1Introduction DylanPossamaï GuillaumeRoyer NizarTouzi Ontherobustsuperhedgingofmeasurableclaims

ISSN:1083-589X in PROBABILITY

On the robust superhedging of measurable claims

Dylan Possamaï

Guillaume Royer

Nizar Touzi

Abstract

The problem of robust hedging requires to solve the problem of superhedging under a nondominated family of singular measures. Recent progress was achieved by van Handel, Neufeld, and Nutz. We show that the dual formulation of this problem is valid in a context suitable for martingale optimal transportation or, more generally, for optimal transportation under controlled stochastic dynamics.

Keywords:Robust hedging ; quasi-sure stochastic analysis.

AMS MSC 2010:93E20 ; 91B30 ; 91B28.

Submitted to ECP on April 10, 2013, final version accepted on December 9, 2013.

1 Introduction

An important attention is focused on the problem of robust superhedging in the recent literature. Motivated by the original works of Avellaneda [1] and Lyons [2], the first general formulation of this problem was introduced by Denis and Martini [4] by considering the hedging problem under a nondominated family of probability measures on the canonical space of continuous trajectories. Since the hedging problem involves stochastic integration, [4] used the capacity theory to develop the corresponding quasi- sure stochastic analysis tools, i.e. stochastic analysis results holding simultaneously under the considered family of non-dominated measures.

The next progress was achieved by Soner, Touzi and Zhang [13] who introduced a restriction of the set of non-dominated measures so as to guarantee that the predictable representation property holds true under each measure. However, [13] placed strong regularity conditions on the random variables of interest in order to guarantee the measurability of the value function of some dynamic version of a stochastic control problem, and to derive the corresponding dynamic programming principle.

By using the notion of measurable analyticity, Nutz and van Handel [11] and Neufeld and Nutz [9] extended the previous results to general measurable claims by introducing some conditions that the non-dominated family of singular measures must satisfy.

The main objective of this paper is to extend the approach of Neufeld and Nutz [9]

so as to introduce some specific additional constraints on the family of probability mea- sures, and to weaken the integrability condition on the random variables of interest.

∗CEREMADE, Université Paris Dauphine, France. E-mail:[email protected]

†CMAP, École Polytechnique Paris, France. E-mail:[email protected]

‡CMAP, École Polytechnique Paris, France. E-mail:[email protected]Research supported by the ChairFinancial Risksof theRisk Foundationsponsored by Société Générale, and the ChairFinance and Sustainable Developmentsponsored by EDF and Calyon.

Such an extension is crucially needed in the recent problem of martingale transporta- tion problem [5, 6], where the superhedging problem allows for the static trading of any Vanilla payoff in addition to the dynamic trading of the underlying risky asset. Assum- ing that the financial market, with this enlarged possibilities of trading, satisfies the no-arbitrage condition leads essentially to the restriction of the family of probability measures to those under which the canonical process is a uniformly integrable mar- tingale. The main problem is that this restriction violates the conditions of [9] on one hand, and that the integrability conditions in [9] are not convenient for the stochastic control approach of [5, 6].

The paper is organized as follows. Section 2 introduces the main probabilistic frame- work. The robust superhedging problem is formulated in Section 3, where we also re- port our main result, together with the comparison to [9]. Section 4 contains the proof of the duality result. Finally, some extensions are reported in Section 5.

2 Preliminaries

2.1 Probabilistic framework LetΩ :={ω∈C [0, T],R^d

:ω0= 0}be the canonical space equipped with the uni- form norm||ω||^T_∞:= sup_0≤t≤T|ωt|. F will always be a fixedσ-field onΩwhich contains all our filtrations. We then denote B the canonical process, P⁰ the Wiener measure, F:={Ft}0≤t≤T the filtration generated byB andF⁺:={F_t⁺,0≤t≤T}, the right limit ofF^whereF_t⁺:=∩_s>tF. We will denote byM(Ω)the set of all probability measures on Ω. We also recall the so-called universal filtrationF^∗:={F_t^∗}_0≤t≤T defined as follows

F_t^∗:= \

P∈M(Ω)

F_t^P,

whereF_t^Pis the usual completion underP^.

For any subset E of a finite dimensional space and any filtration X^on (Ω,F), we denote byH⁰(E,X)the set of allX-progressively measurable processes with values in E. Moreover for allp >0 and for allP∈ M(Ω), we denote byH^p(P, E,X)the subset ofH⁰(E,X)whose elementsH satisfyE^Ph

RT

0 |Ht|^pdti

<+∞.The localized versions of these spaces are denoted byH^ploc(P, E,X).

For any subsetP ⊂M(Ω), aP−polar set is aP−negligible set for allP∈ P, and we say that a property holdsP−quasi-surely if it holds outside of aP−polar set. Finally, we introduce the following filtrationG^P :={G_t^P}_0≤t≤T which will be useful in the sequel

G_t^P :=F_t^∗+∨ N^P, t < T andG_T^P :=F_T^∗∨ N^P, whereN^P is the collection ofP-polar sets.

For allα∈H¹loc(P0,S^>0d ,F), whereS^>0d is the set of positive definite matrices of size d×d, we define the probability measure on(Ω,F)

P^α:=P⁰◦(X_.^α)⁻¹whereX_t^α:=

Z t

0

α^1/2_s dBs, t∈[0, T], P⁰−a.s.

We denote byPS the collection of all such probability measures on (Ω,F). We recall from Karandikar [7] that the quadratic variation processhBiis universally defined un- der anyP∈ P_S, and takes values in the set of all nondecreasing continuous functions from R^{+ to} S^>0d . We will denote its pathwise density with respect to the Lebesgue measure byba. Finally we recall from [14] that every P ∈ PS satisfies the Blumenthal zero-one law and the martingale representation property.

Our focus in this paper will be on the following subset ofPS.

Definition 2.1. Pmis the sub-class ofPS consisting of allP∈ PS such that the canon- ical processBis aP−uniformly integrable martingale.

2.2 Regular conditional probability distributions

In this section, we recall the notion of regular conditional probability distribution (r.c.p.d.), as introduced by Stroock and Varadhan [16]. Let P ∈ M(Ω) and consider someF-stopping timeτ. Then, for everyω∈Ω, there exists an r.c.p.d. P^ωτ satisfying:

(i)P^ωτ is a probability measure onFT.

(ii) For eachE∈ F_T, the mappingω→P^ωτ(E)isF_τ-measurable.

(iii) P^ωτ is a version of the conditional probability measure of P ^onFτ, i.e., for every integrableFT-measurable r.v.ξwe haveE^P[ξ|Fτ](ω) =E^Pωτ[ξ],P−a.s.

(iv)P^ωτ(Ω^ω_τ) = 1, whereΩ^ω_τ :={ω⁰∈Ω : ω⁰(s) =ω(s), 0≤s≤τ(ω)}.

We next introduce the shifted canonical space and the corresponding notations.

•For0≤t≤T, denote byΩ^t:={ω∈C([t, T],R) : w(t) = 0}the shifted canonical space, B^t the shifted canonical process onΩ^t, P^t0 the shifted wiener measure, F^{t the} shifted filtration generated byB^t.

•For0≤s≤t≤T,ω∈Ω^s, define the shifted pathω^t∈Ω^t,ω^t_r:=ωr−ωtfor all r∈ [t, T].

•For0≤s≤t≤T,ω∈Ω^s, define the concatenation pathω⊗tωe∈Ω^sby:

(ω⊗tω)(r) :=e ωr1_[s,t)(r) + (ωt+ωer)1_[t,1](r) for all r∈[s, T].

• For0≤s≤t≤T, for anyF_T^s-measurable random variableξonΩ^s, and for each ω∈Ω^s, define the shiftedF_T^t-measurable random variableξ^t,ωonΩ^tby:

ξ^t,ω(eω) :=ξ(ω⊗teω) for all ωe∈Ω^t.

•The r.c.p.d. P^ωτ induces naturally a probability measureP^τ,ωonF_T^τ(ω)such that the P^τ,ω-distribution ofB^τ(ω)is equal to theP^ωτ-distribution of{B_t−B_τ(ω), t∈[τ(ω), T)}. It is then clear that for every integrable andFT-measurable random variableξ,

E^Pωτ[ξ] =E^Pτ,ω[ξ^τ,ω].

For the sake of simplicity, we shall also callP^τ,ω the r.c.p.d. ofP^.

•Finally, we introduce for all(s, ω)∈[0, T]×Ω:

PS(s, ω) :=

(

P^s0◦Z · s

α^1/2_u dB_u^s−1

, with Z T

s

|αu|du <+∞, P^s0−a.s.

)

Pm(s, ω) :={P∈ Ps(s, ω)s.t. B^sis a uniformly integrable martingale}.

Remark 2.2. We are abusing notations here. In order to suit to the definition of [9]

and [11], we should have considered the concatenation of PS(s, ω) (resp. Pm(s, ω)) defined above with the dirac mass onω_0≤t≤sto ensure that elements ofPS(s, ω)(resp.

Pm(s, ω))are probabilities on Ω, and not onΩ^s. The reader should note that the link between these two definition is obvious and we will implicitly identify these families.

It is clear that the families(PS(s, ω))(s,ω)∈[0,T]×Ωand(Pm(s, ω))(s,ω)∈[0,T]×Ωare adap- ted in the sense that PS(s, ω) = PS(s,eω)and Pm(s, ω) = Pm(s,ω)e , wheneverω|_[0,s] = ω|e[0,s].

3 Superreplication and duality

3.1 Problem formulation and main results

Throughout this paper, we consider some scalarG_T-measurable random variableξ. For any(s, ω)∈[0, T]×Ω, we naturally restrict the subsetPmandPm(s, ω)to:

P_m^ξ :=

P∈ Pm: E^P[ξ⁻]<+∞

P_m^ξ(s, ω) :=

P∈ Pm(s, ω) : E^P[(ξ^s,ω)⁻]<+∞ .

Notice that such a restriction can be interpreted as suppressing measures which induce arbitrage opportunities in our market.

Our main interest is on the problem of superreplication under model uncertainty and the corresponding dual formulation. Given some initial capitalX0, the wealth process is:

X_t^H :=X0+ Z t

0

HsdBs, t∈[0, T], whereH∈ H^ξ, the set of admissible trading strategies defined by:

H^ξ:=

H ∈H⁰(R^d,G^Pm)∩H²loc(P,R^d,G^Pm), X^His aP−supermartingale,∀P∈ P_m^ξ . The main result of this paper is the following.

Theorem 3.1. Letξbe an upper semi-analytic r.v. withsup_P∈PmE^P[ξ⁺]<+∞. Then V(ξ) := inf

X₀: X_T^H ≥ξ, P_m^ξ −q.s. for someH ∈ H^ξ = sup

P∈P_mE^P[ξ].

Moreover, existence holds for the primal problem, i.e. V(ξ) +RT

0 HsdBs ≥ξ, P_m^ξ−q.s.

for someH∈ H^ξ.

Remark 3.2. Suppose that the random variable ξ⁻ is P−integrable for all P ∈ Pm. Then,P_m^ξ =Pm, and the corresponding set of admissible strategiesH^ξ =: His inde- pendent ofξ. Under the condition sup_P∈P∞E^P[ξ⁺] < ∞, it follows from the previous theorem that:

inf

X0: X_T^H ≥ξ, Pm−q.s. for someH ∈ H = sup

P∈Pm

E^P[ξ].

Remark 3.3. When it comes to which filtration the trading strategies are admissible, we can actually do a little bit better thanG^Pm, and consider the universal filtrationF^∗ completed by thePm-polar sets, instead of its right limit. Indeed, letX be a process adapted toG^Pm, then following the arguments in Lemma2.4of [14], we defineXe by

Xet:= lim sup

↓0

1

Z t

t−

Xsds.

Then,Xecoincidesdt×P−a.e. withX, for anyP∈ PS and is adapted toF^∗completed by thePm-polar sets. For simplicity, we however refrain from considering this extension.¹ The problem of superhedging under model uncertainty was first considered by Denis and Martini [3] using the theory of capacities and the quasi-sure analysis. The set of probability measures considered in [3] is larger than PS, and whether existence of an optimal hedging strategy strategy holds or not in the framework of [3] is still an open problem. Later, Soner, Touzi and Zhang [14] considered the same problem but

1We would like to thank Marcel Nutz for pointing this out.

with a strict subset of PS satisfying a separability condition, which allowed them to recover the existence of an optimal strategy. The same approach is adapted in Galichon, Henry-Labordère and Touzi to obtain the duality result of Theorem 3.1 for uniformly continuousξ. Recently, Neufeld and Nutz [9] introduced a new approach which avoids the strong regularity condition on ξ. In the next subsection, we briefly outline their approach and explain why it fails to cover our framework.

3.2 The analytic measurability approach

We now introduce the general framework of [11] and [9]. Let P be a non-empty subset ofPS, with corresponding "shifted" setsP(s, ω), satisfying:

Condition 3.4. Lets∈R^+,τ a stopping time such thatτ≥s,ω ∈Ω, andP∈ P(s, ω). Setθ:=τ^s,ω−s.

(i)The graph{(P⁰, ω) :ω∈Ω, P⁰ ∈ P(t, ω)} ⊆M(Ω)×Ωis analytic.

(ii)We haveP^θ,ω∈ P(τ, ω⊗sω)forP^-a.e.ω∈Ω.

(iii)Ifν: Ω→ B(Ω)is anFθ-measurable kernel andν(ω)∈ P(τ, ω⊗sω)forP^-a.e.ω∈Ω, then the following measureP∈ P(s, ω):

P(A) :=

Z Z

(1_A)^θ,ω(ω⁰)ν(dω⁰;ω)P(dw), A∈ F.

Theorem 3.5(Theorem 2.3 in [9]). Suppose{P(s, ω)}_(s,ω)satisfies Condition 3.4. Then, for any upper semi-analytic mapξwithsup_P∈PE^P[|ξ|]<+∞, we have:

inf

X₀: X_T^H≥ξ P −q.s. for someH∈ H = sup

P∈PE^P[ξ].

For the purpose of the application of this result to the problem of martingale optimal transportation, see [5, 6], the last result presents two inconveniences:

- The integrability condition of the previous theorem from [9] turns out to be too strong.

The weaker integrability conditions in our Theorem 3.1 is crucial for the analysis con- ducted in [5, 6].

- The set of probability measures of interest is the smaller subsetPm. We shall verify below thatPmsatisfies Conditions 3.4(i) and (ii), but fails to satisfy (iii). Therefore, we need to extend the results of [9] in order to address the case ofPm.

Example 3.6. [P_m does not satisfy Condition 3.4 (iii)] For simplicity, let d = 1. Let s∈(0, T),t≥s,ω∈ΩandP=P^s0∈ PB(s, ω). Now considerω∈Ω^s. The family(Pⁱ)_i∈N is defined by

∀i∈N, Pi=P^t0◦ Z ·

0

σ^1/2_i dB_u^t −1

.

where(σi)_i∈Nis a sequence of positive numbers which will be chosen later. We consider the following partition(Ei)_i∈NofFt

∀i∈N, E_i:={ωs.t.ω_t∈(−i−1,−i]∪[i, i+ 1)}.

We then introduce theFt-measurable kernelν(ω)(A) :=P+∞

i=01_Ei(ω)Pi(A),and we de- finePas in Condition 3.4(iii) fromP^andν. We now show thatE^P[|BT|] = +∞for some convenient choice of the sequence(σⁱ). In particular this shows thatP∈ P/ _m.

E^P[|BT|] =E^Ph

E^P[|BT||Ft]i

=E^Ph

E^{Pt,ω⊗s ω}[|BT|]i

=E^P

"_+∞

X

i=0

1_Ei(ω)E^Pih

|B_T^t,ω⊗sω|i

#

=E^P

"_+∞

X

i=0

1_Ei(ω) Z +∞

−∞

|σ_iu+B_t(ω)|e^−u2/2

√2π du

#

=

+∞

X

i=0

f_i(σ_i),

where, for alli,

f_i(σ) := σ 2π

Z −i

−i−1

Z +∞

−∞

u+ty σ

e^{−u2 +y}

2

2 dudy+

Z i+1

i

Z +∞

−∞

u+ty σ

e^{−u2 +y}

2

2 dudy

.

Notice thatfi(σ) −→ ∞, as σ → ∞. Then there exists σi > 0 such that fi(σi) ≥ 1. Hence,E^P[|BT|] = +∞, wherePis defined using this family of coefficients.

Proposition 3.7. P_mandP_m^ξ verify Condition 3.4 (i) and (ii).

Proof. We only provide the proof forP_m, the result forP_m^ξ follows by direct adaptation.

We first verify Condition 3.4 (ii). LetP∈ Pm, and consider an arbitraryF-stopping time τ, andF^τ-stopping timeσ. By Lemma A.1 in [8], there exists someF^τ-stopping timeσ˜ such that for everyω,˜σ^τ,ω=σ. Then, we have forP^-a.e. ω

E^Pτ,ω[|B_σ^τ|]≤E^Pτ,ω[|B_σ^τ,ω_˜τ,ω|] +|Bτ(ω)|=E^Pτ[|Bσ˜|](ω) +|Bτ|(ω)<+∞, where we used the fact thatP∈ Pm. Similarly, we have forP^-a.e.ω:

E^Pτ,ω[B_σ^τ] =E^Pτ,ω[B_σ^τ,ω_˜τ,ω−B_τ(ω)] =E^Pτ[B_σ˜](ω)−B_τ(ω) = 0.

By the arbitrariness ofτandσ, this completes the verification of Condition 3.4 (ii).

To verify Condition 3.4 (ii), we adapt an argument from [9]. We define the following map

ψ:H¹loc(P⁰,S^>0d ,F)→M(Ω), α7→P^α=P⁰◦ Z ·

0

α^1/2_s dBs

−1

.

From [9] (see Lemmas 3.1 and 3.2), we know that it is sufficient to show thatP_m⊂ M(Ω)is the image of a Borel space (i.e. a Borel subset of a Polish space) under a Borel map. For that we show thatH⁰(S^>0d ,F)is Polish and H¹loc(P⁰,S^>0d ,F) ⊂ H⁰(S^>0d ,F) is Borel. The first part is already given in Lemma 3.1 of [9]. We then need to show that the mapψ:H¹loc(P⁰,S^>0d ,F)→M(Ω)is Borel, which is a direct consequence of Lemma 3.2 in [9].

It then only remains to prove thatH¹m(P0,S^>0d ,F)⊂H⁰(S^>0d ,F) is Borel,where H¹m(P⁰,S^>0d ,F) :=

α∈H⁰(S^>0d ,F) : sup

τ E^P0

|X_τ^α|1_|Xα τ|≥n

n→+∞−→ 0

.

It is clear that

H¹m(P0,S^>0d ,F) = \

p∈N^∗

[

N∈N

\

n≥N

α∈H⁰(S^>0d ,F) : sup

τ E^P0[|X_τ^α|1_|Xτ^α_|>n]≤ 1 p

.

and

α∈H⁰(S^>0d ,F) : sup

τ E^P0[|X_τ^α|1_|Xα

τ|>n]≤1 p

=ψ⁻¹

P∈ PS : sup_τE^P0[|B_τ|1_|Bτ|>n]≤¹_p .

It then suffices to show that for anyn∈N, the following functionfnis Borel measurable:

fn : P7→sup

τ E^P[|Bτ|1_|Bτ|>n].

We actually show that this function is lower semi-continuous. ForK, l >0, define:

φ(x) =|x|1_|x|>n and φK,l(x) =|x| ∧K(|x| −n)⁺−(|x| −n−l)⁺

l , x∈R^d.

We emphasize that φ_K,l is uniformly continuous and bounded. Let then P ∈ PS and consider some sequence(Pⁱ)_i≥0which converges weakly toP. We representP^{i and}P byαiandα. Remember that for anyPe ∈ PS, associated to someαe, we have

fn(eP) = sup

τ E^P0h

|X_τ^eα|1_|Xαe τ|>n

i .

The weak convergence ofP^itoPis equivalent to the convergence in law ofX^αi toX^α. Hence, for allτ, we have

i→+∞lim E^P0[φ_K,l(X_τ^αi)] =E^P0[φ_K,l(X_τ^α)], from which we deduce easilylim inf

i→+∞sup_τE^P0[φK,l(X_τ^αi)]≥E^P0[φK,l(X_τ^α)].As this is true for allK > 0 and l > 0, and since the functionφK,l is non-decreasing in K and non- increasing inl, we also have

lim inf

i→+∞sup

τ E^P0[φ(X_τ^αi)]≥E^P0[φK,l(X_τ^α)].

LettingKgo to+∞andlto0on the right-hand side above, we deduce using monotone convergence and taking supremum inτ

lim inf

i→+∞sup

τ E^P0[φ(X_τ^αi)]≥sup

τ E^P0[φ(X_τ^α)]

Thenfnis lower semicontinuous and thus measurable.

4 The duality result

In this section we show our main result Theorem 3.1. We will assume throughout thatξis upper semi-analytic. For that purpose, we introduce the dynamic version of the dual problem:

Yt(ξ)(ω) := sup

P∈Pm(t,ω)E^P[ξ^t,ω], t∈[0, T], ω∈Ω.

We first observe thatYt is measurable with respect to the universal filtrationF_t^∗, as a consequence of Step1 in the proof of Theorem 2.3 in [11], since Condition 3.4(i) holds true forPm.

Lemma 4.1. Letτbe anF-stopping time. Then, for allP∈ Pm: Y_τ(ξ) = ess sup^P

P⁰∈Pm(τ,P)

E^P0[ξ|Fτ], P−a.s.

wherePm(τ,P) ={P⁰ ∈ Pm : P⁰=P^onFτ}.

Proof. The inequality≥is trivial asY_τisF_τ^∗-measurable and measures extend uniquely to universal completions, which means that ifP^and P⁰ coincide onF_τ, they also coin- cide onF_τ^∗ (see Step 3 of the proof of Theorem 2.3 in [11] for similar arguments). We then focus on ≤. Fix some P ∈ Pm. We recall that following the same construction as in Step 2 of the proof of Theorem 2.3 in [11], for any > 0, we can construct a Fτ-measurable kernelν : Ω→M(Ω)such that:

E^ν(ω)[ξ^τ,ω]≥ Y_τ(ω)−

1_{{−∞<Yτ(ω)<∞}}+⁻¹1_{{Yτ(ω)=∞}}− ∞1_{{Yτ(ω)=−∞}}, (4.1) and such thatν(ω)∈ Pm(τ, ω),P^-a.s.

We then consider the probability Pe ∈ PS(τ,P) associated to ν through Condition 3.4(iii), where this last assertion uses that Condition 3.4(iii) is verified forPS thanks to

Theorem 2.4 in [9]. However there is no guarantee thatPe ^{belongs to}Pm, and the rest of this proof overcomes this difficulty by using a suitable approximation.

Step 1:Construction of an approximationνⁿ. Define, for alln≥1, νⁿ(ω) := ν(ω)1_{Eν(ω)[|B^τ_T|]≤n}+P^τ,ω1_{Eν(ω)[|B_T^τ|]>n}.

Clearly,νⁿ is a measurable kernel, and sincePm is stable by bifurcation, we also have νⁿ(ω)∈ P_m(τ, ω). Observe thatE_n :={ω∈Ω : E^ν(ω)[|B_T^τ|]≤n}, n≥1,is an increasing sequence inFτwithP(En) −→

n→+∞1, as a consequence of the fact thatE^ν(ω)[|B^τ_T|]<+∞, P−a.s. We now define the measureP^{n by:}

Pⁿ(A) :=

Z Z

(1A)^τ,ω(ω⁰)νⁿ(dω⁰;ω)P(dw), A∈ F.

We first show thatPⁿ ∈ P_m(τ,P). The fact thatPⁿ coincides withP^onF_τ is clear by construction. Next, we compute that:

E^Pn[|BT|] =E^Pn

|BT|1E_n+|BT|1E_n^c

=E^Pnh

E^νn(ω)[|B_T^τ,ω|]1E_n

i

+E^Pnh

E^νn(ω)[|B_T^τ,ω|]1E_n^c

i

=E^Pnh

E^ν(ω)[|B_T^τ,ω|]1E_n

i

+E^Pnh

E^Pτ,ω[|B_T^τ,ω|]1E^c_n

i

≤E^Ph

E^ν(ω)[|B_T^τ|] +|Bτ| 1En

i +E^Ph

E^Pτ,ω[|B_T^τ,ω|]1E_n^c

i

≤E^P[|Bτ|] +n+E^P[|BT|]<+∞.

To prove the martingale property ofB underPⁿ, we consider an arbitraryF^-stopping timeσ, and we compute that:

E^Pn[Bσ] =E^Pn[Bσ1_σ≤τ+Bσ1σ>τ] =E^P[Bσ1_σ≤τ] +E^Pn[Bσ1σ>τ], by the fact thatPⁿ=P^onF_τ. We continue computing

E^Pn[B_σ1_σ>τ] =E^Pnh

E^(Pn)τ,ω[B_σ^τ,ωτ,ω]1_σ>τi

=E^Pnh

E^νn(ω)[B_σ^ττ,ω +B_τ(ω)]1_σ>τi

=E^Pn[Bτ1σ>τ] =E^P[Bτ1σ>τ],

where the last equality uses the definition ofνⁿ which ensures thatνⁿ(ω)∈ Pm(τ, ω), P-a.s. Therefore, we haveE^Pn[B_σ] =E^P[B_σ∧τ] = 0,sinceBis a martingale underP^.

Step 2:By (4.1), we have for everyω

E^νn(ω)[ξ^τ,ω]≥(Y_τ(ω)−)∧⁻¹1_En+E^Pτ,ω[ξ^τ,ω]1_Ec n. Then for anyω∈Ω\N^P, for someP^{-null set}N^P

E^Pn[ξ|Fτ](ω)≥(Yτ(ω)−)∧⁻¹1E_n(ω) +E^P[ξ|Fτ](ω)1E^c_n(ω).

Hence, for anyω∈Ω\N^P, for alln≥0 ess sup^P

P⁰∈Pm(τ,P)

E^P0[ξ|Fτ](ω)≥(Y_τ(ω)−)∧⁻¹1_En(ω) +E^P[ξ|Fτ](ω)1_Ec n(ω).

We emphasize thata priori, the right-hand side above is onlyF_τ^∗-measurable. However, ifP^and P⁰ coincide onFτ, they also coincide onF_τ^∗, since measures extend uniquely on universal completions. Therefore the above inequality does indeed holdP−a.s.

Since the sequenceE_nincreases toΩ(up to someP-null set which we implicitly add toN^P), for anyω∈Ω\N^P, there existsN(ω)∈Nsuch that ifn≥N(ω), thenω ∈E_n. Therefore, takingnlarge enough, we have

ess sup^P

P⁰∈Pm(τ,P)E^P0[ξ|Fτ](ω)≥(Yτ(ω)−)∧⁻¹. (4.2) If Yτ(ω) = −∞, then by the inequality proved at the beginning, the left-hand side above is also equal to−∞. Hence the result in this case. If Yτ(ω) = +∞, then (4.2) implies directly that the left-hand side is also+∞by arbitrariness of > 0. Finally, if Y_τ(ω)is finite, the desired result follows from (4.2) by arbitrariness of.

We then continue with a version of the tower property in our context.

Proposition 4.2. LetP∈ P_m^ξ, andσ, τ twoF-stopping times withσ≤τ. Then,P−a.s.

Yσ(ξ) = ess sup

P⁰∈Pm^ξ(σ,P)

E^P0

"

ess sup

P⁰⁰∈Pm^ξ(τ,P⁰)

E^P00[ξ|Fτ]|Fσ

#

= ess sup

P⁰∈Pm(σ,P)E^P0

"

ess sup

P⁰⁰∈Pm(τ,P⁰)E^P00[ξ|Fτ]|Fσ

# ,

where for anyF-stopping timeιand anyP∈ P_m^ξ P_m^ξ(ι,P) :=

P⁰∈ P_m^ξ s.t.P⁰ =P^onFι .

ProofWe considerP∈ P_m^ξ. Exactly as in the proof of Lemma 4.1, we can construct a measurable kernelνⁿfrom a kernelν such that:

•νⁿisFτ-measurable.

•Pⁿ∈ Pm(τ,P)wherePⁿ(A) =RR

(1A)^τ,ω(ω⁰)νⁿ(dω⁰;ω)P(dw), A∈ F.

•νis aFτ-measurable kernel such that (4.1) holds.

•En={νⁿ =ν}is an increasing sequence such thatP(En) −→

n→+∞1. We then compute for any >0

ess sup

P⁰∈Pm(σ,P)

E^P0[ξ|F_σ] ≥ E^Pn[ξ|F_τ] ≥ E^P[(Y_τ−)∧⁻¹1_En−E^P[ξ⁻|F_τ]1_Ec

n|F_σ], P−a.s.

Recall thatE^P[ξ⁻]<∞. Then, it follows from the dominated convergence theorem that E^P[ξ⁻1E^c_n|Fσ]−→0, asn→ ∞,P−a.s. Also, sinceYτ ≥ −E^P[ξ⁻|Fτ]∈L¹(P), it follows from Fatou’s lemma that:

E^P[(Y_τ−)∧⁻¹|Fσ]≤ ess sup

P⁰∈Pm(σ,P)

E^P0[ξ|Fσ]P−a.s.

Finally, when→0, we haveP-a.s., with the last equality being obvious E^P[Yτ|Fσ]≤ ess sup

P⁰∈P_m(σ,P)E^P0[ξ|Fσ] = ess sup

P⁰∈Pm^ξ(σ,P)

E^P0[ξ|Fσ].

2 Proposition 4.3. Assume thatsup_P∈PmE^P[ξ⁺]<∞. Then, for anyP∈ P_m^ξ, the process {Yt(ξ), t≤T}is aP-supermartingale.

Proof. In view of Proposition 4.2, we already have the tower property. It only remains to show the integrability ofYt(ξ)for allt ∈ [0, T]. For that we only need to show that Yt(ξ⁺)is integrable. We fix0 ≤t ≤T andP∈ P_m^ξ. Then Yt(ξ⁺)(ω)∈R⁺× {+∞}. Let us then show that the family{E^P0[ξ⁺|Ft], P⁰ ∈ Pm(t,P)}is upward directed.

We consider P1 and P2 in Pm(s,P). The set A := {E^P2[ξ⁺|Ft] ≤ E^P1[ξ⁺|Ft]}, is F_t-measurable. ThenP:=P11_A+P21_Acis an element ofP_m(t,P)such that:

E^P[ξ⁺|F_t] =E^P1[ξ⁺|F_t]∨E^P2[ξ⁺|F_t] We then have an increasing sequenceP^{n of}Pm(t,P)such that:

E^Pn[ξ⁺|Ft]% ess sup

P⁰∈Pm(s,P)E^P0[ξ⁺|Ft], P−a.s.,

and by the monotone convergence theoremlimn→+∞E^Pn[ξ⁺] =E^P[Yt(ξ⁺)].Hence E^P[Y_t(ξ⁺)]≤ sup

P∈P_mE^P[ξ⁺]<+∞, for allP∈ Pm.

We now have all ingredients to follow the classical line of argument for the

Proof of theorem 3.1. For the sake of simplicity, the dependence ofY inξwill be omit- ted.

(i) We first show that right-limiting processYt :=Yt+, t ≤ T,is a(G^Pm,P)super- martingale for all P ∈ P_m^ξ. By Proposition 4.3 and the fact that for any P ∈ PS, F^P is right-continuous, containsG^andPhas the predictable representation property (see [14]),Y is a(F^∗,P)supermartingale for everyP∈ P_m^ξ. Then applying [3] (see Theorem VI.2), we have thatY is well definedP^{-a.s. and}Y is a right continuous(G^Pm,P)super- martingale for allP∈ P_m^ξ.We also notice the important fact that for allP∈ P_m^ξ we have Yt≤YtP-a.s. In particular,Y0≤Y0,andY0is constant becauseG₀^Pm is trivial.

(ii) We next construct the optimal trading strategy. By the Doob-Meyer decomposi- tion (see Theorem 13 page 115 in [12]), there exists a pair of processes(H^P, K^P)where H^Pbelongs toH²loc(P,R^d,G^Pm)andK^PisP−integrable and non-decreasing, such that:

Yt=Y0+ Z t

0

H_s^PdBs−K_t^P, t∈[0, T], P−a.s.

SinceY is right continuous, it follows from Karandikar [7] that the family H^P can be aggregated by some processHb in the sense thatHb =H^P,dt×dP-a.s. for allP∈ P_m^ξ. Thus, for every P ∈ P_m^ξ, the local martingale R

HdBb is bounded from below by the martingaleE^P[ξ|G_t^Pm]. Hence this is a supermartingale which ensures thatHb is inH^ξ and superreplicates the claimξP_m^ξ-quasi-surely.

5 Extensions

5.1 The case ofPb

In this section, we show that Theorem 3.1 together with the previous arguments in its proof, hold for the following example, which is important in the context of second- order BSDEs as introduced in [13]. We recall thathBiis well defined pathwise and that its density is denoted byba.

Definition 5.1. Pb is the sub-class ofPS consisting of allP∈ PS such that:

a_P≤ba≤a_P, dt×dP−a.s. for somea_P, a_P∈S^>0d .

We emphasize that our Example 3.6 also shows directly that Pb does not satisfy Condition 3.4(iii). We now prove the three main technical results of this paper in this case.

Proposition 5.2. Pbverifies Condition 3.4 (i) and (ii).

Proof. (ii) has already been proved in [15]. Let us now prove (i). We observe that:

Pb= [

a,a∈S^>0_d (Q)

{P∈ PS : a≤aˆ≤a dt¯ ×dP−a.s.}.

By Proposition 3.1 in [11], we know that all these sets satisfy Condition 3.4(i). Since a countable union of analytic set is analytic, we obtain the result.

It remains to introduce a suitable sequence of approximations of measurable kernels as in the proof of Lemma 4.1 and Proposition 4.2.

Proposition 5.3. The results of Lemma 4.1, Proposition 4.2 and 4.3 and Theorem 3.1 are valid if we replaceP_mbyP_b.

Proof. We only redefine an approximated kernel family adapted toP_b, which allows, by the same arguments as in Lemma 4.1 and Proposition 4.3, to obtain the duality result of Theorem 3.1. Letτ be aF-stopping time andν theFτ-measurable kernel obtained by the same construction as in Lemma 4.1 and Proposition 4.2. For P ∈ Pb we are interested in the measurePdefined by:

P(A) = Z Z

(1A)^τ,ω(ω⁰)ν(dω⁰;ω)P(dw).

ThenP^{is in}PS and there is someαs.t.P=P⁰◦(X_.^α)⁻¹. DefineP^nby:

Pⁿ:=P⁰◦ Z ·

0

(α^1/2_s 1_s≤τ+πn(α^1/2_s )1s>τ)dBs) −1

, n≥1,

where πn is the projection on B_S>0

d (0, n)\B_S>0

d (0,1/n), where B_S>0

d (x, r) denotes the closed ball ofS^>0d centered at xwith radius r. Then P^{n belongs to} P_b. Observe also that the sets

En:=n

ω∈Ω, P^nτ,ω

=ν(ω)o ,

are inFτ and define an increasing covering ofΩ. We then build the "right" approxima- tionPeⁿ, ensuring all the convergences in the proofs, byPeⁿ =Pⁿ1En+P1E^c_n. Pe^{n is in} Pb, and associated to theFτ-measurable kernelν˜ⁿ(ω) :=ν(ω)1_ω∈En+P^τ,ω1_ω∈Ec

n. Then we can reproduce exactly the same proofs as in the case ofP_m.

5.2 A general framework

As the reader may have realized, our proofs in the case of Pm and Pb are very similar, and essentially rely on the construction of a suitable approximated kernel. In this subsection, we consider a generic subsetP of PS (and the corresponding shifted familiesP(s, ω)), and we give a general condition, (weaker than Condition 3.4) under which our results still hold true. We recall that such a family is said to be stable by bifurcation if for anyF-stopping times0≤σ≤τ, ω ∈Ω,AFτ-measurable, P^1andP² inP(σ, ω), we have

P=P11_A+P21_Ac ∈ P(σ, ω).

Condition 5.4. Lets∈R+,τ ≥sa stopping time,ω∈Ω,P∈ P(s, ω)andθ:=τ^s,ω−s. (i)The graph{(P⁰, ω) :ω∈Ω, P⁰ ∈ P(t, ω)} ⊆M(Ω)×Ωis analytic.

(ii)We haveP^θ,ω∈ P(τ, ω⊗sω)forP^-a.e.ω∈Ω. (iii)P is stable by bifurcation.

(iv) If ν : Ω → M(Ω) is an Fθ-measurable kernel and ν(ω) ∈ P(τ, ω⊗sω) for P^-a.e.

ω ∈ Ω, then there existsνⁿ : Ω → M(Ω), which is a Fθ-measurable kernel such that P(νⁿ=ν) −→

n→∞1and the following measurePⁿ ∈ P(s, ω): Pⁿ(A) =

Z Z

(1A)^θ,ω(ω⁰)νⁿ(dω⁰;ω)P(dw), A∈ F.

Remark 5.5. Notice that Condition 5.4 is weaker than Condition 3.4. Indeed, Condition 3.4(iii) implies directly that the setP is stable by bifurcation. Moreover, considering the constant kernelsνⁿ := ν, it also implies Condition 5.4(iv). Furthermore, as shown in our previous proofs, the setsPmandPbsatisfy Condition 5.4 but not Condition 3.4.

Similarly to our previous notations, we introduce the sets H^ξ(P) and P^ξ. In this context, we obtain a new version of Theorem 3.1:

Theorem 5.6. LetP(s, ω)be a family of probability measures satisfying Condition 5.4.

Letξbe an upper semi-analytic r.v. withsup_P∈PE^P[ξ⁺]<+∞. Then V(ξ) := inf

X₀: X_T^H≥ξ, P^ξ−q.s. for someH ∈ H^ξ(P) = sup

P∈PE^P[ξ].

Moreover, existence holds for the primal problem, i.e. V(ξ) +RT

0 H_sdB_s ≥ξ, P^ξ−q.s.

for someH∈ H^ξ(P). Proof. If we defineEen:=n

ω∈Ω : (Pⁿ)^θ,ω=ν(ω)o

and then recursively E0:=Ee0and for alln≥1,En:=En

[ Ee_n−1, then E_n is an increasing sequence such that P(E_n) −→

n→+∞ 1. We can then use the Fτ-measurable kernel νⁿ to define a probability measure Peⁿ exactly as in the proof of Proposition 5.3. We can then use exactly the same arguments as in our previous proofs.

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