On the quasi-sure superhedging duality with frictions

We prove the superhedging duality for a discrete-time financial market with proportional transaction costs under portfolio constraints and model uncertainty. Frictions are modeled through solvency cones as in the original model of [Kabanov, Y., Hedging and liquidation under transaction costs in currency markets. Fin. Stoch., 3(2):237-248, 1999] adapted to the quasi-sure setup of [Bouchard, B. and Nutz, M., Arbitrage and duality in nondominated discrete-time models. Ann. Appl. Probab., 25(2):823-859, 2015]. Our results hold under the condition of No Strict Arbitrage and under the efficient friction hypothesis.


Introduction
It is more often the rule, rather than the exception, that socio-economics phenomena are influenced by a strong component of randomness. Starting from the pioneering work of Knight (see e.g. [21]) a distinction between risk and uncertainty has been widely accepted with respect to the nature of such a randomness. We often call a situation risky if a probabilistic description is available (e.g. the toss of a fair coin). In contrast, we call a situation uncertain if it cannot be fully described in probabilistic terms. Simple reasons could be the absence of an objective model (e.g. the result of a horse race; see [6] and the references therein) or the lack of information (e.g. the draw from an urn whose composition is unknown). The classical literature in mathematical finance has been mainly focusing on risk and the attention to problems of Knightian uncertainty has been drawn only relative recently starting from [4]. In particular, fundamental topics such as the theory of arbitrage and the related superhedging duality have been systematically studied in frictionless discrete-time markets in [8,12] in a quasi-sure framework and in [1,15,17] in a pointwise framework.
Under risk, the classical model of a discrete-time market with proportional transaction costs has been introduced in [20]. The model is described by a collection of cones K := {Kt}t=0,...,T which determines: i) admissible strategies; ii) solvency requirements; iii) pricing mechanisms. More precisely, the latter are called consistent price systems and they are essentially martingale processes taking values in the dual cones K * t . Instances of such models have been considered, in the uncertainty case, in [5,7,13,14,19], nevertheless, the problem of establishing a quasisure superhedging duality has remained open. Recently, such a duality has been derived in [11] using a randomization approach (see also [2,9,18] for other applications) under the condition No Arbitrage of the Second Kind (NA2(P)). The idea is to construct a fictitious frictionless price processŜ for which: i) the superhedging price in the frictionless market coincides with the one with frictions; ii) the class of consistent price systems is in a one to one correspondence with the A G-measurable map U , defined on a space X and taking values in the power set of a space Y , is called a multifunction (or random set) and it is denoted by U : X ⇒ Y . L 0 (G; U ) denotes the class of G-measurable selectors of U which are defined on dom U := {x ∈ X | U (x) = ∅}. For U : X1 × X2 ⇒ R d and x ∈ X1 fixed, the notation U (x; ·) refers to the random set U viewed as a (multi)function of X2. Given a class of probabilities R ⊂ P(X2), the (conditional) quasi-sure support of U (x; ·), denoted by supp R U (x; ·), is the smallest closed set F ⊂ R d such that U (x; ·) ⊂ F , R-q.s. For a collection of random sets U := (Ut) T t=0 adapted to a filtration G, we denote by L 0 (G−; U ) the class of processes H such that Ht+1 ∈ L 0 (Gt; Ut) for every t = 0, . . . T − 1. Finally, for two R d -valued processes H and S, we set (H • S)t := t−1 u=0 Hu+1 · (Su+1 − Su).
-We set F := BΩ and F u its universal completion. Similarly, the filtrations F = {Ft}t∈I and F u = {F u t }t∈I are given by Ft := BΩ t and F u t its universal completion. -for each t ∈ I, Pt is a random set of probabilities on Ω1 with analytic graph and P0 is non-random. We set, P = {P0 ⊗ · · · ⊗ PT −1 | Pt ∈ L 0 (F u t ; Pt), ∀t ∈ {0, . . . , T − 1}}, The class L 0 (F u t ; Pt) is non-empty from Jankov-Von Neumann Theorem (see [10,Proposition 7.49]) so that P is well defined through Fubini's Theorem.

Main result
We consider the general model of financial markets with proportional transaction costs introduced in [20]. The model is fully described by a collection of random convex closed cones K := {Kt}t∈I ⊂ R d with d ≥ 2, called solvency cones. These represent the sets of positions, in terms of physical units of d underlying assets, which can be liquidated in the zero portfolio at zero cost. We assume that any position with non-negative coordinates is solvent, i.e., R d + ⊂ Kt. The set −Kt represents the class of portfolios which are available at zero cost. We assume that Kt is Ft-measurable for any t ∈ I with K0 non-random. Following standard notation, for a cone K ⊂ R d we denote by K * := {x ∈ R d | x · k ≥ 0 ∀k ∈ K} its dual cone and by K • := −K * its polar cone. We generalize the model of [20] by introducing some constraints on the admissible positions in the market. These are modeled by a collection C := {Ct}t∈I ⊂ R d of random convex closed cones such that every Ct is Ft-measurable. A zero-cost strategy η := (ηt)t∈I is said to be admissible if it satisfies ηt ∈ At for any t ∈ I, where In words, η satisfies the constraints impose by (Ct)t∈I and it is obtained as the sum of portfolios which are available at zero cost. We denote by H K the class of admissible strategies and omit the dependence on C as it will be fixed throughout the paper.
The first assumption is known as efficient friction hypothesis. The second one means that it is allowed to not trade between two periods. The latter is obviously satisfied in the unconstrained case, that is Ct ≡ R d for any t ∈ I.
The following is called No Strict Arbitrage condition in the literature.
. The interpretation is that (Z, Q) defines a frictionless arbitrage-free price process which is compatible with the model of transaction costs defined by {(Kt, Ct)}t∈I. We shortly denote by S the set of SCPS and by S 0 the class of normalized SCPS, namely, those satisfying Z d t = 1 for any t ∈ I.
We are now ready to state the main result of the paper. Let G : Ω ⇒ R d be a Borel measurable random vector which represents the terminal payoff of an option in terms of physical units of the underlying assets. The superhedging price of G is given by Moreover, the superhedging price is attained when π K (G) < ∞.
The proof of Theorem 2.4 is given in Section 4. The main difficulty is to establish Theorem 2.4 when only dynamic trading is allowed. In Theorem 4.12 below we extend the duality to the case where also buy and hold positions in a finite number of options are allowed.
In the following it would be more convenient to extend the original market with an extra unconstrained component. More precisely, one could consider the marketK withKt := Kt ×R+ andCt := Ct × R for t ∈ I, which also satisfies Assumption 2.1. It is easy to see that π K (G) = πK(Ḡ) withḠ = [G; 0]. On the dual side, i : Remark 2.5. Without loss of generality, we may assume that (Kt, Ct) are already in the form described above, for any t ∈ I.
We adapt some results of [7] to the case of portfolio constraint. These will be useful in the next sections. Consider the collection of random setsK := {Kt}t∈I, defined via a backward recursion as follows. We letK * T := K * T and where, for ω ∈ Ωt fixed, Γt(ω) := supp Pt(ω)K * t+1 (ω; ·). We defineKt as the dual ofK * t for any t ∈ I.
The following are generalizations of Lemma 6 and Proposition 4 in [7] to the present setting. The proofs are analogous and we postpone them to the Appendix. Lemma 2.6.K * t has analytic graph for every t ∈ I. Proposition 2.7. If K satisfies Assumption 2.1 and NA s (P), the same holds forK. In particular, int(K * t ) = ∅ P-q.s. for all t ∈ I.

The randomization approach
In this section we construct an enlarged measurable space (Ω,F,F u ,F,F u ) endowed with a suitable class of probabilitiesP. On this space, we construct a price processŜ = (Ŝt)t∈I which represent a frictionless financial market with the property thatŜt ∈K * tP -q.s. for any t ∈ I (Corollary 3.5 below) and which is arbitrage free (Proposition 3.9 below).
We next construct the price processŜ. Recall that, for any t ∈ I, Kt is Borel-measurable and, thus, also K * t . Moreover, int(K * t ) is non-empty by Assumption 2.1. From [13, Lemma A.1] there exists St ∈ L 0 (Ft; int(K * t )). Since K * t ⊂ R d + we can normalize St with respect to, e.g., the last component, so that St takes values in We define a Borel-measurable price processŜ as, where the last component serves as a numéraire. The rest of the section is devoted to the construction of the desired set of probability measureŝ P. For every t ∈ I, we define the random sets Θt(ω) := {θ ∈ R d−1 |Ŝt(ω, θ) ∈ int(K * t (ω))}, ω ∈ Ωt. Proof. In the proof we will repeatedly use the fact that the class of analytic sets is closed under countable union and intersection and that the image of an analytic set through a Borel-measurable function is again analytic.
Step 1. For any t ∈ I, consider the random setK * ,d−1 where the projection is taken over the first d − 1 coordinates andK * ,0 t is the analogous of (3.1) forK. Observe that, From Lemma 2.6, graph(K * t ) is analytic and so is the intersection in (3.4). As the projection is a continuous map, we conclude that graph(K * ,d−1 t ) is analytic.
Step 2. We now show that the set is analytic, for an arbitrary v ∈ R d−1 and t ∈ I. This together with Lemma 5.1 in the Appendix yields the claim, as graph(Θt) is the intersection of countably many analytic sets of the from A v t .
Observe that the function f : whereS is the process given by the first d − 1 components of S. SinceS is Borel-measurable, graph(St) is a Borel set. Moreover, from Step 1, graph(K * ,d−1 t + v) is an analytic set. As f is Borel-measurable, we conclude that A v t is analytic.
Corollary 3.2. For any t ∈ I, the random set has analytic graph.
Proof. The graph of δΘ t is the image of the graph of Θt through the map (ω, θ) → (ω, δ θ ) which is an embedding (see [3,Theorem 15.8]). Since the image of an analytic set through a continuous function is again analytic, the thesis follows.
Recall that the map πΩ 1 : P(Ω1) → P(Ω1), which associate, to everŷ P ∈ P(Ω1), its marginal on Ω1 is Borel measurable (see [3,Theorem 15.14]). Therefore, as in the proof of [11, Lemma 2.12 (i)], the set is also analytic. To conclude observe that graph(Pt) is the intersection of the two previous sets.
We now such that this class is non-empty on a sufficiently rich set of events.
Proof. Fix t ∈ I−1. For t = −1 there is nothing to show as Θ0 is non-random and non-empty from Proposition 2.7. Suppose 0 ≤ t ≤ T − 1. Let Dt+1 := dom(Θt+1) which is analytic from Lemma 3.1. As in the proof of Proposition 3.3, the function φ : is upper semianalytic. We deduce that the set is analytic and, thus, also its projection on Ωt. Denote by N ′ t := (proj Ωt (Bt)) C ∈ F u . We show that, under NA s (P), N ′ t is P-polar. To see this observe that is P-polar from Proposition 2.7. Suppose that there exists P ∈ P such that P(N ′ t ) > 0 and denote by {Pt}t∈I its disintegration. By definition of Bt, Pt(ω, D C t+1 ) > 0 for every ω ∈ N ′ t , therefore, the random variable It remains to show that N ′ t = Nt. The inclusion ⊂ follows from the definition of Bt. Take now Pt an F u t -measurable selector of Bt and δ θ t+1 ∈ L 0 (F u t+1 ; δΘ t+1 ), where δΘ t+1 is defined in Corollary 3.2. Since Pt(ω, dom(Θt+1)) = 1 for any ω ∈ (N ′ t ) C , we can extend δ θ t+1 arbitrarily on the complement of dom(Θt+1) and, with a slight abuse of notation, we still denote it by δ θ t+1 . The product measure Pt ⊗ δ θ t+1 belongs toPt(ω) for any ω ∈ (N ′ t ) C . This shows (N ′ t ) C ⊂ (Nt) C and the thesis follows.
Corollary 3.5 shows that the role of the parameter θ for the price processŜ is to "span" the dual cones given by the backward recursion (2.3).
We setP := {P−1 ⊗ · · · ⊗PT −1 |Pt ∈ L 0 (F u t ;Pt), ∀t ∈ I−1}, The class is well defined and constructed via Fubini's Theorem as done for P. Indeed, from Lemma 3.4, the set Nt is P-polar, thus, we can extend arbitrarily anyPt to a universally measurable kernel which, with a slight abuse of notation, we still denote byPt.
By construction, we have that the probability of the set of trajectories taking values in the interior ofK * t is equal to 1, i.e., We finally show that starting from a model (K, P) satisfying NA s (P), the induced frictionless market (Ŝ,P) satisfies the no arbitrage condition of Definition 3.7 below. Definition 3.6. We say that a process H is an admissible strategy if Ht+1 ∈ L 0 (F u t ; Ct) and the self-financing condition (Ht+1 − Ht) ·Ŝt = 0P-q.s. is satisfied, for any 0 ≤ t ≤ T − 1. The class of admissible strategies is denoted byĤ r .
In order to use the frictionless duality results of [8] we need to verify Assumption 3.1 and 5.1 in that paper. Note that the set Ht, in the notation of [8], corresponds to the set of constraints Ct considered here. Under NA s (P), Corollary 3.5 and Proposition 2.7, imply that where, for a set U ⊂ R d , span(U ) denotes its linear hull. We deduce that the sets Ht, Ht(P) and CH t (P) in [8], they all coincideP-q.s. with the first d − 1 components of the set Ct. Since Ct is a convex closed cone, Assumption 3.1 i)-ii) and 5.1 i) are met. By [8,Remark 5.2] it is sufficient to verify Assumption 5.1 ii). In particular we show that ] is a Charathéodory map, namely, it is continuous in x when (ω,P) are fixed and it is measurable in (ω,P) when x is fixed. From [22,Example 14.15], the random set F ((ω,P), Ct(ω)) is again Borel-measurable. Finally, At restricted to D is again Borel-measurable since, for any c ∈ R, The following is Theorem 3.2 of [8] which is also valid in our context. For ω ∈ Ωt fixed, NA(Pt(ω)) corresponds to NA(P) for the one period market (Ŝt(ω),Ŝt+1(ω; ·)).
Theorem 3.8. The following are equivalent: Proof. The only difference from the proof of [8, Theorem 3.2] is thatPt(ω) might have empty values on the P-polar set Nt ∈ F u . Recall graphPt is analytic by Proposition 3.3. Thus, also ). In our framework, the above set has to be intersected with dom(Pt) which is analytic and, therefore, the intersection is again universally measurable. The same proof yields that Lemma 3.4]. The universally measurable kernels Qt defining Q ∈Q are constructed outside a P-polar set and, in particular, they are chosen as selectors of a set Ξ with dom(Ξ) = (N ′ t ) C . In our framework, the same Ξ satisfies dom(Ξ) = (Nt ∪ N ′ t ) C , which is still universally measurable and P-polar. The same proof allows to conclude. iii) ⇒ i) is standard. Proposition 3.9. NA s (P) implies NA(P).

The Superhedging duality
This section is devoted to the proof of Theorem 2.4. To this aim we compare both the primal and the dual problem with the randomized counterpart in the frictionless market induced byŜ and constructed in Section 3. Using duality results known for the frictionless case we obtain the result.
Equality of the primal problems. We first observe that using admissible strategies with respect to K or with respect toK yields the same superhedging price.
Proof. Since Kt ⊂Kt for any t ∈ I, the inequality (≥) is trivial. Let now (y,η) ∈ R × HK be a superhedge for G. We show that there exists η ∈ H K such that ηT =ηT and, thus, (y, η) is a superhedge for G. By definition, we can writeηT = T t=0 −kt for somekt ∈ L 0 (F u t ;Kt), for any t ∈ I. We observe that, from (2.3), From [7,Lemma 8],kt = f + g with f ∈ L 0 (F u t ; Kt) and g ∈ L 0 (F u t ;Kt+1 ∩ Ct). Iterating the same procedure up to time T − 1 and recalling thatKT = KT we obtain that Moreover, g t s := T u=s+1 f t u belongs to L 0 (F u t ;Ks+1 ∩ Cs) for s = t, . . . , T − 1. Note that f t s is defined only for s ≥ t. We set f t s = 0 for s < t, so that we can rewritekt = T s=0 f t s .
Define now kt := t s=0 f s t and ηt := t u=0 −ku for t ∈ I. Clearly ηT =ηT so that We are only left to show that ηt ∈ L 0 (F u t ; Ct) for any t ∈ I. To this aim observe that for t = T it follows from ηT =ηT . For t = 0, . . . , T − 1, we have where, for the second equality we exchanged the order of summation in the first term and used the definition of g s t in the second term. The above equation reads as ηt =ηt + t s=0 g s t . By construction, g s t ∈ Ct P-q.s. Moreover, the admissibility ofη implies thatηt ∈ Ct P-q.s. By recalling that Ct is a convex cone, the thesis follows.
We now consider the superhedging problem in the frictionless market defined byŜ. Note that a trading strategy inĤ r (see Definition 3.6) could in principle depend on the variable θ. As this variable is only fictitious a genericF u -predictable process cannot consistently identify an element in H K . To this aim we need to reduce the class of admissible strategies to those which only depend on the variable ω. (4.1) We denote byĤ the set of all self-financing consistent strategies.
s. On the other hand, a consistent strategy depends only on the ω variable and hence the position in the numéraire needs to be able to cover the worst case scenario for the price ofŜt, which explains (4.1). We show below that, for any consistent strategy, the left hand side of (4.1) is measurable.
Depending on the choice of the admissible strategies, two corresponding superhedging prices of a random variable g can be computed in the enlarged market: π r (g) := inf y ∈ R | ∃H ∈Ĥ r such that y + (H •Ŝ)T ≥ g,P-q.s. , (4.2) π(g) := inf y ∈ R | ∃H ∈Ĥ such that y + (H •Ŝ)T ≥ g,P-q.s. . We want to show that the superhedging price of G is equal to the superhedging price of G ·ŜT in the frictionless market, using only consistent strategies.
Towards this aim, let us first elaborate on the self-financing condition for consistent strategies. For any 0 ≤ t ≤ T −1, let ∆Ht := Ht+1 −Ht and define F (ω, We also observe that the self-financing condition (4.1) can be rewritten as follows.
For the rest of the proof we suppose that k d t is pointwise finite for any t ∈ I (indeed, the P-q.s. version k d t 1 {k d t <∞} is again universally measurable). From (4.4), kt(ω) · x ≥ 0 for any x ∈K * ,0 t , which implies kt(ω) ∈Kt(ω). We now rewrite the superhedging property of (y, H) in terms of (y, η). For any (ω, θ) outside aP-polar set, we have (ω, θ). (4.5) We claim that this implies: To prove the claim observe that the first term in (4.5) depends on θ only through the last component θT , whereas, the second term in (4.5) depends only on the first T − 1 components of θ. Fix n ∈ N, P ∈ P. From Lemma 4.4 and (4.4), there existsP n ∈P such thatP n | Ω = P and Since n ∈ N and P ∈ P are arbitrary, we deduce that (4.6) holds and the claim is proven.

It remains to show that for
Suppose that, by contradiction, there exists a set A and a probability P ∈ P such that P(A) > 0 and ξ(ω) / ∈KT (ω) for any ω ∈ A. Without loss of generality, we may take a Borel measurable version of ξ. Recall thatKT = KT is assumed to be Borel measurable, so that ) is Borel measurable from [13,Lemma A.1]. Moreover, its projection on Ω contains A. Since B is Borel, from Jankov-Von Neumann Theorem, there exists a universally measurable map sT : Ω → R d with graph sT ⊂ B. Since graph sT ⊂ graph(int K * T ) we can normalize with respect to the last component and from [22,Theorem 14.16] there exists F u T -measurable random vector θT satisfying ξ(ω) ·ŜT (ω,θT (ω)) < 0, ∀ω ∈ A.

Proof of Theorem 2.4.
We are now ready to prove the main result of the section. Note that from Lemma 4.1 and 4.5, we can only deduce the equality of the primal problems if one restricts to consistent trading in the enlarged market (compare with (4.3)). It remain to show that the same price is obtained with randomized strategies as defined in (4.2), in other words, we need to prove thatπ(G ·ŜT ) = π r (G ·ŜT ). Denote by U SA(Ωt, t) the class of g :Ωt → R upper semianalytic functions which depends on θ only through θt, i.e., g(ω, θ) = g(ω ′ , θ ′ ), ∀(ω, θ), (ω ′ , θ ′ ) ∈Ωt, with ω = ω ′ and θt = θ ′ t .
The one-period case. We obtain first the results for T = 1 which will constitute the building blocks for the general case.
Proof. The inequalityπ(g) ≥π r (g) is trivial. For the converse, let Bn(0) be the closed ball in R d with center in 0 and radius n ∈ N. The intersectionK * ,0 0 ∩ Bn(0) is a compact set of the form O n × {1} for O n a compact subset of R d−1 . Recall the definition ofP0 from (3.5) and let P−1 ∈ P(Ω1) be arbitrary 1 . We definê Denote byπn andπ r n the analogous ofπ andπ r in equations (4.2) and (4.3) withP n replacinĝ P and note that, by construction, {πn(g)}n and {π r n (g)}n are increasing sequences bounded from above byπ(g) andπ r (g) respectively. We use now a minimax argument as in [11] to deduce thatπ =π r n (g). 1 Recall that the coneK * ,0 0 is non-random. Thus, in the enlarged market, the only relevant variable is θ.
To justify the above it is sufficient to observe that the function is convex in H for θ fixed and affine in θ for H fixed. We can thus apply the minimax theorem of [23, Corollary 2].
Proof of Theorem 2.4 for T = 1. From Proposition 3.9, NA s (P) implies NA(P) for the enlarged market. From Lemma 4.1, Lemma 4.5 and Proposition 4.7, π K (G) =π r (G ·ŜT ) which is the superhedging price of G ·ŜT , in the enlarged market. We show that Indeed, the second equality follows from [8,Theorem 4.3] after observing that, when C0 is a cone, A Q in the aforementioned paper is finite if and only ifQ ∈Q. The third equality follows from Proposition 4.6 and the last inequality follows fromS 0 ⊂ S 0 . The converse inequality follows from standard arguments. From [8,Theorem 4.3] an optimal superhedging strategy exists in the enlarged market, when the price is finite. The proofs of Lemma 4.1 and 4.5 provide the construction of an optimal strategy in the original market.
The multi-period case. From [8,Lemma 3.4],Qt as in (4.7) has analytic graph for every (4.12) with Θt as in (3.3). It is possible to show that gt ∈ U SA(Ωt, t). Indeed, the measurability property follows exactly from the same argument as in the first lines of the proof of [12,Lemma 4.10]. Moreover, gt depends on θ only throughŜt, thus, only through θt (see (3.2)).
Recall now that the sum of two upper semi-analytic functions is again upper semi-analytic (see e.g. [10,Lemma 7.30]). SinceŜt is Borel measurable we deduce that gt − h ·Ŝt is an upper semi-analytic function of (ω, h, θ). From Lemma 3.1 and [10, Proposition 7.47] we deduce that g ′ t is upper semi-analytic.
LetPt be the set of probabilities on Ωt × R d−1 ×Ω1 given bỹ Recall that the random setsPt and δΘ from Corollary 3.2 have analytic graph. Since the map x → δx is an embedding and the map (P, Q) → P ⊗ Q is continuous (see [10,Lemma 7.12]), it follows that alsoPt has analytic graph.
is a universally measurable normal integrand.
Proof. Denote by fP(ω, h, x) the functions on the right hand side of (4.14) for which the supremum is taken. From [22,Corollary 14.41] we need to check: a) for any (ω, h) ∈ Ωt × R d , the function f (ω, h, ·) is lower semi-continuous.
b) for any x ∈ R d there exists ε ′ > 0 such that, for all ε ∈ (0, ε ′ ), the function is universally measurable, where Bε(x) denotes the closed ball of radius ε centerd in x.
a) Since fP(ω, h, ·) is continuous for everyP and the pointwise supremum of continuous functions is lower semi-continuous, the claim follows. b) Consider an arbitrary ε > 0. We first show that for any (ω, h), This follows from the application of a minimax Theorem (see e.g. [23,Corollary 2]). Bε(x) is a compact set and, for fixedx, the map fP(ω, h,x) is linear (hence concave) inP. On the other hand {P ≪Pt(ω)} is a convex set and, for fixedP, the map fP(ω, h,x) is affine (hence convex) and continuous inx.
To conclude the proof it is enough to show that f2 is Borel measurable. To see this observe that the function E P [x · (Ŝt+1(ω; ·) −Ŝt(ω; ·))] is measurable in (ω, P) and continuous inx, namely, it is a Carathéodory map. From [22,Example 14.15] its composition with Bε(x) yields a Borel-measurable random set A such that sup A = −f2. To conclude, observe that for an arbitrary c ∈ R, is a Borel set from the measurability of A.
Remark 4.9. Note that, for any ω ∈ Ωt, the right hand side of (4.14) is equal to the inf{K ∈ R | X ≤ KPt(ω)-q.s}, where X is the random variable inside the expectation. In particular, this is equal to the minimal amount, at time t, for which the strategy x is a superhedge for gt+1 given that h is the strategy used at time t − 1. Moreover, by construction ofPt, the strategy x with the initial amount f (ω, h, x), is a (conditional) superhedging strategy which depends only on the event ω and not on the event (ω, θ). In the terminology of Definition 4.2, this construction provides consistent strategies.
Recall that NA(Pt(ω)) is the conditional version of NA(P) (see Theorem 3.8).
Proposition 4.10. Let 0 ≤ t ≤ T − 1 and assume NA (Pt(ω)). There exists a universally measurable map ϕ : Ωt × R d → R d and a P-polar set N such that for any Proof. Define the consistent conditional superhedging price of gt+1 given (ω, h) as the map The fact that g ′ t (ω, h) is the consistent conditional superhedging price of gt+1, follows from the same minimax argument of Proposition 4.7 and [8,Theorem 4.3]. Moreover, from Theorem 3.8, NA(Pt(ω)) holds outside a polar set N . Again from [8,Theorem 4.3], g ′ t (ω, h) > −∞ on N C × R d and the infimum is attained. It remain to show that a superhedging strategy can be chosen in a measurable way. From Lemma 4.8 the map f defined in (4.14) is a universally measurable normal integrand. From [22,Proposition 14.33] and recalling that g ′ t is upper semi-analytic (hence universally measurable), the map } is a closed-valued universally measurable random set. The desired ϕ is any measurable selector of Ψ (which exists from, e.g. [22,Corollary 14.6]).
We now choose g = G ·ŜT , which is Borel-measurable by assumption. From Lemma 4.1, Lemma 4.5, Proposition 4.6 and (4.15), we deduce Again, the converse inequality follows by standard arguments and the duality follows (recall also the discussion before Remark 2.5). Finally, the attainment property in the frictionless market follows from [8, Theorem 6.1]. The proofs of Lemma 4.1 and Lemma 4.5 provide the construction of an optimal strategy in the original market.

The case with options.
We now consider the case where a finite number of options ϕ1, . . . , ϕe are available for semistatic trading. In this section we show that this case can be embedded in the previous one. For any k = 1, . . . , e, we assume that ϕ k : Ω → R d is a Borel-measurable function representing the terminal payoff of an option, in terms of physical units of an underlying d-dimensional asset. Any ϕ k has bid and ask price at time 0 denoted, respectively, by b k and a k . We set Φ := [ϕ1; · · · ; ϕe; −ϕ1; · · · ; −ϕe] with corresponding prices p := (a1, . . . , ae, −b1, . . . , −be) T . Φ takes values in R d×m and p ∈ R m with m := 2e. For ease of notation we relabel the options and incorporate their price in the payoff so that Φ = [φ1 − p1e d ; · · · ; φm − pme d ].
In addition, we suppose that we are given a dynamic trading market (K, C) satisfying all the hypothesis of Section 2. An admissible strategy is of the formη := (η, α) with η ∈ H K a dynamic strategy and α ∈ R m + . Definition 4.11. We say that NA s Φ (P) holds if NA s (P) holds for the dynamic trading market (K, C) and ηT + Φα ∈ KT P-q.s implies α = 0.
The set of priorsP is obtained from the collectionPt :=Pt ⊗ P(R m ). The set of constraints in the frictionless market is obtained as C × R m + . The set of randomized and consistent strategies in the frictionless market are defined as before and denoted here asH r andH. Similarly for the corresponding superhedging pricesπ r andπ. We here define the semi-static consistent superhedging price asπ Φ(g) := inf y ∈ R | ∃H ∈H such that y + (H •S)T ≥ gP-q.s. and H k t = H k 1 for any k = d + 1, . . . , m, t = 1, . . . , T .
We only need to show the following. Proof. The inequality (≤) is clear as any strategy for the right hand side is also allowed for the left hand side. For the inequality (≥), suppose that (y,H) ∈ R ×H is a superhedge for G ·ŜT . LetH = (H, h), where H is the vector of the first d components and h is the vector of the last m components. Recall thatH ∈H is consistent, i.e, it only depends on the ω variable. Let At := ∪ m k=d+1 {ω ∈ Ωt |H k t+1 =H k 1 } and lett be the first time 0 ≤ t ≤ T − 1 such that P(At) > 0 for some P ∈ P. The superhedging property reads as, Take now ξ a measurable selector of {x ∈ R m | x · ht +1 < 0}. This exists from [7,  as the random set corresponds to the interior of the polar cone of ht +1 . Since P is fixed we might take a Borel-measurable version of ξ. Let (Pt)t=0,...,T −1 be the kernel decomposition of P, extended arbitrarily to t = −1. Fix x ∈ R m and δ θt an arbitrary selector of δΘ t from Corollary 3.2, for any t ∈ I. For any λ > 0, define the probability kernels thus, it is analytic. Moreover, the set {(ω, ω) ∈ Ω × Ω} is a Borel subset of Ω × Ω. To conclude