Exact replication under Delta constraints ∗Jean-François Chassagneux†

Romuald Elie‡

Idris Kharroubi§This version: August 2012

Abstract
This paper deals with the super-replication of non path-dependent European claims
under additional convex constraints on the number of shares held in the portfolio.
The corresponding super-replication price of a given claim has been widely studied
in the literature and its terminal value, which dominates the claim of interest, is theso-called facelift transform of the claim. We investigate under which conditions thesuper-replication price and strategy of a large class of claims coincide with the exact
replication price and strategy of the facelift transform of this claim. In dimension
1, we observe that this property is satisfied for any local volatility model. In any
dimension, we exhibit a necessary and sufficient condition for this property, which
combines the dynamics of the stock together with the characteristics of the closed
convex set of constraints. The obtention of this condition relies on the introduction of
the notion of first order viability property for linear parabolic PDEs. We investigate
in details several practical cases of interest: multidimensional Black Scholes model,
non-tradable assets or short selling restrictions.

Keywords: super-replication, portfolio constraints, viability, facelift, BSDEs.

MSC Classification (2010): 93E20, 91G20, 60H30∗

Research supported by the ANR grant LIQUIRISK and the Chair Finance and Sustainable Development.
†
Department of Mathematics, Imperial College London. j.chassagneux@imperial.ac.uk‡
CEREMADE, Université Paris-Dauphine. elie@ceremade.dauphine.fr§
CEREMADE, Université Paris-Dauphine. kharroubi@ceremade.dauphine.fr

1

1

Introduction

In a complete financial market, the absence of arbitrage opportunities leads to the definition of the unique fair price of any contingent claim. As detailed e.g. in [11], thisuniqueness property disappears under the addition of convex constraints on the admissible portfolio strategies. This implies the existence of a closed interval of arbitrage-free
prices. A commonly considered prudential pricing methodology consists in selecting theupper bound of this interval. This so-called super-replication price coincides with theminimal initial amount of money required to constitute an admissible portfolio strategy
satisfying the constraints and whose terminal value dominates the claim of interest. The
super-replication price under convex delta constraints has been thoroughly studied in theliterature. In [5], the authors obtain a dual representation of the super-replication price interms of a well chosen set of risk neutral probabilities. In [6], closely related to the previous
work, the super-replication price process is shown to be the unique solution of a Backward
Stochastic Differential Equation (BSDE) with constraints on the gain process Z. All theseworks mainly rely on probabilistic and duality arguments. In a Markovian setting, thesuper-replication is characterised using direct dynamic programming arguments and PDEtechnics, see [17, 1].
In [3], the authors observe that, for the classical Black Scholes model, the super-replication
price of a claim under convex delta constraints coincides with the unconstrained replication
price of a so-called facelift transform of this claim. They consider constraints in terms ofnumber of shares in dimension 1, wealth proportion or money amount in any dimension,and exhibit the three corresponding facelifting procedures. In more general Markovian
models, the super-replication price function under convex portfolio constraints of a non
path-dependent claim interprets as the smallest function above the unconstrained price ofthe claim, which is stable under the corresponding facelift transform, see e.g. [1]. The goalof this paper is to state a necessary and sufficient condition under which the noteworthy
result of [3] extends to general local volatility models in dimension d.
To exhibit this condition presented in Theorem 3.1 below, we rely on a BSDE representation of the replicating strategy. We show in Proposition 4.1 that the replicating strategy
is the unique solution of a multidimensional linear BSDE with terminal value ∇h(XT ),where h is a smooth payoff function and X denotes the assets price process.
If h satisfies the portfolio constraint, i.e. ∇h is valued in some convex set K, the condition given in Theorem 3.1 ensures that the solution of a multi-dimensional linear BSDE
with terminal value ∇h(XT ) is valued in a convex set K. Namely, the super-replication
price of the claim with payoff h(XT ) under convex delta constraints coincides with its
unconstrained replication price. It is crucial to observe that we cannot rely on classicalviability results for BSDEs [4] since the class of possible terminal value for the BSDE is
restricted here to gradient type terminal conditions of the form ∇h(XT ). The condition2

obtained in [4] is thus only a sufficient condition for our problem. Contrary to this paper,
we take advantage here of the linear structure of the problem. It appears that the studyof the viability condition for the convex set K boils down to the study of the viability
condition for the tangent half-spaces to K. This makes the proof - in some sense - simpler.
Our approach allows us also to remark that, under the exhibited condition of interest, thesuper-replication price of an American option with exercise payoff h(X) under convex delta
constraints also coincides with its unconstrained replication price.
In practice, the payoff function h does not satisfy that ∇h is valued in K nor any smoothness property. Nevertheless, our main result still holds using the facelift transform of h.
The proofs in the general case are based on regularization technics.
We also discuss in this paper various practical cases which are of interest in Finance,
see Section 3. We first observe that the result of [3] in the Black Scholes model does notextend to the consideration of a financial market with d > 1 assets. The hypercubes are
the only convex set of constraints for which facelifting the payoff allows to get rid of theportfolio constraints in a multidimensional Black Scholes model. This property extends
also to most of the common local volatility models, in which each asset follows its own
dynamics. In particular, hypercubes include the consideration of non-tradable assets or no
short sell restrictions. More specifically, we observe that the only model dynamics in which
no short sell restrictions on Asset 1 can be relaxed using a facelifting procedure are the one
for which the quadratic covariations between the other assets do not depend on Asset 1.
We also exhibit model dynamics conditions related to other type of portfolio constraints:
illiquid assets, bound on the total number of allowed positions on the financial market etc.
We finally observe that no reasonable model dynamics with positive asset value, allows
to satisfy in dimension d > 1 the exact replication property for any closed convex set of
constraints.
The rest of this paper is organized as follows. In Section 2, we specify the problem
formulation and exhibit the main properties of the super-replication methodology andrelated facelift operator. In Section 3, we produce the main result of the paper which
gives a tractable analytical necessary and sufficient condition, ensuring that the exact
replication property holds for a large class of payoff functions. We describe practicalexamples of interest and provide a simple probability-change argument in dimension d = 1.
Focusing on regular payoff functions h stable under the facelifting procedure, Section 4 isdedicated to the proof of the necessary and sufficient condition for the so-called first orderviability property. Namely, the first order viability property ensures that the gradient of
the solution of a linear PDE lies in a convex set K on [0, T ] as soon as it does at time T .
This section revisits arguments of [4] in our framework. Section 5 provides the proof of themain theorem and details in particular the corresponding regularization procedure. The
Appendix collects useful properties of the facelift transform and some technical proofs.

3Notations. Any element x ∈ Rd will be identified to a column vector with i-th component
xi and Euclidian norm |x|, (ei )1≤i≤d is the canonical basis of Rd . We denote by Md the setof matrices with d lines and d columns, and Sd the subset of symetric elements of Md . Fora matrix A ∈ Md , Tr[A] denotes its trace, A.j its j-th column, Ai. its i-th row, and Aij thei-th term of A.j . The transpose of a matrix or a vector y will be denoted y > , hx, yi denotes
the product x> y whenever it is defined for x and y vectors or matrices. For a function h
2 h the d-dimensional row vector and the matrix infrom Rd to R, we denote by ∂x h and ∂xx
Md defined by∂x h(x) = ∂h(x)∂x1,...,∂h(x) ∂xd

2∂xxh(x) =andh ∂ 2 h(x) i∂xi ∂xj 1≤i,j≤d,x ∈ Rd .

>dWe shall also note ∂i h(x) := ∂h(x)∂xi , for 1 ≤ i ≤ d and ∇h(x) = [∂x h(x)] , x ∈ R .
Cb1 (Rd , Rq ) denotes the set of function from Rd to Rq which are differentiable with continuous and bounded first derivatives. Finally, we denote by λ the Lebesgue-measure and fora given closed convex set, dK is the (non-negative) distance function to this set.

2

Super-replication and facelift properties

In this section, we introduce the market model and formulate the super-replication problem
under Delta constraints, namely when the number of shares constituting the portfolio mustremain in a closed convex set.

2.1

The market model

We consider a financial market defined on a probability space (Ω, F, P), endowed with ad-dimensional brownian motion W . For 0 ≤ t ≤ T , we denote by Ft = (Fst )t≤s≤T thecompletion of the filtration generated by (Ws − Wt )s≥t . We denote by P t the σ-fieldof progressively measurable processes associated to Ft . In the sequel, we interpret theprobability P as a pricing measure.
We suppose that the financial market is composed of d risky assets and one non-risky
asset, whose interest rate is assumed to be 0 for ease of presentation. Up to considering
discounted processes, all the results of the paper extend straightaway to financial market
with deterministic interest rates.
For an initial condition (t, x) ∈ [0, T ] × Rd , where x represents the vector value of thed assets at time t, the vector of risky asset price process is described by the diffusion(Xst,x )s∈[t,T ] defined as the unique solution of the stochastic differential equation
Z s
t,x
Xs
= x+
σ(Xrt,x )dWr ,s ∈ [t, T ] ,
(2.1)
t

4

where σ : Rd → Md is the volatility function. The Dynkin second order linear differential
operator associated to the dynamics (2.1), denoted by Lσ , is given by1  2
Lσ ϕ(t, x) := ∂t ϕ(t, x) + Tr ∂xx
ϕ(t, x)σ(x)σ > (x) , (t, x) ∈ [0, T ] × Rd ,2
for any ϕ ∈ C 1,2 ([0, T ] × Rd ).
Throughout this paper, we work under the condition that the function σ is Cb1 (Rd , Md )and shall sometimes use the following assumption:- (Hσ) The matrix σ(x) is uniformly elliptic for all x ∈ Rd i.e.
σσ > ≥ Id ,for some  > 0.

Starting with an initial wealth y ≥ 0 at time t ∈ [0, T ], an investment strategy is describedby a P t -measurable process ∆ = (∆1 , . . . , ∆d )> valued in Rd , where ∆is represents thenumber of shares of asset i detained at time s ∈ [t, T ]. We denote by At,x the set of self
financing strategies ∆ such thatZ T
|σ(Xrt,x )∆r |2 dr < ∞ , P − a.s.
t

The portfolio process corresponding to an initial wealth y at time t ∈ [0, T ] and a selffinancing strategy ∆ ∈ At,x is denoted (Yst,x,y,∆ )s∈[t,T ] and satisfies
Z st,x
Yst,x,y,∆ = y +∆>t≤s≤T .
r dXr ,t

The set Abt,x of admissible strategies is given by the strategies ∆ in Abt,x such that theportfolio value Y t,x,y,∆ is bounded from below by a constant.

2.2

Super-replication under constraints

Due to regulatory or structural reasons, we suppose that the possible investment strategies
are restricted to take their values in a deterministic closed convex subset K of Rd s.t.
0 ∈ K.
For any (t, x) ∈ [0, T ] × Rd , the subset of admissible constrained strategies AK
t,x is thendefined byb
AKt,x := ∆ ∈ At,x : dK (∆) = 0 , P ⊗ λ − a.e. .
Observe that the constraint is not imposed on the portfolio value but on the investment
strategy itself. The addition of constraints on the investment strategy implies that exact
replication of a given contingent claim is not always possible, see e.g. [11]. Here, we intend
to focus on super-replication strategies.
5

Definition 2.1 (Super-replication valuation) For a measurable function h : Rd → R
bounded from below, we define the super-replication price of the contingent claim h(XTt,x )under K constraint at time t by


Z T
t,x
h
K
>
t,x
vK (t, x) = inf y ∈ R : ∃∆ ∈ At,x , y + ∆s dXs ≥ h(XT ) P − a.s. ,tfor any (t, x) ∈ [0, T ] × Rd .
The super-replication price of contingent claims has been widely studied in the literature
see e.g. [5, 17, 1]. In our context, a complete characterization of the super-replication
h proved
price under constraint is given in [2] and we will use a supersolution property of vK
therein.
Let us recall that the support function δK of the convex set K defined byδK (x) := sup hx, yi ,x ∈ Rd ,y∈Kwhose domain is denoted K̃ := {ζ ∈ Rd | δK (ζ) < +∞}.
Using the support function of K, we define the following global differential operator related
to the constraints:
CK (p) :=inf|ζ|=1,ζ∈K̃δK (ζ) − ζ 0 p .

Let us also introduce
Definition 2.2 (Facelift) The facelift operator FK for the admissibility set K maps any
measurable function h : Rd → R, to its facelift transform FK [h], defined by
FK [h](x) :=sup h(x + y) − δK (y) = sup h(x + y) − δK (y) ,y∈Rdx ∈ Rd .

y∈K̃

We collect in the Appendix, Section 6.1, some useful properties of the Facelift transform.
In the sequel, we shall use the following assumption related to the payoff function h andits facelift transform:(Hh) The function h is lower semi-continuous, bounded from below and such thathi
E |FK [h](XTt,x )|2 < ∞ ,(t, x) ∈ [0, T ] × Rd .
Let us now recall the following result proved in [2].

6h (t, x) is a viscosity
Proposition 2.1 The super-replication price under K-constraint vKsupersolution of the following PDEmin{ −Lσ u(t, x) , CK (∇u(t, x)) } = 0 ,(t, x) ∈ [0, T ) × Rd

and satisfieshvK(T −, x) ≥ FK [h](x) ,x ∈ Rd ,provided that h is l.s.c, with linear growth and bounded from below.
We conclude this section with a straightforward consequence of the previous result.
Corollary 2.1 Assume that h is l.s.c, with linear growth and bounded from below, thenhihvK(t, x) ≥ E FK [h](XTt,x ) ,(t, x) ∈ [0, T ) × Rd .
For sake of completeness, we provide a proof in the Appendix, see Section 6.2.

3

Relaxing portfolio constraints via terminal facelift

In this section, we investigate under which conditions, super-hedging any claim under
Delta constraints is equivalent to simply hedge the facelift transform of this claim. We
first consider the particular case where the number of shares for each asset is constrained
to stay in-between two constant bounds. In this context, we show that the ‘replication
property’ is always satisfied in the one-dimensional case but not systematically in the multidimensional one. This motivates the second part of this section which presents in Theorem
3.1 a tractable analytical necessary and sufficient condition ensuring this property to holdfor general multi-dimensional convex constraints. We finally focus on several practicalexamples of importance (Black Scholes model, short selling, non-tradable asset, etc.) inorder to emphasize the range of applications for Theorem 3.1, which is the main result of
the paper.

3.1

A motivating example

We consider first a simple practical example where the investor faces constant restrictions
on the number of shares of each asset held in the portfolio. More precisely, the admissibility
set is a closed hypercube given by
Kc := Πdi=1 [−di , ui ] ,where (di , ui ) ∈ [0, ∞]2 , for 1 ≤ i ≤ d .

7

Observe that this form of convex set allows to consider for example the realistic practicalcase where short-selling one or several assets is not allowed. It also covers the natural casewhere some of the assets cannot be traded on the financial market (di = ui = 0).
We first focus on the particular case where only one asset is traded (d = 1). As detailed
in the next proposition, a direct probability change argument shows that super-replicatinga claim under Kc -portfolio constraints simply consists in replicating without constraint the
facelift transform of this claim. For sake of simplicity, we consider here payoff functions withregular facelift transform, but this strong regularity property is relieved in the following
subsection, see Theorem 3.1.
Proposition 3.1 Let d = 1 and h be a payoff function such that FKc [h] ∈ Cb1 (R, R). Then,for any starting point (t, x) ∈ [0, T ] × R, the super-replicating price and hedging strategy
under Kc -constraints of the claim h(XTt,x ) coincides with the exact replicating price andunconstrained hedging strategy of FKc [h](XTt,x ).
Proof. Let (t, x) ∈ [0, T ]×R and consider a payoff function h such that F Kc [h] ∈ C 1 (R, R).
By construction of the facelift transform, ∇FKc [h] is necessarily valued in Kc . We now
consider the exact replicating strategy ∆FKc [h] of FKc [h](XTt,x ) and intend to prove that
∆FKc [h] is valued in Kc on [t, T ]. Due to the regularity of FKc [h], observe that the exact
replicating strategy rewrites"#
t,x
FKc [h]t,x ∇XTt= E ∇FKc [h](XT )Fs ,t≤s≤T ,∆s
∇Xst,xwhere ∇X t,x denotes the tangent process of X t,x and satisfies
Z st,x∇σ(Xrt,x )∇Xrt,x dWr ,t≤s≤T .
∇Xs
= 1+t

Since σ has bounded derivatives, we deduce that (∇Xst,x )t≤s≤T is a positive martingale
with constant expectation equal to 1. Therefore, it also interprets as a probability change∇X
on (Ω, FTt ) and we denote by P∇X the probability defined by dPdP |FTt = ∇XTt,x . This
allows us to rewrite directly
h
i
∇X
F [h]
∆s Kc
= EP
∇FKc [h](XTt,x ) Fst ,t≤s≤T .
Since ∇FKc [h] is valued in the convex set Kc , ∆FKc [h] is also valued in Kc .
The hedging strategy of FKc [h](XTt,x ) being valued in Kc , it coincides with the superhedging strategy under Kc -constraints of h(XTt,x ), see Corollary 2.1. Hence, the superreplicating price of h(XTt,x ) and the replicating price of FKc [h](XTt,x ) also coincide.
2
Remark 3.1 Observe that interpreting the gradient of the stock process as a probabilitychange has already been used for example in [9]. We choose to elicit the case of hypercube
constraints in order to emphasize the simplicity of the proof methodology in this framework.
8

We now turn to the more intricate multi-dimensional case. As detailed in the next
proposition, the previous result easily extends to the particular case where each asset has
its own dynamics, since the previous arguments can be applied on each component of theprice process X.
Proposition 3.2 Let h be a payoff function such that F Kc [h] ∈ Cb1 (Rd , R). Fix (t, x) ∈[0, T ] × Rd and suppose that the dynamics of each asset X t,x,i is given byZ sσi (Xri )dWr ,1≤i≤d,
t≤s≤T .
Xst,x,i = xi +tThen, the super-replicating price and hedging strategy under Kc -constraints of the claimh(XTt,x ) coincides with the replicating price and hedging strategy of FKc [h](XTt,x ).
Proof. Following the same reasoning as in the one-dimensional case, we only require toverify that the exact replicating strategy ∆FKc [h] of FKc [h](XTt,x ) is valued in Kc . As in
the one dimensional case, we have"#i
i ∇X t,x,i


F [h]
T
∆s Kc
= E ∇FKc [h](XTt,x )Fst ,t≤s≤T , 1≤i≤d,∇Xst,x,iwhere ∇X t,x,i is the tangent process of X t,x,i . Due to the particular form of the stock
dynamics, each tangent process ∇X t,x,i is a positive martingale starting from 1. Observe
that, contrary to the one-dimensional proof, we cannot use a common probability change forall the d components of the hedging strategy ∆FKc [h] . Nevertheless, due to the special formof Kc and the fact that ∇FKc [h](Xt, xT ) ∈ Kc , we work separately on each component.
We compute
#
#
"
"
t,x,i
i

∇XTt,x,i
∇X
F
[h]
K
T
Fst ≤ ∆s c
Fst = ui ,−di = E −di≤ E ui∇Xst,x,i∇Xst,x,ifor t ≤ s ≤ T and 1 ≤ i ≤ d. Hence, the replicating strategy ∆FKc [h] is valued in Kc andthe proof is complete.
2
Unfortunately, this nice property does not remain valid for general multi-dimensional
stock dynamics. Consider for example the 2-dimensional case where the dynamics of thefirst asset X 1 is given by
dXt1 = |Xt2 | ∧ σ̄ Xt1 dWt ,with σ̄ > 0 ,and the second asset (the stochastic volatility) is not tradable, i.e. Kc = R × {0}. In thisframework, the super-replicating price of a call (or any convex payoff) option on XT1 issimply the σ̄-volatility Black Scholes price of this call, see e.g. [7].
Hence, even for hypercube type constraints, the exact replication of the facelifted terminal payoff does not always match the constrained super-replication of the payoff. The
purpose of the next section is to investigate the conditions one should impose on the model
dynamics σ and the set K, for this useful property to be satisfied.
93.2

The main result: general convex constraints

We now consider general Delta constraints characterized by a subset K of Rd satisfying
the following assumption :(HK )K is a closed convex subset of Rd with non empty interior and 0 ∈ K.

We consider a stocks’ price process X with general dynamics (2.1). The next theorem
provides a tractable necessary and sufficient condition ensuring that, in order to superreplicate under K-constraints any option with payoff function satisfying (Hh), one simply
needs to replicate the facelift transform of the terminal payoff.
For any point y on the boundary ∂K of K, we denote by NK (y) the set of unitary normal
vectors to K at y i.e.
no
NK (y) :=n ∈ Rd : |n| = 1 and hn, y − y 0 i ≤ 0 ∀y 0 ∈ K .
˘ the set of points y ∈ ∂K where there exists only one normal vector denoted
We define ∂Kn(y), i.e.
˘ := y ∈ ∂K , N (y) = {n(y)} .
∂K(3.1)
˘ is the set of the boundary points where there is a tangent hyperplane,
Equivalently, ∂Ksee [16] for details.
˘ we associate to n(y) a family (n̄` (y), 1 ≤ ` ≤ d) of d vectors
Finally, for any y ∈ ∂K,
such that n̄1 (y) := n(y) and (n̄1 (y), . . . , n̄d (y)) is an orthonormal basis of Rd . We denoteby P (y) the new matrix basis i.e. P (y)e` = n̄` (y), 1 ≤ ` ≤ d. Observe that P (y) is anorthogonal matrix.
When it is clear from context, we shall omit the ’y’ in the above notations, for the reader’sconvenience.
Theorem 3.1 Let us consider the two conditions:(i) For any payoff function h satisfying (Hh) and any (t, x) ∈ [0, T ] × Rd , the superreplicating price and strategy of h(XTt,x ) under K-constraint coincides with the exact
replicating price and unconstrained strategy of the facelifted claim FK [h](XTt,x ).
(ii) The following holds true:hi∂x Tr σσ > (·)n̄` (y)n̄>(y)n(y) = 0 ,k˘for all y ∈ ∂K.
10

2 ≤ k, ` ≤ d ,(3.2)

Under (HK ), we have that (i) implies to (ii). Moreover, if (Hσ) holds then (ii) implies(i).
In order to alleviate the presentation of the paper, the proof of this theorem is postponed to Section 5. Considering first regular payoff functions h which are stable under
the facelifting procedure, the unconstrained hedging strategy of h(XT ) interprets as thesolution of a linear BSDE (or PDE) with terminal condition ∇h(XT ) ∈ K. We introduce
in Section 4 the notion of first order viability for the corresponding BSDE which ensuresthat the solution of the BSDE is valued at any time in K, for any bounded terminal payoff
function of the form ∇h(XT ) lying in K. We then establish in Theorem 4.2 that Condition (ii) above is necessary and sufficient for this newly introduced first order viability
property. The extension of this property to payoff functions satisfying (Hh) is done via aregularization argument presented in Section 5.
Remark 3.2 The previous property extends also naturally to American options, under
additional regularity assumptions. See Remark 4.5 (ii) below for a sketch of proof.
Remark 3.3 Observe that Condition (3.2) is always satisfied if the volatility function σ
is constant. Therefore, the replication property (i) is valid for any closed convex set in theparticular case where the assets’ price X is a multidimensional Brownian motion.
˘ Condition (3.2) can be rewritten fully in the new basis
Remark 3.4 For a fixed y ∈ ∂K,(n̄` (y))1≤`≤d .
Let us define σ̃(·) := P > σ(P ·), then (3.2) simply reads
∂1 σ̃ k. (σ̃ `. )> = 0 ,for all 2 ≤ k ≤ ` ≤ d. If we define X̃ := P > X, then the above condition states that there
is no dependency upon the first component of X̃ of the quadratic covariations of the othercomponents. See below for financial examples.

3.3

Financial applications

We now present financial applications of the main result of the paper. We also precise
the form of the necessary and sufficient condition (3.2) for relevant convex constraints andmodel dynamics in the field of mathematical finance. We successively consider the cases of
illiquid assets, short sell prohibition and restrictions on the total number of positions takenon the financial market. For the particular case of the bidimensional Black Scholes model,
we identify all the constraint types allowing to super-replicate an option via a classical11

replication of the corresponding facelifted payoff. Finally, we emphasize the model dynamics satisfying the viability property for any possible closed convex constraints. Although it
is always the case in dimension d = 1, it appears that no reasonable financial model does
so in dimension d > 1.
Example 1: Non-tradable asset.
In dimension 2, we consider the case where Asset 1 is illiquid and thus cannot be traded.
The corresponding convex set K is the ordinate axis {0} × R. This convex set does notsatisfy Assumption (HK ) since it has an empty interior, but Remark 4.3 below justifies
that Theorem 3.1 is also valid for hyperplanes. The only normal vectors to K are n = (1, 0)
and n = (−1, 0) which lead to the same Condition (3.2), which rewrites

∂1 |σ 21 |2 + |σ 22 |2 = 0 .

(3.3)

This necessary and sufficient condition indicates that the quadratic variation of the second
asset does not depend on the first one. Observe that the condition derived by [4] forclassical viability property rewrites: ∂1 σ 21 = ∂1 σ 22 = 0. This condition is stronger than
(3.3) as expected.
Example 2: No short sell.
Consider a market with two assets where short selling Asset 1 is prohibited i.e. K =R+ × R. Up to their sign, this convex set shares the exact same normal vectors withthe one considered in the previous example. The main observation here is that Condition(3.2) is only related to the border of K so that convex sets with similar borders share thesame viability property. Therefore the prohibition of short selling asset 1 is also related
to Condition (3.3). Moreover, since the corresponding convex sets share the same unit
normal vectors, we emphasize that restricting to portfolios ∆ such that ∆1 ∈ [−a, b] ∩ R
for some a, b ∈ [0, ∞] leads to the same condition (3.3) derived here when (a, b) = (0, ∞).
Similarly, if short selling any of the two assets is prohibited, super-replication reduces tohedging the facelifted claim payoff whenever the stock model satisfies
∂1 |σ 21 |2 + |σ 22 |2 = 0

and



∂2 |σ 11 |2 + |σ 12 |2 = 0 .

(3.4)

In dimension d, when the subset J ⊂ {1, . . . , d} of assets cannot be short sold, the necessary
and sufficient Condition (3.2) rewrites as a constraint on quadratic covariations and takes
the following form:hi∂j σ `1 σ k1 + · · · + σ `d σ kd = 0 ,j ∈ J , `, k ∈ {1, . . . , d} \ {j} .
Example 3: Bound on the number of shares.
We now consider the case where the investor faces a constant upper bound C on the total
number of possible positions he or she can take on the financial market. In dimension 2, thiscorresponds to the consideration of the lozenge convex set K = {(∆1 , ∆2 ) ∈ R2 / |∆1 | +12|∆2 | ≤ C}. Up to their directions, there are two normal vectors for the convex set K:
namely (1, 1) and (−1, 1). Thus, the condition (3.2) rewrites as
(




∂1 |σ 11 − σ 21 |2 + |σ 12 − σ 22 |2 + ∂2 |σ 11 − σ 21 |2 + |σ 12 − σ 22 |2
= 0
 11

 11

.
21
2
12
22
2
21
2
12
22
2
∂1 |σ + σ | + |σ + σ | − ∂2 |σ + σ | + |σ + σ |= 0Observe that this condition is the one given by (3.4), but simply written in a different
orthonormal basis.
Example 4: The multidimensional Black Scholes model.
We now focus on the Black Scholes model in dimension 2 and look towards all the convex
sets K for which super-replicating under K-constraints reduces to replicating the facelifttransform of the claim. We assume that the dynamics of the stocks are given by
dXt = Diag(Xt )ΣdWt ,0≤t≤T ,

with
Σ :=

Σ11
Σ21

Σ12
Σ22

!
and

Diag(Xt ) :=

Xt1
0

0
Xt2

!
,

0≤t≤T ,

where Σ is invertible.
This model being first order viable for all K means that for any (normal) vector (a, b) ∈ R2 ,the condition (3.2) holds true. This rewrites
a∂1 |aΣ21 x2 − bΣ11 x1 |2 + |aΣ22 x2 − bΣ12 x1 |2
+ b∂2 |aΣ21 x2 − bΣ11 x1 |2 + |aΣ22 x2 − bΣ12 x1 |2 = 0,
for any x = (x1 , x2 ) ∈ R2 . Hence a halfspace with unique normal vector (a, b) satisfies the
expected replication property if and only if
ab Σ21 − Σ11 , Σ22 − Σ12 [Σx]> (−b, a)> = 0 ,x ∈ R2 .

Whenever a = 0 or b = 0, i.e. K is an hypercube, we observe that this relation is always
satisfied. Moreover, from Proposition 3.2, we know that the converse is true.
If a 6= 0 and b 6= 0, differentiating the previous expression with respect to the asset value
x leads to(>|b|2 a Σ21 − Σ11 , Σ22 − Σ12 Σ11 , Σ12
= 0,
 21
>
2
21
11
22
12
22
|a| b Σ − Σ , Σ − Σ
Σ , Σ
= 0.
Hence the property is satisfied for any (a, b) ∈ R2 if and only if it holds for (a, b) = (1, 1),
which rewritesΣΣ21 − Σ11 , Σ22 − Σ12

13

>

= 0.

Since Σ is invertible, this condition reads Σ21 = Σ11 together with Σ22 = Σ12 , which
is absurd. Therefore, the only convex sets satisfying the replication property for the bidimensional Black Scholes model with invertible Σ are the hypercubes.
Example 5: Model with unconditional viability property.
We observed in Remark 3.3 that the Brownian motion satisfies the replication property for
any closed convex set K with non empty interior. In dimension 1, Proposition 3.1 statesthat any model satisfies this property. We now intend to identify the models with separate
dynamics and invertible volatily matrix satisfying this property in dimension 2. Hence, we
focus on stock models of the formdXti = σ i1 (Xti )dWt1 + σ i2 (Xti )dWt2 ,0≤t≤T ,

i = 1, 2 .

For any normal vectors (a, b) ∈ R2 , Condition (3.2) rewrites
a∂1 |aσ 21 − bσ 11 |2 + |aσ 22 − bσ 12 |2 + b∂2 |aσ 21 − bσ 11 |2 + |aσ 22 − bσ 12 |2 = 0 .
Hence, a model σ satisfies the replication property for any closed convex set if and only if
ab ∂2 σ 21 − ∂1 σ 11 , ∂2 σ 22 − ∂1 σ 12 σ > (−b, a)> = 0 ,(a, b) ∈ R2 .

Since the volatility function σ is invertible, this condition is equivalent to the relation∂1 σ 11 = ∂2 σ 21

and

∂1 σ 12 = ∂2 σ 22 .

A bi-dimensional model with separate dynamics satisfies the replication property for any
convex set if and only if the two assets share the same linear volatility function up toa constant. Since the volatility matrix requires to be invertible, this constant vector isdifferent from (0, 0), so that the asset values cannot remain non-negative at all times.
Hence, no financial model with positive assets price satisfies the replication property for
any closed convex set in dimension 2.

4

First order viability for the PDE Lσ u = 0

In this section, we introduce a new notion of viability for linear PDEs. Namely, theconstrained super-replication problems considered in this paper entails to find out whether
a hedging strategy with terminal value in K will always do so on the time interval [0, T ].
Since the hedging strategy rewrites as the gradient of a function of X solving the linear
PDE Lσ u = 0, we want to know if the gradient of the solution of this PDE lies in K
on [0, T ] as soon as it does so at time T . This leads to the notion of first order viability
property for the PDE Lσ u = 0, presented in Section 4.1. This notion interprets also interms of viability for linear BSDEs associated to the subclass of gradient type terminal
functions. We verify in Section 4.2 that the PDE Lσ u = 0 is first order viable for a closed
14

convex set K with non empty interior whenever it is first order viable for almost all its
tangent half-spaces. Specializing then our study on first order viability for half-spaces, we
provide in Section 4.3 a geometric condition indicating whether or not the PDE Lσ u = 0
is first order viable for a given half-space. This finally allows us to verify in Section 4.4
that the PDE Lσ u = 0 is first order viable for a closed convex set K if and only if thestructural condition (3.2) on the couple (σ, K) is satisfied.

4.1

First order viability: PDE and BSDE viewpoints

For any h ∈ Cb1 (Rd , R), the price function uh of the European option with terminal payoff
function h and maturity T is given by:hiuh (t, x) := E h(XTt,x ) ,(t, x) ∈ [0, T ] × Rd .
This function interprets as the unique viscosity solution of the parabolic PDE(
Lσ u(t, x) =0for (t, x) ∈ [0, T ) × Rd ,u(T, x) = h(x) for x ∈ Rd ,(4.1)

in the class of continuous functions with polynomial growth, see e.g. [15]. Moreover, byTheorem 3.11 in [12], we deduce that uh is Cb1 on [0, T ] × Rd .
We now introduce the notion of first order viability.
Definition 4.1 The PDE Lσ u = 0 is first order viable for a given set C (or C-first orderviable) if and only if, for any function h ∈ Cb1 (Rd , R) s.t.
∇h(x) ∈ C ,x ∈ Rd ,the function uh defined by (4.1) satisfies∇uh (t, x) ∈ C ,(t, x) ∈ [0, T ] × Rd .

Remark 4.1 Contrary to the classical definition of viability, which requires the function
uh to take value in C, our definition deals with the first derivative of uh which has to bevalued in C.
The first order viability property for the PDE also has a direct interpretation in termsof linear BSDE solution. Indeed, as detailed in the next proposition, ∂x uh (t, x) admits a
BSDE representation, for any (t, x) ∈ [0, T ] × Rd and h ∈ Cb1 (Rd , R).
1

An uniform ellipticity condition for σ appears in the statement of this theorem but this assumption is
not used in the proof and indeed not required.

15

For t ∈ [0, T ], we denote by S 2 [t, T ] the set of Ft -adapted continuous processes (Us )s∈[t,T ]valued in Rd and by H2 [t, T ] the set of Ft -predictable processes (Vs )s∈[t,T ] valued in Md
such that
i
h
i
hZ T
2
|Vs |2 ds < ∞ .
E sup |Us | < ∞ and Ets∈[t,T ]Proposition 4.1 Let h be in Cb1 (Rd , R). For (t, x) ∈ [0, T ] × Rd , we consider the process(∆t,x,h , Λt,x,h ) ∈ S 2 [t, T ] × H2 [t, T ] solution of the BSDE
Z T
Z T
t,x
t,x,h
t,x
t,x,h
∆s
= ∇h(XT ) +Fσ (Xr , Λr )dr −Λt,x,hdWr , s ∈ [t, T ] , (4.2)
rsswhere Fσ : Rd × Md → Rd is defined by
Fσ (x, Λ) :=dX[∂x σ .j (x)]> Λ.j ,x ∈ Rd ,Λ ∈ Md .

(4.3)

j=1

Then we have∆t,x,h= ∇uh (s, Xst,x ) ,sx ∈ Rd .

s ∈ [t, T ] ,Moreover, under (Hσ), we have that Λt,x,h = Γt,x,h σ(X t,x ), for some symetric matrix
valued process Γt,x,h .
Observe that the first order viability for the PDE Lσ u = 0 can be directly rewritten interms of (zero-order) viability property for the linear BSDE (4.2) on a subclass of gradient
payoff functions.
Corollary 4.1 The PDE Lσ u = 0 is first order viable for a given set C if and only if,)t≤s≤T of the BSDE
for any h ∈ Cb1 (Rd , R) such that ∇h belongs to C, the solution (∆t,x,hsd(4.2) belongs P−a.s. to C, for any (t, x) ∈ [0, T ] × R .
Proof of Proposition 4.1. Fix (t, x) ∈ [0, T ] × Rd and h ∈ Cb1 (Rd , R). Let (Y t,x , Z t,x ) ∈S 2 [t, T ] × H2 [t, T ] be the solution to the following BSDE with no driver:
Z T
t,x
t,x
Ys
= h(XT ) −Zrt,x dWr , s ∈ [t, T ] .
sSuch a solution exists and is unique since h ∈ Cb1 (Rd , R) and σ ∈ Cb1 (Rd , Md ). We consider
the inverse of the tangent process ∂x X t,x as well as the tangent process (∂x Y t,x , ∂x Z t,x ) of
(Y t,x , Z t,x ), see e.g. [12]. They have the following dynamics
Z s
d Z s
d
X
X

t,x −1t,x −1.jt,x 2[∂x Xrt,x ]−1 [∂x σ .j (Xrt,x )]dWrj ,[∂x Xs ] = Id +[∂x Xr ][∂x σ (Xr )] dr −t

∂x Yst,x = ∂x h(XTt,x )∂x XTt,x −

j=1

j=1

d Z T
X
j=1

∂x Zrt,x,j dWrj ,

s

16

t

t≤s≤T .

From e.g. Theorem 3.1 in [12], we know that∂x u(t, x) = ∂x Yxt,x .

(4.4)

Recalling that the process (∆t,xs )t≤s≤T is solution to the markovian linear BSDE (4.2)
t,xwith continuous coefficient function, we have that ∆t,xs = v(s, Xs ) for some continuousfunction v, see e.g. Theorem 4.1 in [10]. Observe also that ∆t,xt = v(t, x) is deterministic.
Applying Itô’s formula, we compute using the dynamics of [∂x Xst,x ]−1 and ∂x Y t,x that>[∆t,x
= ∂x Yst,x [∂x Xst,x ]−1 ,s ]t≤s≤T .

(4.5)>
Setting s = t in the above equation and using (4.4), we obtain that ∆t,xt = [∂x u] (t, x).
t,xWe deduce that ∆t,xs = ∇u(s, Xs ), for any s ∈ [t, T ].

When the volatility matrix σ is smooth, one can show, using Feynman-Kac formula,that ∆t,x,h
is a classical solution of a semi-linear PDE and then Λt,x,h= ∂xx u(t, x)σ(x). Intt
the general case, one uses a regularization procedure and the stability property of (linear)
BSDEs to show that Γt,x,h is the limit of symetric matrix and thus symetric itself.
2
The rest of the section is dedicated to the proof of a necessary and sufficient condition for
the first order viability property to hold. It is important to observe that the only possible
terminal conditions for the linear BSDE (4.2) are of the form ∇h(XT ). Hence the viability
characterization for BSDEs derived in [4] does not apply directly here since it requires theconsideration of any terminal condition of the form g(XT ), with g a continuous function.
In the next section, we adapt the arguments developed in [4] using a geometric approach.

4.2

First order viability and half-space decomposition

In this section, we prove that the first order viability for a closed convex set K satisfying(HK ), is characterized by the first order viability of a well chosen collection of half-spaces
H supporting K, i.e. such that K ⊂ H and ∂H ∩ K 6= ∅. For this purpose, we first rewrite
K as the intersection of the corresponding half-spaces and then discuss the related firstorder viability properties.
For y ∈ ∂K, we denote by ρ(y) the radius of the largest closed ball included in K which
is tangent to K at point y i.e.
ρ(y) := sup ρ ≥ 0 | y ∈ B̄(x, ρ) ⊂ K ,for some x ∈ K.

The set of points of ∂K with ρ > 0 corresponds to a subset of points where the convex
˜ namely,surface presents some regularity. We denote this subset by ∂K,˜ := {y ∈ ∂K ,∂K
17

ρ(y) > 0} .

˜ there exists a unique normal vector so that
In particular, observe that for any point in ∂K,˜ ⊂ ∂K˘ ⊂ ∂K, where ∂K
˘ is defined in (3.1). For y ∈ ∂K,
˜ we denote by n(y) the∂K
unique normal vector and by Hy the half-space tangent to K at point y containing K, i.e.
n
o
˜ .
Hy :=
y 0 ∈ Rd , hy 0 − y, n(y)i ≤ 0 ,y ∈ ∂K
Lemma 4.1 Any convex set K satisfying (HK ) rewrites
K =

\

Hy .

(4.6)

˜
y∈∂K

Proof. The proof divides in two steps.
Step 1: Theorem 18.8 in [16] states that x ∈ K if and only ifhx − y, n(y)i ≤ 0 ,˘ .
y ∈ ∂K(4.7)

˘ It is obvious that x ∈ K implies
Let A be a dense subset of ∂K.
hx − y, n(y)i ≤ 0 ,

y∈A.

(4.8)

We are going to verify the converse implication, showing that (4.8) implies (4.7). For this˘ an approximating sequence (yi ) of
to be true, we only need to find for any fixed y ∈ ∂Kpoints in A s.t. yi → y and n(yi ) → n(y).
˘ since A is a dense subset of ∂K,
˘ there exists (yi )i an
Let first observe that for y ∈ ∂K,
approximating sequence of points in A converging to y. Since {n(yi ), i ≥ 0} is compact,
we have that, up to a subsequence still denoted (yi ), (n(yi ))i converges to some unit vector
ν. Moreover, we compute0 ≥ hx − yi , n(yi )i = hx − y, n(yi )i + hy − yi , n(yi )i ,

x∈K,

i≥1.

This implies that hx − y, νi ≤ 0 for any x ∈ K, so that ν is a normal vector for K at y.
˜ we get ν = n(y) and (4.8) holds.
Since y ∈ ∂K,
Step 2: Let B be the unit closed ball and set K α = K ∩ αB for α ∈ N, with α ≥ 1.
˘ \ ∂K˜ and |x| < α, thenHence K = ∪α K α . For α ≥ 1, one observes that if x ∈ ∂K
˘ α \ ∂K˜ α . McMullen [13] shows that ∂K
˘ α \ ∂K˜ α has a zero Rd−1 -lebesgue measure,x ∈ ∂K˜ is dense in ∂K.
˘ Combined withfor any α ≥ 1. Since K = ∪α K α , this implies that ∂K
Step 1, this concludes the proof of the lemma.
2
We now observe that the first order viability property for the convex set K relates to the˜ This nice property allows
first order viability property for every half-space Hy , y ∈ ∂K.
us to restrict our upcoming argumentation to the consideration of viability property for
half-spaces.
18

Proposition 4.2 Let (HK ) be in force. Then, the PDE Lσ u is first order viable for the
˜closed convex set K if and only if it is first order viable for every half-space Hy , for y ∈ ∂K.
The proof of this proposition requires the following lemma, which states the homothetic
stability of the first order viability property.
Lemma 4.2 The PDE Lσ u is first order viable for a closed set C if and only if it is firstorder viable for every closed set λC + y with λ > 0 and y ∈ Rd .
Proof. We fix a set C and choose y ∈ Rd and λ > 0. We suppose that the PDE Lσ u
is first order viable for C and simply need to verify that it is also first order viable for
λC + y. Let h ∈ Cb1 (Rd , R) such that ∂x h> is valued in λC + y and define the functiong ∈ Cb1 (Rd , R) by
g(x) :=

1
h(x) − hx, yi ,λx ∈ Rd .

The gradient of g directly satisfies∂x g(x)> =1
∂x h(x)> − y ∈ C ,λx ∈ Rd .

Since the PDE Lσ u is viable for C we deduce that [∂x ug ]> is valued in C. Moreover, weeasily check that the function (t, x) 7→ λug (t, x) + hx, yi solves the PDE
Lσ u = 0 on [0, T ) × Rd ,u(T, .) = h on Rd .

(4.9)

From uniqueness of the solution to (4.9), we getuh (t, x) = λug (t, x) + hx, yi ,(t, x) ∈ [0, T ] × Rd .

Since [∂x ug ]> is valued in C, we deduce that [∂x uh ]> is valued in λC + y. The arbitrarinessof h indicates that Lσ u is first order viable for λC + y and concludes the proof.
2
We now to turn to the proof of Proposition 4.2.
Proof of Proposition 4.2. The proof divides in two steps, corresponding to each implication.
˜
Step 1: Assume that the PDE Lσ u = 0 is first order viable for any Hy , with y ∈ ∂K.
We deduce from this viability property and the representation of K given in Lemma 4.1
˜
that, for any h ∈ Cb1 (Rd , R) with ∂x h> valued K, [∂x uh ]> is valued in every Hy , y ∈ ∂K.
Using again the representation of K given in Lemma 4.1, we conclude that K is first orderviable.
Step 2: Assume that the PDE Lσ u = 0 is first order viable for K.
19˜ Up toWe intend to prove that it is also first order viable for any Hy with y ∈ ∂K.
considering K − {y} according to Lemma 4.2, we suppose that y = 0.
˜ and denoting by n the normal vector to K at point 0, there exists
By definition of ∂K
R0 > 0 such that the ball B̄(−R0 n, R0 ) ⊂ K is tangent to K at 0. We pick h ∈ Cb1 (Rd , R)with ∂x h> valued in H0 . We choose any arbitrary ε > 0 and intend to prove that [∂x uh ]>is valued in H0 + εn.
Since ∂x h> is bounded, there exists Rε such that ∂x h> is valued in B̄((ε − Rε )n, Rε ).
Therefore, we deduce that∂x h> (x) ∈ B̄((ε − Rε )n, Rε ) =RεRεK + εn ,B̄(−R0 n, R0 ) + εn ⊂R0R0x ∈ Rd .

Since the PDE Lσ u is first order viable for K, Lemma 4.2 indicates that it is also firstRε
Rε
K + εn and therefore [∂x uh ]> is valued in R
K + εn. But, since theorder viable for R
00
half-space H0 is convex cone, we haveRε
K + εn ⊂R0RεH0 + εn = H0 + εn .
R0Thus [∂x uh ]> is valued in H0 + εn, for any ε > 0. Hence, [∂x uh ]> is also valued in H0 .
˜
Therefore, the PDE Lσ u = 0 is first order viable for any hyperplane Hy , with y ∈ ∂K.
2

4.3

First order viability property for half-spaces

The aim of this section is to prove the following result.
Theorem 4.1 If the PDE Lσ u = 0 is first order viable for the half-space H with normal
unit vector n thenhn, Fσ (x, γσ(x))i = 0 ,∀x ∈ Rd , γ ∈ Sd

such thathn, γi = 0 . (4.10)

Moreover, whenever (Hσ) holds, the converse is valid.
The proof of this theorem is done in two steps below, proving each assertion separately,
namely Proposition 4.3 and Proposition 4.4. The proofs follow ideas developed in [4] for(zero-order) viability properties on BSDEs, but are much simpler due to the considerationof half-spaces.
Remark 4.2 (i) The condition hn, Fσ (x, γσ(x))i ≤ 0 has a natural geometric interpretation. For a given terminal condition in H, it indicates that the ∆-component of thesolution of the BSDE (4.2) with generator Fσ is pushed inside the half-space H wheneverit comes near its boundary. This condition takes the form hn, Fσ (x, γσ(x))i = 0 in ourcontext since Fσ given in (4.3) is linear with respect to its second variable.
20

(ii) Assume (Hσ) holds true. The BSDE (4.2) satisfies the classical (zero-order) viability
property for the half-space H if and only if
hn, Fσ (x, γσ(x))i = 0 ,∀x ∈ Rd , γ ∈ Md

such thathn, γi = 0 .

This condition is the one given in [4] adapted to our context, it is stronger than (4.10). It
is important to observe that the restriction to terminal conditions of the form ∇h(XTt,x )translates into the only consideration of symetric matrices γ.
Before proving Theorem 4.1, we state that Condition (4.10) and Condition (3.2) for
half-spaces with normal vector n, are the same.
Lemma 4.3 Condition (4.10) is equivalent tohi∂x Tr σσ > (·)n̄` n̄>
n=0,k

2≤k≤`≤d.

(4.11)

recalling that (n, n̄2 , . . . , n̄d ) is an orthonormal basis of Rd .
Proof. We denote by P the new basis matrix associated to (n(y), n̄2 (y), . . . , n̄d (y)), recall
that P is an orthogonal matrix. We then introduce the family (k` )1≤k≤`≤d of elements of
Sd , given byk` := n̄` (y)n̄k (y)> + n̄k (y)n̄` (y)>>>:= P (e` e>
k + ek e` )Pfor 1 ≤ k ≤ ` ≤ d. This family is a basis of Sd .
Moreover, it is straightforward to show that the family (k` )2≤k≤`≤d is a basis of {γ ∈Sd , s.t. n> γ = 0}. If we assume that>
(4.12)Fσ (·, k` σ) = ∂x [Tr σσ > n̄k n̄>` ] ,it is then clear that Condition (4.10) is equivalent to Condition (4.11).
The following computations prove (4.12):Fσ (·, k` σ) =dX∂x [σ .j ]> k` σ .j

j=1

=

d
X

∂x [σ .j ]> n̄` (y)n̄k (y)> σ .j + ∂x σ .j (x)> n̄k (y)n̄` (y)> σ .j

j=1

=

d
X

∂x [n` (y)> σ .j n̄k (y)> σ .j ]>

j=1

=

d
X

∂x [(P > σ)`j (P > σ)kj ]> = ∂x [(P > σ)`. ((P > σ)k. )> ]>j=1>> > > >>>>= ∂x [e>` P σ(ek P σ) ] = ∂x [n̄` σσ n̄k ]>= ∂x [Tr σσ > n̄k n̄>
` ] .

21

2
We now proceed with the proof of the necessary part of Theorem 4.1.
Proposition 4.3 If the PDE Lσ u = 0 is first order viable for a half-space H with unit
normal vector n, then Condition (4.10) holds for n.
In order to derive this proposition, we use the following technical lemma whose proof isgiven in the Appendix, see Section 6.3.
Lemma 4.4 Let (t, x, n, γ) ∈ [0, T ] × Rd × ∂ B̄(0, 1) × Sd such that hn, γi = 0. If the
PDE Lσ u = 0 is first order viable for a half-space H with unit normal vector n, then ∆t,x
belongs to H where (∆t,x , Λt,x ) is the solution on [t, T ] of the BSDE (4.2) associated to theterminal condition γ(XTt,x − x).
Proof of Proposition 4.3. Using Lemma 4.2, we can assume w.l.o.g. that 0 ∈ ∂H.
In order to verify that Condition (4.10) holds for the vector n ∈ ∂ B̄(0, 1), we pick any
x ∈ Rd and γ ∈ Sd such that hn, γi = 0. Since one can choose either γ or −γ and the map
γ 7→ Fσ (x, γσ(x)) is linear, we only need to check thathn, Fσ (x, γσ(x))i ≤ 0 .

(4.13)

We pick ε ∈ [0, T ] and denote by X ε solution on [T − ε, T ] of the SDE (2.1) starting in x
at time T − ε and by (∆ε , Λε ) the solution on [T − ε, T ] of the BSDE (4.2) associated tothe terminal condition γ(XTε − x). We deduce from hn, γi = 0 that hn, γ(XTε − x)i = 0 sothat ∆εT = γ(XTε − x) ∈ ∂H. This implies that
Z T
Z T
ε
ε
ε
hn, Λεs idWs ,T −ε≤t≤T .
hn, Fσ (Xs , Λs )ids −hn, ∆t i =ttSince the PDE Lσ u = 0 is first order viable for the half-space H, Lemma 4.4 indicates that∆ε belongs to H and thushn, ∆εt i ≤ 0 ,T −ε≤t≤T .

(4.14)

ˆ ε on [T − ε, T ] by
Let us define the process ∆ˆ ε = γ(X ε − x) + Fσ (x, γσ(x))(T − t) −
∆
t
T

Z T

γσ(Xsε )dWs ,T −ε≤t≤T .

tSince hn, γi = 0, we compute directlyˆ ε i = hn, Fσ (x, γσ(x))iε .
hn, ∆
T −ε

(4.15)

ˆ ε i/ε is non-positive for εIn particular, observe that (4.13) is satisfied as soon as hn, ∆T −εsmall enough.
22ˆ ε , γσ(X ε )) are solutions on [T − ε, T ] of BSDEs with the same
Besides, (∆ε , Λε ) and (∆terminal condition γ(XTε − x) and respective drivers (Fσ (Xs , .))s and Fσ (x, γσ(x)). The
stability property for BSDE, see for e.g. Prop 4.1 in [4], reads
"Z
2 #
T
ˆ ε − ∆ε |2 ≤ CEE|∆
|Fσ (X ε , γσ(X ε )) − Fσ (x, γσ(x))|ds
,
T −ε

T −ε

s

T −ε

s

where C is a non-negative constant, which may change from line to line and does notdepend on ε.
The Cauchy Schwartz inequality together with the Lipschitz property of the driver withrespect to its second variable leads toˆ εT −ε − ∆εT −ε |2 ≤ Cε
E|∆

Z T
T −ε



E |γσ(Xsε ) − γσ(x)|2 + |Fσ (Xsε , γσ(x)) − Fσ (x, γσ(x))|2 ds≤ Cε2 (ε + αε ) ,(4.16)

where the last inequality follows from classical estimates on the forward diffusion X ε on
[T − ε, T ], and αε is given by#"αε := EsupT −ε≤s≤T|Fσ (Xsε , γσ(x)) − Fσ (x, γσ(x))|2.

Observe from the Markov property of the process X ε that αε rewrites
αε = E sup |Fσ (Xs0,x , γσ(x)) − Fσ (x, γσ(x))|2 .
0≤s≤εSince σ ∈ Cb1 (Rd , Rd ), the function Fσ (., γσ(x)) given in (4.3) is continuous and bounded.
Therefore, the continuity of the process X 0,x together with the dominated convergence
theorem ensures that αε goes to 0 as ε does so. Thus, we deduce from (4.16) that
ˆε
∆ε
∆
T −ε
− T −ε
ε
ε

−→ 0

as

ε→0.

L2

ˆ ε /ε and ∆ˆ ε /ε share a.s. the same limit.
Up to a subsequence, this implies that ∆
T −ε
T −ε
Therefore (4.14) together with (4.15) provideˆ εt ihn, ∆εt i
hn, ∆
= lim
≤ 0,
ε→0
ε→0
ε
ε

hn, Fσ (x, γσ(x))i = lim

2which concludes the proof.

Remark 4.3 Observe that the same line of arguments indicates that (4.10) is satisfied
for a given vector n ∈ ∂ B̄(0, 1) as soon as the PDE Lσ u = 0 is first order-viable for an
hyperplane ∂H with normal vector n. One simply needs to work with conditions of theform hn, .i = 0 instead of hn, .i ≤ 0 in the above proof.
23

Moreover, observe that ∂H rewrites H ∩ H 0 with H 0 a half-space with normal vector−n, and (4.10) is automatically satisfied for −n as soon as it is valid for n. Therefore,
Proposition 4.4 below indicates that the PDE Lσ u = 0 is first order-viable for both half
spaces H and H 0 whenever (4.10) holds for n. Thus this condition is also necessary andsufficient in order to ensure that the PDE Lσ u = 0 is first order viable for any hyperplanewith normal vector n.
Proposition 4.4 Suppose that Condition (4.10) holds for the vector n ∈ ∂ B̄(0, 1). Then,
under (Hσ), the PDE Lσ u = 0 is first order viable for any half-space H with normal vectorn.
Proof. Without loss of generality, we assume that 0 ∈ ∂H, recalling Lemma 4.2.
Let h be a function in Cb1 (Rd , R) with gradient valued in H. Let X and (∆, Λ) denote therespective solutions of the SDE (2.1) and the BSDE (4.2) associated to any fixed starting
point (t, x) in [0, T ] × Rd and terminal condition h(XT ). We intend to prove that ∆ isvalued in H on [t, T ]. The first order viability of Lσ u = 0 for H is then a direct consequenceof Corollary 4.1.
Using Proposition 4.1, we have, for t ≤ s ≤ T
Z T
Z T
n> ∆s = n> ∆T +n> Fσ (Xu , Γu σ(Xu ))du −n> Γu σ(Xu )dWu ,s(4.17)swhere Γ is valued in Sd . Recalling the proof of Lemma 4.3, we write Γu :=P
and observe that n> Γu = d`=1 γu1` n> 1` = γu1. P > .

Pk`1≤k≤`≤d γu k`Moreover, since ∆hT = ∇h(XT ) ∈ H, we get from (4.17) that
Z T

>

X

n ∆s ≤
sγuk` ∂x [Tr σσ > n̄k n̄>` ](Xu )ndu −Z T

γu1. P > σ(Xu )dWu ,(4.18)

s1≤k≤`≤dfor all s ∈ [t, T ], recalling (4.12).
Using Condition (4.10) in its equivalent form (4.11), we obtain>n ∆s ≤

Z TXdsγu1` ∂x [Tr σσ > n̄1 n̄>` ](Xu )ndu −Z T

γu1. P > σ(Xu )dWu ,(4.19)s`=1

for s ∈ [t, T ]. Defining the vector valued process θu s.t. θu` = ∂x [Tr σσ > n̄1 n̄>` ](Xu )n,1 ≤ ` ≤ d, we computeZ Tn> ∆s ≤ −γu1. P > σ(Xu )(dWu − [P > σ(Xu )]−1 θu du) , t ≤ s ≤ T ,(4.20)s

Under (Hσ), applying Girsanov Theorem, we know that there exists an equivalent probR
ability measure Pθ , under which the process W − [P > σ(Xu )]−1 θu du is a Brownian motion.
24

Taking the expectation under this new probability measure in (4.20), we obtainn> ∆s ≤ 0,t≤s≤T ,
2

which concludes the proof.

Remark 4.4 Let H be a half-space with unit normal vector n and τ a stopping time valued
in [0, T ]. Whenever Condition (4.10) holds for n, we observe that the same arguments asin the above proof show that the first component of the solution of the BSDE (4.2) lies in
H on [0, τ ], if it belongs to H at time τ .

4.4First order viability for general convex sets

As established in Proposition 4.2, the first order viability of a closed convex set K withnon empty interior is characterized by the first order viability of supporting half-spaces Hy
˜ We shall verify in this section that it is also characterized
tangent to K at points y ∈ ∂K.
in terms of first order viability on the largest class of supporting hyper-spaces Hy associated˘ More importantly, we derived in Theorem 4.1 a necessary and sufficient
to any y ∈ ∂K.
analytical condition ensuring the PDE Lσ u = 0 to be first order viable for a given halfspace. Combining these observations provides therefore a similar condition for any closed
convex set K with non empty interior.
Theorem 4.2 Let (HK ) be in force. If the PDE Lσ u = 0 is first order viable for theconvex set K, then the condition (3.2), which rewrites equivalently
hn(y), Fσ (x, γσ(x))i = 0 ,˘ , γ ∈ Sd
for all x ∈ Rd , y ∈ ∂K

such that hn(y), γi = 0 ,(4.21)

is satisfied. Besides, whenever (Hσ) holds, the converse is valid.
Proof. The proof is performed in several steps.
Step 1: (4.21) implies the first order viability property.
˘
We assume in this step that (4.21) is satisfied and (Hσ) holds. Then, for any y ∈ ∂K,
Theorem 4.1 indicates that the PDE Lσ u = 0 is first order viable for the supporting half˜ andspace Hy . In particular, it is first order viable for any half-space Hy with y ∈ ∂K
Proposition 4.2 implies that it is first order viable for the convex K.
Step 2: The first order viability property implies (4.21).
Assume that the PDE Lσ u = 0 is first order viable for K. Proposition 4.2 indicates that˜ and Theorem 4.1 implies that
this is also true for any half-space Hy with y ∈ ∂K
hn(y), Fσ (x, γσ(x))i = 0 ,˜ , γ ∈ Sd
∀x ∈ Rd , y ∈ ∂K
25such thathn(y), γi = 0 .

˘It remains to check that this relation is also valid for a given y ∈ ∂K.
˘
˘ is the limit of a
Let y be in ∂K.
As observed in the proof of Lemma 4.1, y ∈ ∂K
˜ s.t. n(yp ) → n(y). Fix now some (x, γ) ∈ Rd × Sd
sequence (yp )p∈N of points lying in ∂Ksatisfying hn(y), γi = 0. Then there exists a sequence (γp ) in Sd such that hn(yp ), γp i = 0
for any p ≥ 1 and γ = limp→∞ γp . Indeed, since γ ∈ Sd , there exists O ∈ Md s.t.
OO> = O> O = Id and O> γO = D where D is a diagonal matrix with D1,1 = 0. The
matrix O corresponds to the new basis matrix from the canonical basis to an orthonormal
basis B 0 = (n(y), n2 , . . . , nd ). Then consider the basis Bp0 obtained by applying the GramSchmidt orthonormalization procedure to the basis (n(yp )p , n2 , . . . , nd ) and denote by Op
the new basis matrix for Bp0 for p ≥ 1. Since the Gram-Schmidt orthonormalization
procedure is continuous, we get that limp→∞ Op = O. Then define the sequence (γp )p in
Sd byγp := Op> DOp ,p≥1.

From the definition of the basis Bp0 and since D1,1 = 0 we have hn(yp ), γp i = 0 for all p ≥ 1.
Moreover, since limp→∞ Op = O, we have limp→∞ γp = γ.
˜ and hn(yp ), γp i = 0, we get
Now, since yp ∈ ∂K
hn(yp ), Fσ (x, γp σ(x))i = 0 ,for any p ≥ 1. Letting p go to infinity, we obtain hn(y), Fσ (x, γσ(x))i = 0 and (4.21) holds
˘ × Rd × Sd such that hn(y), γi = 0.
also for (y, x, γ) ∈ ∂K
2
Remark 4.5 (i) As observed in Remark 4.4, Condition (3.2) also ensures that, if the firstcomponent ∆ of the solution to the BSDE (4.2) lies in H at a given stopping time τ valued
in [0, T ], ∆ remains in K on [0, τ ]. Therefore Condition (3.2) is necessary and sufficient toensure that the BSDE (4.2) satisfies the first order viability property for K on any random
time interval [0, τ ], with τ stopping time smaller than T .
(ii) Consider for example an American option whose exercise payoff is h(X t,x ) on [t, T ].
We assume that h ∈ Cb1 (Rd , R) with ∇h valued in K. We denote by τ ∗ the optimal stopping
time. Under some regularity assumptions, it is known that the Delta of the option at timet,xt is ∆t,x
t , where the terminal condition in (4.2) is now random and given by ∇h(Xτ ∗ ) ∈ K,
see e.g. [8] Theorem 2.3 and the references therein. One can then apply (i) above toconclude that Theorem 3.1 holds true for American Option, under strengthened regularity
assumption.

5

Proof of Theorem 3.1

In this Section we prove the main result of this paper, i.e. Theorem 3.1, using the resultsof Section 4 together with some regularization arguments. We prove each implication26

separately.

5.1(i) =⇒ (ii)Let h be any function in Cb1 (Rd , R) such that ∇h is valued in K. Our goal is to show that∇uh is valued in K when (i) holds true (h is not necessarily bounded from below). If this is
the case, then the PDE Lσ u = 0 is first-order viable and the statement is a straightforward
consequence of Theorem 4.2.
We now construct an approximating sequence of functions (hn )n≥0 s.t. hn satisfies (Hh)and ∇uhn (t, x) → ∇uh (t, x), as n → ∞.
To this end, we introduce, for n ≥ 0,Z z1F[0,1] [1[−n,∞) ](y)dy − n +,fn : z ∈ R 7→
2−∞
where F[0,1] is the facelift operator on R associated to the convex set [0, 1]. We then define
the sequence (hn )n by hn = fn ◦ h for all n ≥ 0. We notice that fn is lower bounded andfn (y) = y for all y ≥ −n, and all n ≥ 0. We compute that ∇hn = fn0 (h)∇h and since
fn0 ∈ [0, 1], ∇hn ∈ [0, ∇h] ⊂ K. Thus hn satisfies (Hh). Moreover, we have hn (y) → h(y)and ∇hn (y) → ∇h(y), as n → ∞, for all y ∈ Rd . Using the representation of Proposition
4.1 and usual stability arguments for BSDEs, we obtain that ∇uhn (t, x) → ∇uh (t, x), as
n → ∞. Under (i), we have that ∇uhn takes its values in K and so does ∇uh .
25.2(ii) =⇒ (i)Step 1: Replication strategy for FK [h](XTt,x ).
Let fix (t, x) ∈ [0, T ) × Rd . Under (Hh), h is bounded from below. Hence, Lemma 6.1 (v)ensures that FK [h](XTt,x ) is also bounded from below. Using the martingale representation
Theorem, we have
Z Tt,x
FK [h]u(t, x) = FK [h](XT ) −Zst,x dWs ,twhich allows us to define the replicating financial strategy (∆s )s∈[t,T ] for FK [h](XTt,x ) by∆>:= Zst,x σ −1 (Xst,x ) ,ss ∈ [t, T ] .

Since Z t,x ∈ H2 [t, T ] and σ −1 (X t,x ) is a continuous process, this strategy is obviously
admissible.
Step 2: Viability of the replication strategy of FK [h].
We now prove that the replicating strategy Φ is admissible. The main difficulty relies here
in the lack of regularity of the payoff function under assumption (Hh). We therefore use
an approximation argument and proceed in two substeps.
27Substep 2.a: Regularization of h.
Under (Hh), we consider the Lipschitz-regularization (hn ) of h given in Lemma 6.2. We
introduce the sequence (n Z t,x ) given by
Z T
t,x
n t,x
FK [hn ]Zs dWs ,n≥1.
u(t, x) = FK [hn ](XT ) −t

Using (Hh), Lemma 6.2 and the dominated convergence theorem, we easily obtainhilim E |FK [h](XTt,x ) − FK [hn ](XTt,x )|2 = 0n→∞

and thus n Z t,x → Z t,x in H2 [t, T ].
Defining the process n ∆> := n Z t,x σ −1 (X t,x ) we directly deduce that up to a subsequencen∆ → ∆,P ⊗ λ a.e. on Ω × [t, T ] .

(5.1)

Substep 2.b: Regularization of FK [hn ].
Fix n ∈ N and let ϕ be a compactly supported smooth probability density function on Rd .
We define the sequences of function (ϕk )k≥1 and (Fn,k )k≥1 from Rd to R by ϕk : x 7→
k d ϕ(kx) andZ
Fn,k : x 7→ ϕk ∗ FK [hn ](x) =ϕk (x + y)FK [hn ](x)dy ,Rdfor any k ≥ 1. Let us introduce k,n Z given by the martingale representation Theorem
Z T
t,x
Fn,k
k,n t,xu(t, x) = ϕk ∗ FK [hn ](XT ) −Zs dWs ,n, k ≥ 1 .
t

As in the previous step, we define the sequence of processes k,n ∆> := k,n Z t,x σ −1 (X t,x ).
We observe that, up to a subsequence, k,n ∆ → n ∆ a.e. on Ω × [t, T ] as k goes to ∞.
Besides, Theorem 3.1 in [12] directly implies thatk,n∆s = ∇uFn,k (s, Xst,x ) ,t≤s≤T ,n, k ≥ 1 .

(5.2)

Now, since FK [hn ] is a bounded Lipschitz function, combining Rademacher Theorem with
Lemma 6.3 (i) we have CK ∂x FK [hn ](x) ≥ 0, for almost every x ∈ Rd , which means∇FK [hn ] ∈ K ,a.e. on Rd ,We also observe that, since FK [hn ] is Lipschitz cotinuous, we have from the dominatedconvergence theorem∂j Fn,k = ϕk ∗ ∂j ∇FK [hn ],1≤j≤d.

Now, since K is closed and convex, we obtain∇Fn,k ∈ K ,28n, k ≥ 1 .

(5.3)

Applying Theorem 4.2, we deduce that (∇uFn,k (t, .))n,k and thus (n,k ∆)n,k are valued inK, recalling (5.2).
Substep 2.c: Viability of the replicating strategy of FK [h].
For any n ≥ 1, since k,n ∆ → n ∆ a.e. as k goes to infinity, the closeness of K implies thatn ∆ is valued in K a.e.
A similar argument yields that ∆ is also valued in K, recalling Step 1. Since ∆ is anadmissible strategy, we conclude that ∆ ∈ AKt,x .
Step 3: Identification of the super replicating price of h and the replicating price of FK [h].
h (t, x) of h(X t,x ),Substep 2.c yields that uFK [h] (t, x) dominates the super-replication price vKTrecalling Definition 2.1. The proof is concluded using Corollary 2.1.
2

6

Appendix

6.1

Facelift properties

The first lemma collects some useful properties of the facelift transform. Lemma 6.2 is anapproximation result and Lemma 6.3 is a (minimal) PDE characterisation of the facelift.
Lemma 6.1 (i) If h is lower semi-continuous, then FK [h] is also l.s.c..
(ii) If h(x) ≥ g(x) for all x, then FK [h](x) ≥ FK [g](x), for all x ∈ Rd .
(iii) If 0 ∈ K and h(.) = c with c a given constant then FK [h] = h.
(iv) FK [h ∨ g] = FK [h] ∨ FK [g].
(v) FK [h] ≥ h and FK [h] = FK [FK [h]].
Proof. Property (i) holds true since δK is assumed to be continuous on its domain,
the supremum of l.s.c functions being l.s.c as well. Properties (ii)-(iii)-(iv) are obvious
consequences of the Definition 2.2 of the facelift transform. Property (v) follows from thefact that 0 ∈ K̃ and the following computation
FK [FK [h]](x) = sup FK [h](x + y2 ) − δK (y2 )y2 ∈Rd=suph(x + y2 + y1 ) − δK (y1 ) − δK (y2 ) ≤ FK [h](x) ,y1 ,y2 ∈Rd

2for any x ∈ Rd .

Lemma 6.2 Assume that h is lower semi-continuous and bounded from below by −mh ,for some mh ≥ 0.
29

Then, there exists an increasing sequence (hn )n≥1 of bounded Lipschitz function, uniformly
bounded from below by −mh converging to h and such that FK [hn ] ↑ FK [h].
As a by-product, we obtain that FK [h] is lower semi-continuous.
Proof. We define the sequence of functions (gn )n bygn (x) = inf h(y) + n|x − y| ,x ∈ Rd ,y∈Rdfor n ≥ 1. It is clear that the sequence (gn )n is nondecreasing, that −mh ≤ gn ≤ h and gn
is n-Lipschitz continuous for all n ≥ 1.
We now prove that (gn )n converges pointwise to h. Fix some x ∈ Rd . Since h is l.s.c andbounded from below there exists a sequence (xn )n in Rd such thatgn (x) = h(xn ) + n|x − xn | ,

n≥1.

(6.1)

Since h is bounded from below by −mh , we deducen|x − xn | ≤ h(x) − h(xn ) ≤ h(x) + mh ,n≥1,

so that limn→∞ xn = x. Together with (6.1) and the lower semi continuity of h, this yields
lim gn (x) ≥ lim inf h(xn ) ≥ h(x) .

n→∞

n→∞

Thus, gn (x) ↑ h(x) as n ↑ ∞, for all x ∈ Rd .
Define now the sequence of functions (hn )n by
hn (x) := gn (x) ∧ n ,x ∈ Rd , ,n≥1.

Since gn is Lipschitz continuous and bounded from below, hn is bounded and Lipschitz
continuous, for all n ≥ 1. Moreover, since (gn )n is nondecreasing and converges pointwise
to h, we also get that (hn )n is nondecreasing and converges pointwise to h.
It remains to prove the convergence of FK [hn ] to FK [h]. For any x ∈ Rd , we simply
observe thatFK [h](x) = sup h(x + u) − δK (u) =
u∈K̃

sup

hn (x + u) − δK (u) = lim ↑ FK [hn ](x) .
n→∞n≥1,u∈K̃

2
Lemma 6.3 Let h be a Lipschitz continuous function from Rd to R.
(i) Assume that FK [h] is locally bounded, then FK [h] is a viscosity super-solution ofmin{CK (∂x u), u − h} = 0

on

Rd .

(ii) Let v be a differentiable super-solution of (6.2), thenv(x) ≥ FK [v](x) ≥ FK [h](x) ,x ∈ Rd .

In particular, if h is differentiable and ∇h ∈ K, then FK [h] = h.
30

(6.2)

Proof. Step 1: Proof of (i).
We first notice that, since h is Lipschitz continuous, FK [h] is also Lipschitz continuous.
Let x̄ ∈ Rd and φ ∈ C 1 (Rd , R) a test function such that
0 = FK [h](x̄) − φ(x̄) = (strict) min (FK [h] − φ)(x) .
x∈Rd

(6.3)

Observe that Lemma (6.1) (v) implies FK [FK [h]] = FK [h], so thaty ∈ Rd .

FK [h](x̄) ≥ FK [h](x̄ + y) − δK (y) ,Using (6.3), we deduceφ(x̄) ≥ φ(x̄ + y) − δK (y) ,y ∈ Rd .

In particular, for y = εζ where ε > 0 and ζ ∈ K̃ with |ζ| = 1, we obtainφ(x̄) − φ(x̄ + ζ)≤ −δK (ζ) ,y ∈ K̃ .

Letting ε goes to 0 yields δK (ζ) − ∂x φ(x̄)ζ ≥ 0. Since ζ is arbitrarily chosen in K̃, thisyields CK (∂x φ)(x̄) ≥ 0.
Step 2: Proof of (ii).
Let v be a differentiable supersolution of (6.2). We then have
δK (y) − ∂x v(x + ty)0 y ≥ 0 ,(t, x, y) ∈ [0, T ] × Rd × K̃ .

Fix now x̄ ∈ Rd . We get from the previous inequality
Z T∂v(x̄ + ty) δK (y) −dt ≥ 0 ,∂t
0y ∈ Rd .

Therefore, we computev(x̄) ≥ v(x̄ + y) − δK (y) ,y ∈ Rd .

Taking the supremum over y, we obtain v(x̄) ≥ FK [h](x̄).
Suppose now that h is differentiable and [∂x h]> ∈ K. Since we already know thatFK [h] ≥ h, we conclude FK [h] = h.
2

6.2

Proof of Corollary 2.1

h by
For sake of clarity, let us define ṽKhhṽK(t, x) := vK(t, x)for (t, x) ∈ [0, T ) × Rd ,hhṽK(T, x) := vK(T −, x)

31

for x ∈ Rd .

Recall also the definition of uFK [h] and (uFK [hn ] )n in the proof of Theorem 3.1, Section 5.2,
Step 1 and Substep 2.a and the fact thatFK [h](x) = lim ↑ FK [hn ](x)n∞anduFK [h] (t, x) = lim ↑ uFK [hn ] (t, x) ,n∞for (t, x) ∈ [0, T ] × Rd .
From the left-hand equality of the above statement and Proposition 2.1, we deduce thath is a viscosity super-solution ofṽK(−Lσ u(t, x) =0for (t, x) ∈ [0, T ) × Rd ,(6.4)
u(T, x) = FK [hn ](x) for x ∈ Rd .
Since FK [hn ] is Lipschitz continuous, it is also well known (see e.g. [14]) that uFK [hn ] is aviscosity solution of (6.4), for any n ≥ 1.
The PDE (6.4) satisfies the assumptions of Theorem 4.4.5 in [15], which provides a strong
comparison theorem for viscosity solutions with polynomial growth. Since the functions
h have linear growth, this yields
uFK [hn ] and ṽKhṽK(t, x) ≥ uFK [hn ] (t, x) ,(t, x) ∈ [0, T ] × Rd ,for any n ≥ 1. The proof is concluded letting n go to infinity.

6.3

2

Proof of Lemma 4.4

We fix (t, x) ∈ [0, T ] × Rd . We first notice that, since γ ∈ Sd , the terminal condition
γ(XTt,x − x) can be written under the form γ(XTt,x − x) = ∂x h̄(XTt,x ) with h̄ defined byh̄(x0 ) =1 0
hx − x, γ(x0 − x)i ,2x 0 ∈ Rd .

However, we cannot directly conclude from the H-first order viability of Lσ u = 0 that∆t,x belongs to H, since the terminal payoff function h does not belong to Cb1 (Rd , R). We
therefore construct a sequence (hp )p valued in Cb1 (Rd , R) approximating h.
Since hn, γi = 0, we can write γ = γ + + γ − with γ + and γ − two elements of Sd which
are respectively non-negative and non-positive and satisfy hn, γ + i = hn, γ − i = 0.
For all p ≥ 1, define the function hp by0− 0
hp (x0 ) = h+p (x ) + hp (x ) ,x0 ∈ Rd ,where the function h+p is defined by(0h+p (x )=

p
p | γ + (x0 − x)| − p2

if

1 + 0
0
2 hγ (x − x), (x − x)i

if

32

p
γ + (x0 − x)| ≥ p ,p| γ + (x0 − x)| ≤ p ,|and h−p is defined by(0h−p (x )=

p
−p | −γ − (x0 − x)| − p2

if

1 − 0
0
2 hγ (x − x), (x − x)iif

for all x0 ∈ Rd . We then easily check that
 p √γ ± (x0 −x)
0 >| ±γ ± (x0 −x)|∂x h±=p (x )
 γ ± (x0 − x)

if
if

p
| −γ − (x0 − x)| ≥ p ,p| −γ − (x0 − x)| ≤ p ,p| ±γ ± (x0 − x)| ≥ p ,p| ±γ ± (x0 − x)| ≤ p .

for all x0 ∈ Rd . Therefore we get from the dominated convergence theorem thath2i
E ∂x hp (XTt,x ) − ∂x h(XTt,x )−→ 0 ,as p → ∞ .

(6.5)

Observe that hp ∈ Cb1 (Rd , R) and ∂x hp is valued in H, for all p ≥ 1. Since the PDE
Lσ u = 0 is first order viable for H, we deduce from Proposition 4.1 that∆ps ∈ H ,t≤s≤T ,

p≥1,

(6.6)

where (∆p , Λp ) is the solution on [t, T ] of the BSDE (4.2) associated to the terminal
condition ∂x hp (XTt,x )> .
We get from (6.5) and classical estimates on BSDEs thath2ip
E sup ∆t,x−∆−→ 0 ,ss

as p → ∞ .

t≤s≤TSince H is closed, (6.6) together with the previous convergence imply that ∆t,x is valued
in H.

2

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