growth.

(1.16)

We are now in a position to state our main result. From now on v γ (T, x) stands for lim

(t , x ) → (T, x) t < T v γ (t , x )
whenever it is well defined.

Theorem 1.4. The value function v γ is continuous with linear growth. Moreover, v γ is the unique viscosity solution with linear growth of

F [ϕ]1 [0,T ) + (ϕ -ĝ)1 {T } = 0 on [0, T ] × R.
(1.17)

We conclude this section with additional remarks.

Remark 1.5. Note that ĝ can be uniformly continuous without g being continuous.

Take for instance g(x) = 1 {x≥K} with K ∈ R, and consider the case where γ > 0

is a constant. Then, ĝ(x) = [1 {x≥xo} γ 2 (x -x o ) 2 ] ∧ 1 with x o := K -(2/γ) 1 2 .
Remark 1.6. The map ĝ inherits the linear growth of g. Indeed, let c 0 , c

1 ≥ 0 be constants such that |g(x)| ≤ w(x) := c 0 + c 1 |x|. Since ĝ ≥ g by construction, we have ĝ-≤ w. On the other hand, since γ ≥ ι > 0, by (1.13), it follows from the arguments in [23, Lemma 3.1] that ĝ ≤ (w -Γ) conc + Γ, in which Γ(x) = ιx 2 /2.

Now, one can easily check by direct computations that

(w -Γ) conc = (w -Γ)(x o )1 [-xo,xo] + (w -Γ)1 [-xo,xo] c
with x o := c 1 /ι. Hence, (w -Γ) conc + Γ has the same linear growth as w.

Remark 1.7. As will appear in the rest of our analysis, one could very well introduce a time dependence in the impact function f and in γ. Another interesting question studied by the second author in [START_REF] Loeper | Option Pricing with Market Impact and Non-Linear Black and Scholes PDEs[END_REF] and [START_REF] Loeper | Solution of a fully non-linear Black and Scholes equation coming from a linear market impact model[END_REF] concerns the smoothness of the solution and how the constraint on ∂ 2 xx v gets naturally enforced by the fast diffusion arising when

1 -f ∂ 2 xx v is close to 0.
Remark 1.8 (Existence of a smooth solution to the original partial differential equation). When the pricing equation (1.17) admits smooth solutions (cf. [START_REF] Loeper | Option Pricing with Market Impact and Non-Linear Black and Scholes PDEs[END_REF] and [START_REF] Loeper | Solution of a fully non-linear Black and Scholes equation coming from a linear market impact model[END_REF]) that allow to use the verification theorem, then one can construct exact replication strategies from the classical solution. By the comparison principle of Theorem 2.11 below, this shows that the value function is the classical solution of the pricing equation, and that the optimal strategy exists and is an exact replication strategy of the option with payoff function ĝ. We will explain in Remark 2.18 below how almost optimal super-hedging strategies can be constructed explicitly even when no smooth solution exists.

Remark 1.9 (Monotonicity in the impact function). Note that the map λ ∈ R →

σ 2 (x)M 1-λM is non-decreasing on {λ : λM < 1}, for all (t, x, M ) ∈ [0, T ] × R × R.
Let us now write v γ as v f γ to emphasize its dependence on f , and consider another impact function f satisfying the same requirements as f . We denote by v f γ the corresponding super-hedging price. Then, the above considerations combined with Theorem 1.4 and the comparison principle of Theorem 2.11 below imply that v f γ ≥ v f γ whenever f ≥ f on R. The same implies that v f γ ≥ v in which v solves the heat-type equation

-∂ t ϕ - 1 2 σ 2 ∂ 2 xx ϕ = 0 on [0, T ) × R,
with terminal condition ϕ(T, •) = g (recall that ĝ ≥ g). See Section 4.2 for a numerical illustration of this fact.

Viscosity solution characterization

In this section, we provide the proof of Theorem 1.4. Our strategy is the following.

1. First, we adapt the partial differential equation smoothing technique used in [START_REF] Bouchard | Stochastic target games and dynamic programming via regularized viscosity solutions[END_REF] to provide a smooth supersolutions v ,K,δ γ of (1.17) on [δ, T ] × R, with > 0, from which super-hedging strategies can be constructed by a standard verification argument. In particular, v ,K,δ γ ≥ v γ on [δ, T ]×R. Moreover, this sequence has a uniform linear growth and converges to a viscosity solution vγ of (1.17) as δ, → 0 and K → ∞. See Section 2.1.

2. Second, we construct a lower bound v γ for v γ that is a supersolution of (1.17). It is obtained by considering a weak formulation of the super-hedging problem and following the arguments of [8, Section 5] based on one side of the geometric dynamic programming principle, see Section 2.2. It is shown that this function has linear growth as well.

3. We can then conclude by using the above and the comparison principle for (1.17) of Theorem 2.11 below:

v γ ≥ vγ but v γ ≤ v γ ≤ vγ so that v γ = vγ = v γ
and v γ is a viscosity solution of (1.17), and has linear growth.

4. Our comparison principle, Theorem 2.11 below, allows us to conclude that v γ is the unique solution of (1.17) with linear growth.

As already mentioned in the introduction, unlike [START_REF] Cheridito | The multi-dimensional superreplication problem under gamma constraints[END_REF], we could not prove the required geometric dynamic programming principle that should directly lead to a subsolution property (thus avoiding to use the smoothing technique mentioned in 1. above). This is due to the strong interaction between the hedging strategy and the underlying price process through the market impact. Such a feedback effect is not present in [START_REF] Cheridito | The multi-dimensional superreplication problem under gamma constraints[END_REF].

A sequence of smooth supersolutions

We first construct a sequence of smooth supersolutions v ,K,δ γ of (1.17) which appears to be an upper bound for the super-hedging price v γ , by a simple verification argument. For this, we adapt the methodology introduced in [START_REF] Bouchard | Stochastic target games and dynamic programming via regularized viscosity solutions[END_REF]: we first construct a viscosity solution of a version of (1.17) with shaken coefficients (in the terminology of [START_REF] Krylov | On the rate of convergence of finite-difference approximations for bellmans equations with variable coefficients[END_REF]) and then smooth it out with a kernel. The main difficulty here is that our terminal condition ĝ is unbounded, unlike [START_REF] Bouchard | Stochastic target games and dynamic programming via regularized viscosity solutions[END_REF]. This requires additional non trivial technical developments.

Construction of a solution for the operator with shaken coefficients

We start with the construction of the operator with shaken coefficients. Given > 0 and a (uniformly) strictly positive continuous map κ with linear growth, that will be defined later on, let us introduce a family of perturbations of the operator appearing in (1.17):

F κ (t, x, q, M ) := min x ∈D κ (x) min -q - σ 2 (x )M 2(1 -f (x )M ) , γ(x ) -M ,
where

D κ (x) := {x ∈ R : (x -x )/κ(x ) ∈ [-, ]}. (2.1)
For ease of notation, we set

F κ [ϕ](t, x) := F κ (t, x, ∂ t ϕ(t, x), ∂ 2 xx ϕ(t, x)),
whenever ϕ is smooth.

Remark 2.1. For later use, note that the map

M ∈ (-∞, γ(x)] → σ 2 (x)M 2(1-f (x)M )
is non-decreasing and convex, for each x ∈ R, recall (1.13). Hence,

(q, M ) ∈ R × (-∞, γ(x)] → F κ (•, q, M
) is concave and non-increasing in M , for all ≥ 0. This is fundamental for our smoothing approach to go through.

We also modify the original terminal condition ĝ by using an approximating sequence whose elements are affine for large values of |x|. Lemma 2.2. For all K > 0 there exists a uniformly continuous map ĝK and

x K ≥ K such that • ĝK is affine on [x K , ∞) and on (-∞, -x K ] • ĝK = ĝ on [-K, K] • ĝK ≥ ĝ • ĝK -Γ is concave for any C 2 function Γ satisfying ∂ 2 xx Γ = γ.
Moreover, (ĝ K ) K>0 is uniformly bounded by a map with linear growth and converges to ĝ uniformly on compact sets.

Proof. Fix a C 2 function Γ• satisfying ∂ 2 xx Γ• = γ. By definition, ĝ -Γ• is concave. Let us consider an element ∆ + (resp. ∆ -) of its super-differential at K (resp. -K). Set ĝ• K (x) :=ĝ(x)1 [-K,K] (x) + ĝ(K) + (∆ + + ∂ x Γ• (K))(x -K) 1 (K,∞) (x) + ĝ(-K) + (∆ -+ ∂ x Γ• (-K))(x + K) 1 (-∞,-K) (x).
Consider now another C 2 function Γ satisfying ∂ 2 xx Γ = γ. Since Γ• and Γ differ only by an affine map, the concavity of ĝ

• K -Γ is equivalent to that of ĝ• K -Γ• . The concavity of the latter follows from the definition of ĝ• K , as the superdiffential of ĝ• K -Γ• is non-increasing by construction. In particular, ĝ• K -Γ• ≥ ĝ -Γ• and therefore ĝ• K ≥ ĝ. We finally define ĝK by ĝK = min{ĝ • K , (2c 0 + c 1 | • | -Γ• ) conc + Γ• }, (2.2)
with c 0 > 0 and c 1 ≥ 0 such that We now set

-c 0 ≤ ĝ(x) ≤ c 0 + c 1 |x|, x ∈ R, recall
ĝ K := ĝK + (2.3)
and consider the equation

F κ [ϕ]1 [0,T ) + (ϕ -ĝ K )1 {T } = 0. (2.4)
We then choose κ and • ∈ (0, 1) such that κ ∈ C ∞ with bounded derivatives of all orders, inf κ > 0 and κ

= |ĝ K | + 1 on (-∞, -x K ] ∪ [x K , ∞), -1/ • < ∂ x κ < 1/ • , (2.5)
in which x K ≥ K is defined in Lemma 2.2. We omit the dependence of κ on K for ease of notations.

Remark 2.3. For later use, note that the condition |∂ x κ| < 1/ • ensures that the map x → x + κ(x) and x → x -κ(x) are uniformly strictly increasing for all 0 ≤ ≤ • . Also observe that x n → x and x n ∈ D κ (x n ), for all n, imply that x n converges to an element x ∈ D κ (x), after possibly passing to a subsequence. In particular, F κ is continuous.

When κ ≡ 1 and ĝ K ≡ ĝ + , (2.4) corresponds to the operator in (1.17) with shaken coefficients, in the traditional terminology of [START_REF] Krylov | On the rate of convergence of finite-difference approximations for bellmans equations with variable coefficients[END_REF]. The function κ will be used below to handle the potential linear growth at infinity of ĝ. The introduction of the additional approximation ĝ K is motivated by the fact that the proof of Proposition 2.7 below requires an affine behavior at infinity. As already mentioned, these additional complications do not appear in [START_REF] Bouchard | Stochastic target games and dynamic programming via regularized viscosity solutions[END_REF] because their terminal condition is bounded.

We now prove that (2.4) admits a viscosity solution that remains above the terminal condition ĝ on a small time interval [T -c K , T ]. As already mentioned, we will later smooth this solution out with a regular kernel, so as to provide a smooth supersolution of (1.17).

Proposition 2.4. For all ∈ [0, • ] and K > 0, there exists a unique continuous viscosity solution v ,K γ of (2.4) that has linear growth. It satisfies

v ,K γ ≥ ĝK + /2, on [T -c K , T ] × R, (2.6
)

for some c K ∈ (0, T ). Moreover, {[v ,K γ ] + , ∈ [0, • ], K > 0}
is bounded by a map with linear growth, and

{[v ,K γ ] -, ∈ [0, • ], K > 0} is bounded by sup g -.
Proof. The proof is mainly a modification of the usual Perron's method, see [START_REF] Crandall | User's guide to viscosity solutions of second order partial differential equations[END_REF]Section 4]. a. We first prove that there exists two continuous functions w and w with linear growth that are respectively super-and subsolution of (2.4) for any

∈ [0, • ]. Since ĝ K = ĝK + ≥ g by Lemma 2.2, it suffices to set w := inf g > -∞,
see (1.16). To construct a supersolution w, let us fix η ∈ (0, ι ∧ inf f -1 ) with ι as in (1.13), set Γ(x) = ηx 2 /2 and define g = (ĝ • K -Γ) conc + Γ. Then, g ≥ ĝ • K , while the same reasoning as in Remark 1.6 implies that g shares the same linear growth as ĝ • K , see (2.3) and Lemma 2.2. We then define w by

w(t, x) = g(x) + 1 + (T -t)A in which A := sup σ 2 γ 2(1 -f γ)
.

The constant A is finite, and w has the same linear growth as g, see (

1.1)-(1.13). Since a concave function is a viscosity supersolution of -∂ 2 xx ϕ ≥ 0, we deduce that g is a viscosity supersolution of η -∂ 2 xx ϕ ≥ 0. Then, w is a viscosity supersolution of min -∂ t ϕ -A , η -∂ 2 xx ϕ ≥ 0. Since γ ≥ ι ≥ η, it
remains to use Remark 2.1 to conclude that w is a supersolution of (2.4).

b. We now express (2.4) as a single equation over the whole domain [0, T ] × R using the following definitions

F ,K κ,+ (t, x, r, q, M ) := F κ (t, x, q, M )1 [0,T ) + max F κ (t, x, q, M ), r -ĝ K (x) 1 {T } F ,K κ,-(t, x, r, q, M ) := F κ (t, x, q, M )1 [0,T ) + min F κ (t, x, q, M ), r -ĝ K (x) 1 {T } .
As usual

F ,K κ,± [ϕ](t, x) := F ,K κ,± (t, x, ϕ(t, x), ∂ t ϕ(t, x), ∂ 2 xx ϕ(t, x)).
Recall that the formulations in terms of F ,K κ,± lead to the same viscosity solutions as (2.4) (see Lemma 5.1 in the Appendix). This is the formulation to which we apply Perron's method. In view of a., the functions w and w are sub-and supersolution of F ,K κ,-= 0 and F ,K κ,+ = 0. Define:

v ,K γ := sup{v ∈ USC : w ≤ v ≤ w and v is a subsolution of F ,K κ,-= 0},
in which USC denotes the class of upper-semicontinuous maps. Then, the upper-(resp. lower-)semicontinuous envelope

(v ,K γ ) * (resp. (v ,K γ ) * ) of v ,K γ is a viscosity subsolution of F ,K κ,-[ϕ] = 0 (resp. supersolution of F ,K κ,+ [ϕ] = 0)
(v ,K γ ) * = (v ,K γ ) * , on [0, T ] × R.
Hence, v ,K γ is a continuous viscosity solution of (2.4), recall Lemma 5.1. By construction, it has linear growth. Uniqueness in this class follows from Theorem 2.11 again. c. It remains to prove (2.6). For this, we need a control on the behavior of v ,K γ as t → T . It is enough to obtain it for a lower bound v ,K that we first construct. Let ϕ be a test function such that

(strict) min [0,T )×R (v ,K γ -ϕ) = (v ,K γ -ϕ)(t 0 , x 0 )
for some (t 0 , x 0 ) ∈ [0, T ) × R. By the supersolution property,

min x ∈D κ (x 0 ) {γ(x ) -∂ 2 xx ϕ(t 0 , x 0 )} ≥ 0.
Recalling (1.1) and (1.13), this implies that, for x ∈ D κ (x 0 ),

1 -f (x )∂ 2 xx ϕ(t 0 , x 0 ) ≥ ιf (x ) ≥ ι inf f =: ι > 0.
Using the supersolution property and the above inequalities yields

0 ≤ min x ∈D κ (x 0 ) -∂ t ϕ(t 0 , x 0 ) - σ 2 (x )∂ 2 xx ϕ(t 0 , x 0 ) 2(1 -f (x )∂ 2 xx ϕ(t 0 , x 0 )) ≤ min x ∈D κ (x 0 ) -∂ t ϕ(t 0 , x 0 ) - σ 2 (x ) ∂ 2 xx ϕ(t 0 , x 0 ) -γ(x 0 ) 2(1 -f (x )∂ 2 xx ϕ(t 0 , x 0 )) ≤ -∂ t ϕ(t 0 , x 0 ) - σ2 ∂ 2 xx ϕ(t 0 , x 0 ) 2ι + σ2 γ(x 0 ) 2ι where σ := sup σ.
Denote by v ,K the unique viscosity solution of

-∂ t ϕ - σ2 ∂ 2 xx ϕ 2ι + σ2 γ 2ι 1 [0,T ) + (ϕ -ĝ K )1 {T } = 0. (2.7)
The comparison principle for (2.7) and the Feynman-Kac formula imply that

v ,K γ (t, x) ≥ v ,K (t, x) = E - T -t 0 σ2 γ(S x r ) 2ι dr + ĝ K (S x T -t )
where

S x = x + σ √ ι W.
It remains to show that (2.6) holds for v ,K in place of v ,K γ . The argument is standard. Since ĝK is uniformly continuous, see Lemma 2.2, we can find

B K ε > 0 such that ĝ K (S x T -t ) -ĝ K (x) 1 {|S x T -t -x|≤B K ε } ≤ ε for all ε > 0. We now consider the case |S x T -t -x| > B K ε .
Let C > 0 denote a generic constant that does not depend on (t, x) but can change from line to line. Because, ĝK is affine on [x K , ∞) and on (-∞, -x K ], see Lemma 2.2,

E ĝ K (S x T -t ) -ĝ K (x) 1 {S x T -t ≥x K } ≤ C(T -t) 1 2 if x ≥ x K , and E ĝ K (S x T -t ) -ĝ K (x) 1 {S x T -t ≤-x K } ≤ C(T -t) 1 2 if x ≤ -x K .
On the other hand, by linear growth of ĝ K , if x < x K , then

E ĝ K (S x T -t ) -ĝ K (x) 1 {S x T -t ≥x K } 1 {|S x T -t -x|≥B K ε } ≤ E ĝ K (S x T -t ) -ĝ K (x) 2 1 2 P |S x T -t -x| ≥ |x K -x| ∨ B K ε 1 2 ≤ C (1 + |x|)(T -t) 1 2 |x K -x| ∨ B K ε ≤ C B K ε (T -t) 1 2
.

The (four) remaining cases are treated similarly, and we obtain

E ĝ K (S x T -t ) -ĝ K (x) ≤ C B K ε (T -t) 1 2 + ε.
Since γ is bounded, this shows that

|v ,K (t, x) -ĝ K (x)| ≤ C B K ε (T -t) 1 2 + ε for t ∈ [T -1, T ]. Hence the required result for v ,K . Since v ,K γ ≥ v ,K
, this concludes the proof of (2.6).

For later use, note that, by stability, v ,K γ converges to a solution of (1.17) when → 0 and K → ∞.

Proposition 2.5. As → 0 and K → ∞, v ,K γ converges to a function vγ that is the unique viscosity solution of (1.17) with linear growth.

Proof. The family of functions {v ,K γ , ∈ (0, • ], K > 0} is uniformly bounded by a map with linear growth, see Proposition 2.4. In view of the comparison result of Theorem 2.11 below, it suffices to apply [2, Theorem 4.1].

Remark 2.6. The bounds on vγ can be made explicit, which can be useful to design a numerical scheme, see Section 4.1 below. First, as a by-product of the proof of Proposition 2.4, v ,K γ ≥ inf g. Passing to the limit as → 0 and K → ∞ leads to vγ ≥ inf g =: w.

We have also obtained that

v ,K γ ≤ (ĝ • K -Γ) conc + Γ + 1 + A in which x → Γ(x) = ηx 2 /2 for some η ∈ (0, ι ∧ inf f -1
) with ι as in (1.13), and

A := T sup(σ 2 γ/[2(1 -f γ)]
). On the other hand, (2.2) implies

ĝ • K ≤ 1 + (2c 0 + c 1 | • | -Γ• ) conc + Γ• for Γ• such that ∂ 2 xx Γ• = γ. Then, v ,K γ ≤ 1 + (2c 0 + c 1 | • | -Γ• ) conc + Γ• -Γ conc + Γ + 1 + A ≤ 1 + (2c 0 + c 1 | • | -Γ) conc + Γ -Γ conc + Γ + 1 + A = 1 + 2c 0 + c 1 | • | -Γ conc + Γ + 1 + A =: w and vγ ≤ w.
The function w defined above can be computed explicitly by arguing as in Remark 1.6. Also note that (2.2) and the arguments of Remark 1.6 imply that there exists a constant C > 0 such that

lim sup |x|→∞ |v ,K γ (x)|/(1 + |ĝ K (x)|) ≤ C, for all ∈ [0, • ] and K > 0.
(2.8)

Regularization and verification

Prior to applying our verification argument, it remains to smooth out the function v ,K γ . This is similar to [5, Section 3], but here again the fact that ĝ may not be bounded incurs additional difficulties. In particular, we need to use a kernel with a space dependent window.

We first fix a smooth kernel

ψ δ := δ -2 ψ(•/δ)
in which δ > 0 and ψ ∈ C ∞ b is a non-negative function with the closure of its support [-1, 0] × [-1, 1] that integrates to 1, and such that yψ(•, y)dy = 0.

(2.9)

Let us set

v ,K,δ γ (t, x) := R×R v ,K γ ([t ] + , x ) 1 κ(x) ψ δ t -t, x -x κ(x) dt dx . (2.10)
We recall that κ enters into the definition of F κ and satisfies (2.5).

The following shows that v ,K,δ γ is a smooth supersolution of (1.17) with a space gradient admitting bounded derivatives. This is due to the space dependent rescaling of the window by κ and will be crucial for our verification arguments. Proposition 2.7. For all 0 < < • and K > 0 large enough, there exists δ • > 0 such that v ,K,δ γ is a C ∞ supersolution of (1.17) for all 0 < δ < δ • . It has linear growth and ∂ x v ,K,δ γ has bounded derivatives of any order.

Proof. a. It follows from (2.5) and (2.8) that

lim sup |x|→∞ |v ,K γ (x)|/(1 + |κ(x)|) < ∞.
Direct computations and (2.5) then show that v ,K,δ γ has linear growth and that all derivatives of ∂ x v ,K,δ γ are uniformly bounded. b. We now prove the supersolution property inside the parabolic domain. Since the proof is very close to that of [5, Theorem 3.3], we only provide the arguments that require to be adapted, and refer to their proof for other elementary details. Fix > 0 and set

v (t, x) := v ,K,δ γ (t, (-) ∨ x ∧ ).
We omit the superscripts that are superfluous in this proof. Given k ≥ 1, set

v ,k (z) := inf z ∈[0,T ]×R v (z ) + k|z -z | 2 .
Since v is bounded and continuous, the infimum in the above is achieved by a point ẑ ,k (z) = ( t ,k (z), x ,k (z)), and v ,k is bounded, uniformly in k ≥ 1. This implies that we can find C > 0, independent of k, such that

|z -ẑ ,k (z)| 2 ≤ C /k =: (ρ ,k ) 2 . (2.11)
Moreover, a simple change of variables argument shows that, if ϕ is a smooth function such that v ,k -ϕ achieves a minimum at z ∈ [0, T ) × (-, ), then

(∂ t ϕ, ∂ x ϕ, ∂ 2 xx ϕ)(z) ∈ P-v (ẑ ,k (z)),
where Pv (ẑ ,k (z)) denotes the closed parabolic subjet of v at ẑ ,k (z); see e.g. [START_REF] Crandall | User's guide to viscosity solutions of second order partial differential equations[END_REF] for the definition. Then, Proposition 2.4 implies that v ,k is a supersolution of

min x ∈D κ (x ,k ) min -∂ t ϕ - σ 2 (x )∂ 2 xx ϕ 2(1 -f (x )∂ 2 xx ϕ) , γ(x ) -∂ 2 xx ϕ ≥ 0 on [ρ ,k , T -ρ ,k )×(-+ρ ,k , -ρ ,k ). We next deduce from (2.11) that x ∈ D /2 κ (x) implies - 2 κ(x ) -C /k 1 2 ≤ x ,k (t, x) -x ≤ 2 κ(x ) + C /k 1 2 .
Since inf κ > 0, this shows that x ∈ D κ (x ,k (t, x)) for k large enough with respect to . Hence, v ,k is a supersolution of

min x ∈D /2 κ min -∂ t ϕ - σ 2 (x )∂ 2 xx ϕ 2(1 -f (x )∂ 2 xx ϕ) , γ(x ) -∂ 2 xx ϕ ≥ 0 on [ρ ,k , T -ρ ,k ) × (-+ ρ ,k , -ρ ,k ).
We now argue as in [START_REF] Ishii | On the equivalence of two notions of weak solutions, viscosity solutions and distribution solutions[END_REF]. Since v ,k is semi-concave, there exist ∂ 2,abs xx v ,k ∈ L 1 and a Lebesgue-singular negative Radon measure ∂ 2,sing xx v ,k such that

∂ 2 xx v ,k (dz) = ∂ 2,abs xx v ,k (z)dz + ∂ 2,sing xx v ,k (
dz) in the distribution sense and

(∂ t v ,k , ∂ x v ,k , ∂ 2,abs xx v ,k ) ∈ P-v ,k a.e. on [ρ k , T -ρ k ] × (-+ ρ ,k , -ρ ,k ),
see [START_REF] Jensen | The maximum principle for viscosity solutions of fully nonlinear second order partial differential equations[END_REF]Section 3]. Hence, the above implies that

min x ∈D /2 κ min -∂ t v ,k - σ 2 (x )∂ 2,abs xx v ,k 2(1 -f (x )∂ 2,abs xx v ,k ) , γ(x ) -∂ 2,abs xx v ,k ≥ 0 a.e. on [ρ ,k , T -ρ ,k ) × (-+ ρ ,k , -ρ ,k ), or equivalently, by (2.1), min -∂ t v ,k - σ 2 (x)∂ 2,abs xx v ,k 2(1 -f (x)∂ 2,abs xx v ,k ) , γ(x) -∂ 2,abs xx v ,k (t , x ) ≥ 0
for all x and for a.e.

(t , x ) ∈ [ρ ,k , T -ρ ,k ) × (-+ ρ ,k , -ρ ,k ) such that 2|x -x| ≤ κ(x). Take 0 < δ < ε/2.
Integrating the previous inequality with respect to (t , x ) with the kernel function ψ δ (•, •/κ)/κ, using the concavity and monotonicity property of Remark 2.1 and the fact that ∂ 2,sing xx v ,k is non-positive, we obtain

min -∂ t v δ ,k - σ 2 ∂ 2 xx v δ ,k 2(1 -f ∂ 2 xx v δ ,k ) , γ -∂ 2 xx v δ ,k ≥ 0 (2.12) on [ρ ,k + δ, T -ρ ,k ) × (-x - ,k , x + ,k ), in which v δ ,k (t, x) := R×R v ,k ([t ] + , x ) 1 κ(x) ψ δ t -•, x -• κ(x) dt dx and x + ,k + δ 2 κ(x + ,k ) = -ρ ,k and -x - ,k - δ 2 κ(-x - ,k ) = -+ ρ ,k .
The above are well defined, see Remark 2.3. By Remark 2.3 and (2.11), ±x ± ,k → ±∞ and ρ ,k → 0 as k → ∞ and then → ∞. Moreover, v δ ,k → v ,K,δ γ as k → ∞ and then → ∞, and the derivatives also converge. Hence, (2.12) implies that v ,K,δ γ is a supersolution of (1.17) on [δ, T ) × R. c. We conclude by discussing the boundary condition at T . We know from Proposition 2.4 that

v ,K γ ≥ ĝK + /2, on [T -c K , T ] × R.
Since ĝ is uniformly continuous, see (1.16), so is ĝK , and therefore v ,K,δ γ (T, •) ≥ ĝK on the compact set [-2x K , 2x K ] for δ > 0 small enough with respect to , see Lemma 2.2 for the definition of x K ≥ K. Now observe that x ≥ 2x K and |x -x| ≤ δκ(x) imply that x ≥ 2x K (1 -δc K 1 ) -δc K 0 in which c K 1 and c K 0 are constants. This actually follows from the affine behavior of κ on [x K , ∞), see (2.5) and Lemma 2.2. For δ small enough, we then obtain x ≥ x K . Since ĝK is affine on [x K , ∞), and since ψ is symmetric in its second argument, see (2.9), it follows that

v ,K,δ γ (T, x) ≥ R×R ĝK (x ) 1 κ(x) ψ δ t -T, x -x κ(x) dt dx = ĝK (x)
for all x ≥ 2x K . This also holds for x ≤ -2x K , by the same arguments.

We can now use a verification argument and provide the main result of this section.

Theorem 2.8. Let vγ be defined as in Proposition 2.5. It has linear growth. Moreover, vγ ≥ v γ on [0, T ] × R.

Proof. The linear growth property has already been stated in Proposition 2.5. We now show that vγ ≥ v γ by applying a verification argument to v ,K,δ γ . From now on 0 < ≤ • in which • is as in (2.5). The parameters K, δ > 0 are chosen as in Proposition 2.7.

Fix (t, x) ∈ (0, T ) × R and δ ∈ (0, t ∧ ). Let (X, Y, V ) be defined as in (1.6)-

(1.2)-(1.7) with (x, ∂ x v ,K,δ γ (t, x), v ,K,δ γ (t, x) -∂ x v ,K,δ γ (t, x)x)
as initial condition at t, and for the Markovian controls

â = σ∂ 2 xx v ,K,δ γ 1 -f ∂ 2 xx v ,K,δ γ (•, X) b = ∂ 2 tx v ,K,δ γ + ∂ 2 xx v ,K,δ γ (µ + âσf ) + 1 2 ∂ 3 xxx v ,K,δ γ (σ + âf ) 2 1 -f ∂ 2 xx v ,K,δ γ (•, X).
By definition of F , (1.13) and (1.1), the above is well-defined as the denominators are always bigger than inf f ι > 0. All the involved functions being bounded and Lipschitz, see Proposition 2.7, it is easy to check that a solution to the corresponding stochastic differential equation exists, and that (â, b)

∈ A • . Direct computations then show that Y = ∂ x v ,K,δ γ (•, X). Moreover, the fact that v ,K,δ γ is a supersolution of F [ϕ] = 0 on [t, T ] × R
ensures that the gamma constraint (1.12) holds, for some k ≥ 1, and that

-∂ t v ,K,δ γ (•, X) - 1 2 σ(X)â ≥ 0 on [t, T ).
The last inequality combined with the definition of â implies

1 2 f (X)â 2 ≥ ∂ t v ,K,δ γ (•, X) + 1 2 (σ(X) + f (X)â)â = ∂ t v ,K,δ γ (•, X) + 1 2 (σ â X (X)) 2 ∂ 2 xx v ,K,δ γ (•, X) on [t, T ).
Hence,

V T = v ,K,δ γ (t, x) + 1 2 T t f (X u )â 2 u du + T t ∂ x v ,K,δ γ (u, X u ) dX u ≥ v ,K,δ γ (t, x) + T t dv ,K,δ γ (u, X u ) = v ,K,δ γ (T, X T ) ≥ g(X T ),
in which the last inequality follows from Proposition 2.7 again. It remains to pass to the limit δ, → 0. By Proposition 2.4, v ,K γ is continuous, so that v ,K,δ γ converges pointwise to v ,K γ as δ → 0. By Proposition 2.5, v ,K γ converges pointwise to vγ as → 0 and K → ∞. In view of the above this implies the required result: vγ ≥ v γ . Remark 2.9. Note that, in the above proof, we have constructed a super-hedging strategy in A k,γ (t, x) and starting with |Y t | ≤ k, for some k ≥ 1 which can be chosen in a uniform way with respect to (t, x), while v ,K,δ γ has linear growth.

Comparison principle

We provide here the comparison principle that was used several times in the above. Before stating it, let us make the following observation, based on direct computations. Recall (1.1) and (1.13).

Proposition 2.10. Fix ρ > 0. Consider the map

(t, x, M ) ∈ [0, T ] × R × R → Ψ(t, x, M ) = σ 2 (x)M 2(e ρt -f (x)M )
.

Then, M → Ψ(t, x, M ) is continuous, uniformly in (t, x), on O := {(t, x, M ) ∈ [0, T ] × R × R : M ≤ e ρt γ(x)}.
Moreover, there exists

L > 0 such that x → Ψ(t, x, M ) is L-Lipschitz on O. Theorem 2.11. Fix ∈ [0, • ]. Let U (resp.
V ) be a upper semicontinuous viscosity subsolution (resp. lower semicontinuous supersolution) of

F κ = 0 on [0, T ) × R.
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). Then, Û and V are respectively sub-and supersolution of

min x ∈D κ min ρϕ -∂ t ϕ - σ 2 (x )∂ xx ϕ 2(e ρt -f (x )∂ xx ϕ) , e ρt γ(x ) -∂ xx ϕ = 0 (2.13)
on [0, T ) × R. For later use, note that the infimum over D κ is achieved in the above, by the continuity of the involved functions.

If sup [0,T ]×R ( Û -V ) > 0, then we can find λ ∈ (0, 1)

such that sup [0,T ]×R ( Û - Vλ ) > 0 with Vλ := λ V + (1 -λ)w, in which w(t, x) := (T -t)A + (c U 0 + c U 1 | • | - ι 4 | • | 2 ) conc (x) + ι 4 |x| 2 with c U 0 , c U 1 two constants such that e ρT |U | ≤ c U 0 + c U 1 | • | and A := 1 2 sup σ 2 1 -ι 2 f ι 2 ,
where ι > 0 is as in (1.13). Note that

Vλ (T, •) ≥ Û (T, •), (2.14)
and that w is a viscosity supersolution of (2.13) Vλ is a viscosity supersolution of λγ

+ (1 -λ) ι 2 -∂ 2 xx ϕ ≥ 0.
(2.15)

Moreover, by Remark 2.1, Vλ is a supersolution of (2.13). Define for ε > 0 and n ≥ 1

Θ ε n := sup (t,x,y)∈[0,T ]×R 2 Û (t, x) -Vλ (t, y) - ε 2 |x| 2 + n 2 |x -y| 2 =: η > 0, (2.16)
in which the last inequality holds for n > 0 large enough and ε > 0 small enough. Denote by (t ε n , x ε n , y ε n ) the point at which this supremum is achieved. By (2.14), it must hold that t ε n < T , and, by 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],

lim n→∞ n|x ε n -y ε n | 2 = 0. (2.17)
It remains to apply Proposition 2.10 together with (2.19) for n large enough and ε small enough to obtain

ρ( Û (t ε n , x ε n ) -Vλ (t ε n , y ε n )) ≤ σ 2 (x ε n )N ε n 2(e ρt ε n -f (x ε n )N ε n ) - σ 2 (ŷ ε n )N ε n 2(e ρt ε n -f (ŷ ε n )N ε n ) + O ε n (1)
≤ L |x ε n -ŷε n | + O ε n (1)
for some L > 0 and where O ε n (1) → 0 as n → ∞ and then ε → 0. By continuity and (2.17) combined with Remark 2.3 and (2.20), this contradicts (2.16) for n large enough.

Supersolution property for the weak formulation

In this part, we provide a lower bound v γ for v γ that is a supersolution of (1.17). It is constructed by considering a weak formulation of the stochastic target problem (1.15) in the spirit of [START_REF] Cheridito | The multi-dimensional superreplication problem under gamma constraints[END_REF]Section 5]. Since our methodology is slightly different, we provide the main arguments.

On C(R + ) 5 , let us now denote by ( ζ := (ã, 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

ã = ã0 + • 0 βs ds + • 0 αs d Ws for some ã0 ∈ R, (2.21)
sup

R + | ζ| ≤ k , (2.22)
and

E P sup | ζs -ζs |, r ≤ s ≤ s ≤ s + δ | F• r ≤ kδ. (2.23)
For

φ := (y, ã, b), y ∈ R, we define ( Xx, φ, Ỹ φ, Ṽ x,v, φ) as in (1.6)-(1.2)-(1.7
) associated to the control (ã, b) with time-0 initial condition (x, y, v), and with W in place of W . For t ≤ T and k ≥ 1, we say that P ∈ Gk,γ (t, x, v, y) if

Ṽ x,v, φ T -t ≥ g( Xx, φ T -t ) and -k ≤ γ ã Y ( Xx, φ) ≤ γ( Xx, φ) on R + P -a.s. (2.24)
We finally define

v k γ (t, x) := inf{v = c + yx : (c, y) ∈ R × [-k, k] s.t. Ãk ∩ Gk,γ (t, x, v, y) = ∅}, and
v γ (t, x) := lim inf (k, t , x ) → (∞, t, x) (t , x ) ∈ [0, T ) × R v k γ (t , x ), (t, x) ∈ [0, T ] × R. (2.25)
The following is an immediate consequence of our definitions.

Proposition 2.12.

v γ ≥ v γ on [0, T ) × R.
In the rest of this section, we show that v γ is a viscosity supersolution of (1.17). We start with an easy remark.

Remark 2.13. Observe that the gamma constraint in (2.24) implies that we can find ε > 0 such that

ε 1 + kε -1 ≤ σ ã X ( Xx, φ) ≤ ε -1 + ε -2 and |ã| ≤ ε -1 P -a.s., for all P ∈ Ãk ∩ Gk,γ (t, x, v, y) and k ≥ 1. Indeed, if ã ≥ -σ/f then -k ≤ γ ã Y ≤ γ implies (- kσ 1 + kf ) ∨ (- σ f ) ≤ ã ≤ γσ 1 -γf and ãf + σ ≥ σ/(1 + kf ).
Then our claim follows from (1.1)-(1.13). On the other hand, if σ + ãf < 0, then γ ã Y ≤ γ implies ã ≥ γσ/(1 -f γ) ≥ 0, see (1.13), while ã < -f /σ < 0, a contradiction.

We then show that v k γ has linear growth, for k large enough.

Proposition 2.14. There exists k o ≥ 1 such that {|v k γ |, k ≥ k o } is uniformly bounded from above by a continuous map with linear growth.

Proof. a. First note that Remark 2.9 implies that {(v k γ ) + , k ≥ k o } is uniformly bounded from above by a map with linear growth, for some k o large enough. b. Let us now fix P ∈ Ãk ∩ Gk,γ (t, x, v, y). Using Remark 2.13 combined with (1.1) and the condition that (ã, b, α, β) is P-essentially bounded, one can find P ∼ P under which

• 0 Ỹ φ s d Xx, φ s is a martingale on [0, T -t]. Then, the condi- tion Ṽ x,v, φ T -t ≥ g( Xx, φ T -t ) P-a.s. implies v + E P[ 1 2 T -t 0 ã2 s f ( Xx, φ s )ds] ≥ inf g > -∞, recall (1.16). By Remark 2.13 and (1.1), v ≥ inf g -C > -∞, for some constant C independent of P ∈ ∪ k ( Ãk ∩ Gk,γ (t, x, v, y)). Hence {(v k γ ) -, k ≥ k o } is bounded by a constant.
We now prove that existence holds in the problem defining v k γ and that it is lower-semicontinuous. Proposition 2.15. For all (t, x) ∈ [0, T ] × R and k ≥ 1 large enough, there exists

(c, y) ∈ R × [-k, k] such that v k γ (t, x) = c + yx and Ãk ∩ Gk,γ (t, c + xy, y) = ∅. Moreover, v k γ is lower-semicontinuous for each k ≥ 1 large enough.
Proof. By [20, Proposition XIII.1.5] and the condition (2.23) taken for r = 0, the set Ãk is weakly relatively compact. Moreover, [16, Theorem 7.10 and Theorem 8.1] implies that any limit point (P * , t * , x * , c * , y * ) of a sequence

(P n , t n , x n , c n , y n ) n≥1 such that P n ∈ Ãk ∩ Gk,γ (t n , x n , c n + x n y n , y n ) for each n ≥ 1 satisfies P * ∈ Ãk ∩ Gk,γ (t * , x * , c * + x * y * , y * ). Since v k
γ is locally bounded, by Proposition 2.14 when k ≥ k o , the announced existence and lower-semicontinuity readily follow.

We can finally prove the main result of this section.

Theorem 2.16. The function v γ is a viscosity supersolution of (1.17). It has linear growth.

Proof. The linear growth property is an immediate consequence of the uniform linear growth of {|v k γ |, k ≥ k o } stated in Proposition 2.14. To prove the supersolution property, it suffices to show that it holds for each v k γ , with k ≥ k o , and then to apply standard stability results, see e.g. [START_REF] Barles | Solution de viscosités des équations d'Hamilton Jacobi[END_REF]. a. We first prove the supersolution property on [0, T )×R. We adapt the arguments of [START_REF] Cheridito | The multi-dimensional superreplication problem under gamma constraints[END_REF] to our context. Let us consider a C ∞ b test function ϕ and (t 0 , x 0 ) ∈ [0, T ) × R such that (strict) min

[0,T )×R (v k γ -ϕ) = (v k γ -ϕ)(t 0 , x 0 ) = 0.
Recall that v k γ is lower-semicontinuous by Proposition 2.15. Because the infimum is achieved in the definition of v k γ , by the afore-mentioned proposition, there exists |y 0 | ≤ k and P ∈ Ãk ∩ Gk (t 0 , x 0 , v 0 , y 0 ), such that v 0 := c 0 +y 0 x 0 = v k γ (t 0 , x 0 ) for some c 0 ∈ R. Let us set ( X, Ỹ , Ṽ ) := ( Xx 0 , φ, Ỹ φ, Ṽ x 0 ,v 0 , φ)

where φ = (y 0 , ã, b). Let θ o be a stopping time for the augmentation of the raw filtration F• , and define

θ := θ o ∧ θ 1 with θ 1 := inf{s : | Xs -x 0 | ≥ 1}.
Then, it follows from Proposition 2.17 below that

Ṽθo ≥ v k γ (t 0 + θ o , Xθo ) ≥ ϕ(t 0 + θ o , Xθo ),
in which here and hereafter inequalities are taken in the P-a.s. sense. After applying Itô's formula twice, the above inequality reads: where

:= 1 2 ã2 f ( X) -L ãϕ(t 0 + •, X• ) , m := µ ã, b Y ( X) -L ã∂ x ϕ(t 0 + •, X• ) n := γ ã Y ( X) -∂ 2 xx ϕ(t 0 + •, X• ), with L ã := ∂ t + 1 2 (σ ã X ) 2 ∂ 2 xx
For the rest of the proof, we recall (2.22). Together with (1.1) and Remark 2.13, this implies that σ ã X ( X), σ ã X ( X) -1 and µ ã, b X ( X) are P-essentially bounded. After performing an equivalent change of measure, we can thus find P ∼ P and a P-Brownian motion W such that:

X = • 0 σ ãs X ( Xs )d Ws .
(2.27)

Clearly, both P and W depend on (ã, b, y 0 ). 1. We first show that y 0 = ∂ x ϕ(t 0 , x 0 ), and therefore (2.28)

Let Pλ ∼ P be the measure under which

W λ := W + • 0 λ[σ ãs X ( Xs )] -1 (y 0 -∂ x ϕ(t 0 , x 0 ))ds
is a Pλ -Brownian motion. Consider the case θ o := η > 0. Since all the coefficients are bounded, taking expectation under Pλ and using (2.26) imply

C η ≥ λ(y 0 -∂ x ϕ(t 0 , x 0 )) 2 E Pλ [θ] +E Pλ θ 0 s 0 m r dr + s 0 n r d Xr λ(y 0 -∂ x ϕ(t 0 , x 0 ))ds
for some C > 0. We now divide both sides by η and use the fact that (η∧θ 1 )/η → 1 Pλ -a.s. as η → 0 to obtain

C ≥ λ(y 0 -∂ x ϕ(t 0 , x 0 )) 2 .
Then, we send λ → ∞ to deduce that y 0 = ∂ x ϕ(t 0 , x 0 ). 2. We now prove that

∂ 2 xx ϕ(t 0 , x 0 ) ≤ γ ã0 Y (x 0 ) ≤ γ(x 0 ). (2.29)
We first consider the time change

h(t) = inf{r ≥ 0 : r 0 (σ ãs X ( Xs )) 2 1 [0,θ] (s) + 1 [0,θ] c (s) ds ≥ t}.
Again, σ ã X ( X) and σ ã X ( X) -1 are essentially bounded by Remark 2.13, so that h is absolutely continuous and its density h satisfies

0 < ht ≤ h(t) := (σ ã X ( X)) 2 1 [0,θ] (t) + 1 [0,θ] c (t) -1 ≤ ht (2.30)
for some constants h and h, for all t ≥ 0. Moreover, Ŵ := Xh is a Brownian motion in the time changed filtration. Let us now take θ o := h -1 (η) for some 0 < η < 1.

Then, (2.28) reads Since γ ã Y ( X) ≤ γ( X), this proves (2.29). 3. It remains to show that the first term in the definition of F [ϕ](t 0 , x 0 ) is also non-negative, recall (1.14). Again, let us take θ o := h -1 (η) and recall from 2. that lim η→0 (η ∧ h -1 (θ 1 ))/η = 1 P-a.s. Note that ã being of the form (2.21) with the condition (2.22), it satisfies [8, Condition (A.2)], and so does n. Using [8, Theorem A.2 and Proposition A.3] and (2.31), we then deduce that 0 h(0) -1 2 n 0 ≥ 0. Hence, (2.30) and direct computations based on (1.8) imply

0 ≤ η∧h -1 (θ 1 ) 0 h(s) h(s) ds + η∧h -1 (θ 1 ) 0 s 0 m h(r) h(r)drd Ŵs + η∧h -1 (θ 1 ) 0 s 0 n h(r) d Ŵr d Ŵs . ( 2
0 ≤ 1 2 ã2 0 f (x 0 ) -L ã0 ϕ(t 0 , x 0 ) - 1 2 γ ã0 Y (x 0 ) -∂ 2 xx ϕ(t 0 , x 0 ) (σ ã0 X (x 0 )) 2 = 1 2 ã2 0 f (x 0 ) -∂ t ϕ(t 0 , x 0 ) - 1 2 γ ã0 Y (x 0 )(σ ã0 X (x 0 )) 2 = -∂ t ϕ(t 0 , x 0 ) - 1 2 σ 2 (x 0 ) 1 -f (x 0 )γ ã0 Y (x 0 ) γ ã0 Y (x 0 ) ≤ -∂ t ϕ(t 0 , x 0 ) - 1 2 σ 2 (x 0 ) 1 -f (x 0 )∂ 2 xx ϕ(t 0 , x 0 ) ∂ 2 xx ϕ(t 0 , x 0 ),
in which we use the facts that ∂ 2 xx ϕ(t 0 , x 0 ) ≤ γ ã0 Y (x 0 ) ≤ γ(x 0 ) and z → z/(1f (x 0 )z) in non-decreasing on (-∞, γ(x 0 )] ⊂ (-∞, 1/f (x 0 )), for the last inequality. b. We now consider the boundary condition at T . Since v k γ is a supersolution of γ -∂ 2 xx ϕ ≥ 0 on [0, T ) × R, the same arguments as in [START_REF] Cvitanić | Super-replication in stochastic volatility models under portfolio constraints[END_REF]Lemma 5.1] imply that v k γ -Γ is concave for any twice differentiable function Γ such that ∂ 2 xx Γ = γ. The function v k γ being lower-semicontinuous, the map

x → G(x) := lim inf t → T, x → x t < T v k γ (t , x ) is such that G ≥ g and G-Γ is concave. Hence, G = (G-Γ) conc + Γ ≥ (g-Γ) conc + Γ = ĝ.
It remains to state the dynamic programming principle used in the above proof.

Proposition 2.17. Fix (t, x, v, y) ∈ [0, T ] × R 2 × [-k, k] and let θ be a stopping time for the P-augmentation of F• that takes P-a.s. values in [0, T -t]. Assume that P ∈ Ãk ∩ Gk,γ (t, x, v, y). Then,

Ṽ x,v, φ θ ≥ v k γ (t + θ, Xx,
ω γ ã Y ( Xξr(ω), φ(ω) ) ≤ γ( Xξr(ω), φ(ω) ) on R + = 1, for ω / ∈ N , in which (ξ r , ϑ r , φ) := ( Xx, φ r , Ṽ x,v, φ r , ( Ỹ x, φ r , ã, b)).
This shows that ϑ r (ω) ≥ v k γ (t + r, ξ r (ω)) outside the null set N , which is the required result.

Conclusion of the proof and construction of almost optimal strategies

We first conclude the proof of Theorem 1.4. Proof of Theorem 1.4. Proposition 2.5 and Theorem 2.8 imply that vγ ≥ v γ in which vγ has linear growth and is a continuous viscosity solution of (1.17). On the other hand, Proposition 2.12 and Theorem 2.16 imply that v γ ≤ v γ on [0, T ) × R in which v γ has linear growth and is a viscosity supersolution of (1.17). By the comparison result of Theorem 2.11 applied with = 0, v γ ≥ vγ . Hence,

v γ = v γ = vγ on [0, T ) × R and v γ = vγ on [0, T ] × R (2.32)
Since vγ is continuous, this shows that

lim (t , x ) → (T, x) t < T v γ (t , x ) = vγ (T, x) = v γ (T, x).
Hence, v γ is a viscosity solution of (1.17), with linear growth.

Remark 2.18 (Almost optimal controls). In the proof of Theorem 2.8, we have constructed a super-hedging strategy starting from v ,K,δ γ (t, x). Since v ,K,δ γ (t, x) → vγ (t, x) = v γ (t, x) as δ, → 0 and K → ∞, this provides a way to construct super-hedging strategies associated to any initial wealth v > v γ (t, x).

Adding a resilience effect

In this section, we explain how a resilience effect can be added to our model. In the discrete rebalancement setting, we replace the dynamics (1.4) by

X n = X 0 + • 0 µ(X n s )ds + • 0 σ(X n s )dW s + R n , in which R n is defined by R n = R 0 + n i=1 1 [t n i ,T ] δ n t n i f (X n t n i -) - • 0 ρR n s ds,
for some ρ > 0 and R 0 ∈ R. The process R n models the impact of past trades on the price, the last term in its dynamics is the resilience effect. Then, the continuous time dynamics becomes

X = X 0 + • 0 σ(X s )dW s + • 0 f (X s )dY s + • 0 (µ(X s ) + a s (σf )(X s ) -ρR s )ds R = R 0 + • 0 f (X s )dY s + • 0 (a s (σf )(X s ) -ρR s )ds V = V 0 + • 0 Y s dX s + 1 2 • 0 a 2 s f (X s )ds.
This is obtained as a straightforward extension of [4, Proposition 1.1].

Let v R γ (t, x) be defined as the super-hedging price v γ (t, x) but for these new dynamics and for R t = 0. The following states that v R γ = v γ , i.e. adding a resilience effect does not affect the super-hedging price.

Proposition 3.1. v γ = v R γ on [0, T ] × R.
Proof. 1. To show that v γ ≥ v R γ , it suffices to reproduce the arguments of the proof of Theorem 2.8 in which the drift part of the dynamics of X does not play any role. More precisely, these arguments show that vγ ≥ v R γ . Then, one uses the fact that v γ = vγ , by (2.32). 2. As for the opposite inequality, we use the weak formulation of Section 2.2 and a simple Girsanov's transformation. For ease of notations, we restrict to t = 0. Fix v > v R γ (0, x), for some x ∈ R. Then, one can find k ≥ 1, (c, y) ∈ R × [-k, k] satisfying v = c + yx, and (a, b) ∈ A k,γ (0, x) such that V T ≥ g(X T ), with (V, X, Y, R) defined by the corresponding initial data and controls. We let

a = a 0 + • 0 β s ds + • 0 α s dW s
be the decomposition of a into an Itô process, see Section 1.1. Let Q R ∼ P be the probability measure under which W

R := W - • 0 (ρR s /σ(X s ))ds is a Q R -Brownian motion, recall (1.1). Then, X = X 0 + • 0 σ(X s )dW R s + • 0 f (X s )dY s + • 0 (µ(X s ) + a s (σf )(X s ))ds Y = Y 0 + • 0 (b s + a s ρR s /σ(X s ))ds + • 0 a s dW R s a = a 0 + • 0 (β s + α s ρR s /σ(X s ))ds + • 0 α s dW R s V = V 0 + • 0 Y s dX s + 1 2 • 0 a 2 s f (X s )ds.
Upon seeing (a, b + aρR/σ(X), α, β + αρR/σ(X), W R ) as a generic element of the canonical space C([0, T ]) 5 introduced in Section 2.2, then Q R belongs to Ãk ∩ Gk,γ (t, x, v, y), and therefore v > v γ (0, x). Hence, v R γ (0, x) ≥ v γ (0, x), and thus v R γ (0, x) ≥ v γ (0, x) by (2.32).

Numerical approximation and examples

In this section, we provide an example of numerical schemes that converges towards the unique continuous viscosity solution of (1.17) with linear growth. The scheme is then exemplified using two numerical applications in the case of constant market impact and gamma constraint.

Finite difference scheme

Given a map φ and h := (h t , h x ) ∈ (0, 1) 2 , define

L h 1 (t, x, y, φ) := - φ(t + h t , x) -y h t - σ 2 (x)G h (t, x, y, φ) 2(1 -f (x)G h (t, x, y, φ))
L h 2 (t, x, y, φ) := γ(x) -G h (t, x, y, φ) where G h (t, x, y, φ) := φ(t + h t , x + h x ) + φ(t + h t , x -h x ) -2y h 2

x .

The numerical scheme is set on the grid π h := {(t i , x j ) = (ih t , x + jh x ) : i ≤ n t , j ≤ n x }, with n t h t = T for some n t ∈ N, and n x h x = x -x, for some real numbers x < x. To paraphrase, v h γ is defined on π h as the solution of S(h, t i , x j , v h γ (t i , x j ), v h γ ) = 0 for i < n t , 1 ≤ j ≤ n x -1 (4.1)

v h γ = ĝ on π h ∩ {({T } × R) ∪ ([0, T ] ∩ {x, x})}
where S(h, t, x, y, φ) := ( w -y) ∨ (y -w) ∧ min l=1,2

L h l (t, x, y, φ)

with w and w as in Remark 2.6.

Theorem 4.1. The equation (4.1) admits a unique solution v h γ , for all h := (h t , h x ) ∈ (0, 1) 2 . Moreover, if h t /h 2 x → 0 and h 2 x → 0, then v h γ converges locally uniformly to the unique continuous viscosity solution of (1.17) that has linear growth.

Proof.

The existence of a solution, that is bounded by the map with linear growth | w| + |w|, is obvious. We now prove uniqueness. First observe that L h 2 is strictly increasing in its y-component, and that

∂L h 1 ∂y (t, x, y, φ) = 1 h t + σ 2 (x) h 2
x (1 -f (x)G h (t, x, y, φ)) 2 > 0 on the domain {y : L h 2 (t i , x j , y, φ) ≥ 0}. Uniqueness of the solution follows. It is easy to see that φ → S(•, φ) is non-decreasing, so that our scheme is monotone. Consistency is clear. Moreover, it is not difficult to check that the comparison result of Theorem 2.11 extends to this equation (there is an equivalence of the notions of super-and subsolutions in the class of functions w such that w ≤ w ≤ w). It then follows from [3, Theorem 2.1] that v h γ converges locally uniformly to the unique continuous viscosity solution with linear growth of

( w -ϕ) ∨ (ϕ -w) ∧ F [ϕ] 1 [0,T ) + (ϕ -ĝ)1 {T } = 0.
In view of (2.32), Remark 2.6 and Theorem 1.4, v γ is the unique viscosity solution of the above equation.

Numerical examples: the fixed impact case

To illustrate the above numerical scheme, we place ourselves in the simpler case where f ≡ λ > 0 and γ > 0 are constant. The dynamics of the stock is given by the Bachelier model dX t = σ dW t , with σ := 0.2. In the following, T = 2. First, we consider a European Butterfly option with three strikes K 1 = -1 < K 2 = 0 < K 3 = 1, where K 1 + 1/(2γ) ≤ K 2 ≤ K 3 -1/(2γ). Its pay-off is g(x) = (x -K 1 ) + -2(x -K 2 ) + + (x -K 3 ) + , and the corresponding face-lifted function ĝ can be computed explicitly:

ĝ(x) = γ 2 (x -x - 1 ) 2 1 [x - 1 ,x + 1 ) + (x -K 1 )1 [x + 1 ,K 2 ) +(x -K 1 -2(x -K 2 ))1 [K 2 ,x - 2 ) + γ 2 (x -x + 2 ) 2 + 2K 2 -(K 1 + K 3 ) 1 [x - 2 ,x + 2 ) +(2K 2 -(K 1 + K 3 ))1 [x + 2 ,+∞) ,
where x ± 1 = K 1 ± 1/(2γ) and x ± 2 = K 3 ± 1/(2γ).

In Figure 1, we separately show the effect of the gamma constraint and of the market impact. As observed in Remark 1.9, the price is non-decreasing with respect to the impact parameter λ and bounded from below by the hedging price obtained in the model without impact nor gamma constraint. On the left and right tails of the curves, we observe the effect of the gamma constraint. It does not operate around x = 0 where the gamma is non-positive. The effect of the market impact operates only in areas of high convexity (around x = -1.5 and x = 1.5) or of high concavity (around x = 0). In Figure 2, we perform similar computations but for a call spread option, where g(x) = (x -K 1 ) + -(x -K 2 ) + , with K 1 = -1 < K 2 = 1 such that K 1 + 1/(2γ) ≤ K 2 . The face-lifted function ĝ is given by

ĝ(x) = γ 2 (x -x -) 2 1 [x -,x + ) + (x -K 1 )1 [x + ,K 2 ) + (K 2 -K 1 )1 [K 2 ,+∞)
with x ± = K 1 ± 1/(2γ).

Appendix

The following is very standard, we prove it for completeness.

Lemma 5.1. A upper-semicontinuous (resp. lower-semicontinuous) map is a viscosity subsolution (resp. supersolution) of

F κ [ϕ]1 [0,T ) + (ϕ -ĝ K )1 {T } = 0
if and only if it is a viscosity subsolution (resp. supersolution) of F ,K κ,-[ϕ] = 0 (resp. F ,K κ,+ [ϕ] = 0).

Proof. The equivalence on [0, T ) is evident, we only consider the parabolic boundary {T } × R. Since F ,K κ,+ ≥ F κ and F ,K κ,-≤ F κ , only one implication is not completely trivial. a. Let v be a viscosity supersolution of F ,K κ,+ [ϕ] = 0, and ϕ ∈ C 2 be a test function such that (strict) min

γ -∂ 2

 2 xx ϕ ≥ 0. It is obtained by considering any twice continuously differentiable function Γ such that ∂ 2 xx Γ = γ, and then setting ĝ := (g -Γ) conc + Γ, in which the superscript conc means concave envelope, cf. [23, Lemma 3.1]. 2 Hence, we expect that v γ (T -, •) = ĝ on R.

  Remark 1.6. The function ĝK has the same linear growth as 2c 0 + c 1 | • |, by the same reasoning as in Remark 1.6. Since 2c 0 > c 0 , ĝK = ĝ• K = ĝ on [-K, K]. Furthermore, as the minimum of two concave functions is concave, so is ĝK -Γ for any C 2 function Γ satisfying ∂ 2 xx Γ = γ. The other assertions are immediate.

θ 0 s ds + θ 0 y 0 - 0 m r dr + s 0 n

 0000 ∂ x ϕ(t 0 , x 0 ) + s r d Xr d Xs ≥ 0.(2.26)

  Xr d Xs ≥ 0.

  .31) Since all the involved processes are continuous and bounded, and since (η∧h -1 (θ 1 ))/η → 1 a.s. as η → 0, the above combined with [8, Theorem A.1 b. and Proposition A.3] implies that γ ã0 Y (x 0 ) -∂ 2 xx ϕ(t 0 , x 0 ) = lim r↓0 n h(r) = lim r↓0 n r ≥ 0.

φθ

  ) P -a.s., in which φ := (y, ã, b). Proof. Since v k γ is lower-semicontinuous and all the involved processes have continuous paths, up to approximating θ by a sequence of stopping times valued in finite time grids, it suffices to prove our claim in the case θ ≡ r ∈ [0, T -t]. Let Pω be a regular conditional probability given F• r for P. It coincides with P[•| F• r ](ω) outside a set N of P-measure zero. Then, for all ω / ∈ N , 0 ≤ δ ≤ 1 and r ≥ 0 the conditions (2.21)-(2.22)-(2.23) hold for Pr ω defined on C(R + ) 5 by Pr ω [ω ∈ A] = Pω [ω r+• ∈ A]. Moreover, [9, Theorem 3.3] ensures that, after possibly modifying N , Pr ω Ṽ ξr(ω),ϑr(ω), φ(ω) T -(t+r) ≥ g( Xξr(ω), φ(ω) T -(t+r) ) = 1 and Pr

Figure 1 :

 1 Figure 1: Left: Super-hedging price of the Butterfly option. Dashed line: λ = 0.5, γ = 1.75; solid line: λ = 0, γ = 1.75; dotted line: λ = 0, γ = +∞. Right: Difference with the price associated to λ = 0, γ = +∞. Dashed line: λ = 0.5, γ = 1.75; solid line: λ = 0, γ = 1.75 .

Figure 2 :

 2 Figure 2: Left: Super-hedging price of the Call Spread option. Dashed line: λ = 0.5, γ = 1.75; solid line: λ = 0, γ = 1.75; dotted line: λ = 0, γ = +∞. Right: Difference with the price associated to λ = 0, γ = +∞. Dashed line: λ = 0.5, γ = 1.75; solid line: λ = 0, γ = 1.75 .

[ 0 ,

 0 T ]×R (v -ϕ) = (v -ϕ)(T, x 0 ) = 0,

  with linear growth, recall the continuity property of Remark 2.3 and see e.g. [10, Section 4]. The comparison result of Theorem 2.11 stated below implies that

In[START_REF] Bouchard | Almost-sure hedging with permanent price impact[END_REF], (a, b) is only required to be progressively measurable and essentially bounded. The additional restrictions imposed here will be necessary for our results in Section

2.2.

Obviously, adding an affine map to Γ does not change the definition of ĝ.

Moreover, Ishii's lemma implies the existence of (a ε n , M ε n , N ε n ) ∈ R 3 such that

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

In particular,

Then, by (2.15) and (1.13),

for n large enough and ε small enough, recall (2.17). Hence, the super-and subsolution properties of Vλ and Û imply that we can find u ε n ∈ [-, ] together with ŷε n and xε n such that

.

By Remark 2.1 and (2.18), this shows that

b. Let now v be a viscosity subsolution of F ,K κ,-[ϕ] = 0, and ϕ ∈ C 2 be a test function such that (strict) max

for some x 0 ∈ R. Then, F ,K κ,-[ϕ](T, x 0 ) ≤ 0. By replacing ϕ by φ, defined for α > 0 as φ(t, x) := ϕ(t, x 0 + α(x -x 0 )) + C(T -t),

we obtain a new test function at (T, x 0 ). Since inf γ > 0, recall (1.1), we can take α small enough so that

As in the previous step, we can now choose C > 0 such that

> 0 at (T, x 0 ). Since F ,K κ,-[φ](T, x 0 ) ≤ 0, we conclude that v(T, x 0 ) = φ(T, x 0 ) ≤ ĝ K (x 0 ).