Generalized Duality for Model-Free Superhedging given Marginals
aa r X i v : . [ q -f i n . P R ] S e p Generalized Duality for Model-Free Superhedging given Marginals
Arash Fahim ∗ Yu-Jui Huang † Saeed Khalili ‡ September 17, 2019
Abstract
In a discrete-time financial market, a generalized duality is established for model-free su-perhedging, given marginal distributions of the underlying asset. Contrary to prior studies, wedo not require contingent claims to be upper semicontinuous, allowing for upper semi-analyticones. The generalized duality stipulates an extended version of risk-neutral pricing. To com-pute the model-free superhedging price, one needs to find the supremum of expected values ofa contingent claim, evaluated not directly under martingale (risk-neutral) measures, but alongsequences of measures that converge, in an appropriate sense, to martingale ones. To derive themain result, we first establish a portfolio-constrained duality for upper semi-analytic contingentclaims, relying on Choquet’s capacitability theorem. As we gradually fade out the portfolioconstraint, the generalized duality emerges through delicate probabilistic estimations.
MSC (2010):
Keywords:
Model-free superhedging, semi-static trading strategies, optimal transport, Cho-quet’s capacitability theorem
Given a finite time horizon T ∈ N with T ≥
2, let Ω := R T + = [0 , ∞ ) T be the path space and S bethe canonical process, i.e. S t ( x , x , ..., x T ) = x t for all ( x , x , ..., x T ) ∈ Ω. We denote by P (Ω)the set of all probability measures on Ω. For all t = 1 , ..., T , let µ t be a probability measure on R + that has finite first moment; namely, m ( µ t ) := Z R + ydµ t ( y ) < ∞ . (1.1)The set of admissible probability measures on Ω is given byΠ := (cid:8) Q ∈ P (Ω) : Q ◦ ( S t ) − = µ t , ∀ t = 1 , ..., T (cid:9) , (1.2) ∗ Florida State University, Department of Mathematics, Tallahassee, FL 32306-4510, USA, email: [email protected] . Partially supported by Florida State University CRC FYAP (315-81000-2424) and NationalScience Foundation (DMS-1209519). † University of Colorado, Department of Applied Mathematics, Boulder, CO 80309-0526, USA, email: [email protected] . Partially supported by National Science Foundation (DMS-1715439) and the Universityof Colorado (11003573). ‡ University of Colorado, Department of Mathematics, Boulder, CO 80309-0395, USA, email: [email protected] . M := { Q ∈ Π : S is a Q -martingale } . (1.3)Note that M 6 = ∅ if and only if µ , ..., µ T possess the same finite first moment and increase in theconvex order (i.e. R R + f dµ ≤ R R + f dµ ≤ ... ≤ R R + f dµ T , for convex f : R + → R ); see [20]. Wewill assume M 6 = ∅ throughout this paper.The current setup is motivated by a financial market that involves a risky asset, representedby S , and abundant tradable options written on it. For instance, if the tradable options at time0 include vanilla call options, with payoff ( S t − K ) + , for all t = 1 , · · · , T and K ≥
0, then thecurrent market prices C ( t, K ) of these call options already prescribe the distribution of S t , for each t = 1 , ..., T , under any pricing (martingale) measure. A path-dependent contingent claim Φ : Ω → R can be superhedged by trading the underlying S and holding options available at time 0. Specifically, let H be the set of ∆ = { ∆ t } T − t =1 with ∆ t : R t + → R Borel measurable for all t = 1 , ..., T −
1. Each ∆ ∈ H represents a self-financing (dynamic)trading strategy. The resulting change of wealth over time along a path x = ( x , ..., x T ) ∈ Ω isgiven by (∆ · x ) t := t − X i =1 ∆ i ( x , ..., x i ) · ( x i +1 − x i ) , for t = 2 , ..., T. In addition, by writing µ = ( µ , ..., µ T ), we denote by L ( µ ) the set of u = ( u , ..., u T ) where u t : R + → R is µ t -integrable for all t = 1 , ..., T . Each u ∈ L ( µ ) represents a collection of optionswith different maturities. A semi-static superhedge of Φ consists of some ∆ ∈ H and u ∈ L ( µ )such that Ψ u, ∆ ( x ) := T X t =1 u t ( x t ) + (∆ · x ) T ≥ Φ( x ) , for all x = ( x , ..., x T ) ∈ Ω . (1.4)Such superhedging is model-free : the terminal wealth Ψ u, ∆ is required to dominate Φ on every path x ∈ Ω, instead of P -a.e. x ∈ Ω for some probability P . This is distinct from the standard model-based approach: classically, one first specifies a model, or physical measure, P for the financialmarket, and then superhedges a contingent claim P -a.s. With the pointwise relation (1.4), nomatter which P materializes, Ψ u, ∆ ≥ Φ must hold P -a.s. There is then no need to specify a physicalmeasure P a priori, which prevents any model misspecification.The corresponding model-free superhedging price of Φ is defined by D (Φ) := inf (cid:8) µ ( u ) : u ∈ L ( µ ) satisfies ∃ ∆ ∈ H s.t. Ψ u, ∆ ( x ) ≥ Φ( x ) ∀ x ∈ Ω (cid:9) , (1.5)where µ ( u ) := P Tt =1 R R + u t dµ t . To characterize D (Φ), the minimal cost to achieve (1.4), Beiglb¨ock,Henry-Labord´ere, and Penkner [2] introduce the martingale optimal transport problem P (Φ) := sup Q ∈M E Q [Φ] . (1.6)When Φ is upper semicontinuous, denoted by Φ ∈ USC(Ω), and grows linearly, D (Φ) coincideswith P (Φ). By [12, Proposition 2.1], for each fixed t , as long as K C ( t, K ) is convex and nonnegative, lim K ↓ ∂ K C ( t, K ) ≥−
1, and lim K →∞ C ( t, K ) = 0, the relation “ E Q [( S t − K ) + ] = C ( t, K ) for all K ≥
0” determines the distribution of S t . That is, Π in (1.2) can be expressed as (cid:8) Q ∈ P (Ω) : E Q [( S t − K ) + ] = C ( t, K ) , ∀ t = 1 , · · · , T and K ≥ (cid:9) . roposition 1.1 (Corollary 1.1, [2]) . Given Φ ∈ USC(Ω) for which there exists
K > such that Φ( x ) ≤ K (1 + x + · · · + x T ) ∀ x = ( x , · · · , x T ) ∈ Ω , (1.7) we have D (Φ) = P (Φ) . Model-free superhedging given marginals, pioneered by Hobson [13], has traditionally focusedon specific forms of contingent claims; see e.g. [5], [14], [17], [6], and [9]. The main contributionof [2] is to allow for general, albeit upper semicontinuous, contingent claims, via the superhedgingduality in Proposition 1.1. In deriving this duality, [2] uses upper semicontinuity only once for aminimax argument. It is tempting to believe that upper semicontinuity is only a technical conditionthat can eventually be relaxed.This is, however, not the case. While the model-free duality given marginals in [2] has beenwidely studied and enriched by now (see [10], [1], [11], and [8], among others), the requirement ofupper semicontinuity stands still. Recently, Beiglb¨ock, Nutz, and Touzi [3] has shown that, in fact,upper semicontinuity cannot be relaxed. They provide a counterexample where Φ is lower, but notupper, semicontinuous and the duality D (Φ) = P (Φ) fails. To restore the duality, [3] modifies thedefinition of D (Φ) in (1.5) in a quasi-sure way: the inequality Ψ u, ∆ ≥ Φ is required to hold not pointwise, but M -quasi surely; that is, Ψ u, ∆ ≥ Φ holds outside of a set that is P -null for all P ∈ M .This quasi-sure modification successfully yields the duality D qs (Φ) = P (Φ) for Borel measurable Φ,where D qs (Φ) denotes the modified D (Φ) as described above. This is done in [3] for the two-periodmodel (i.e. T = 2), and in Nutz, Stebegg, and Tan [18] for the multi-period case (i.e. T ∈ N ).In this paper, we approach the failure of D (Φ) = P (Φ) from an opposite angle. We keepthe definition of D (Φ) as in (1.5), and investigate how P (Φ) should be modified to get a generalduality for Borel measurable Φ and beyond. This has two motivations in terms of both theory andapplications.From the theoretical point of view, the pointwise relation (1.4) is inherited from the optimaltransport theory: the dual problem in the Monge-Kantorovich duality is almost identical to D (Φ),except that it involves the simpler pointwise relation P Tt =1 u t ( x t ) ≥ Φ( x ) (i.e. without the term(∆ · x ) T in (1.4)); see [15]. That is, D (Φ) naturally extends the classical dual problem from optimaltransport to the more general setting we focus on. Finding the primal problem corresponding tothis extended dual is of great theoretical interest in itself.More crucially, as D (Φ) represents precisely the minimal cost for model-free superhedging, ifwe modify its definition, although a duality can be obtained (as in [3] and [18]), it will no longeradhere to the model-free superhedging context, thereby losing its financial relevance. In fact, thereare two different applications here. In the context of optimal transport, Φ is a payoff functionthat assigns a reward to each transportation path x = ( x , ..., x T ) ∈ Ω, and every Q ∈ M is anadmissible transportation plan. The goal is to maximize reward from transportation, i.e. to attain P (Φ) in (1.6)—the perspective taken by [3] and [18]. Our goal, by contrast, is to minimize the costof model-free superhedging; all developments should then be centered around D (Φ) in (1.5).Instead of dealing with D (Φ) directly, we impose, somewhat artificially, portfolio constraints.For any N ∈ N , we consider H N := { ∆ ∈ H : | ∆ t | ≤ N, ∀ t = 1 , · · · , T − } , (1.8)and define D N (Φ) as in (1.5), with H therein replaced by H N . That is, D N (Φ) is a portfolio-constrained model-free superhedging price. Thanks to the general duality in Fahim and Huang[11], the corresponding primal problem P N (Φ) can be identified, and there is no duality gap (i.e. D N (Φ) = P N (Φ)) when Φ is upper semicontinuous. The first major contribution of this paper,3heorem 3.1, shows that this portfolio-constrained duality actually holds generally for upper semi-analytic Φ. Specifically, by treating D N and P N as functionals, we derive appropriate upward anddownarrow continuity (Sections 3.1 and 3.2). Choquet’s capacitability theorem can then be invokedto extend D N (Φ) = P N (Φ) from upper semicontinuous Φ to upper semi-analytic ones.Note that the portfolio bound N ∈ N is indispensable here. In the technical result Lemma 3.2,the compactness of the space of semi-static strategies ( u, ∆) ∈ L ( µ ) × H N is extracted from thebound N ∈ N , under an appropriate weak topology. Such compactness then gives rise to theupward continuity of D N ; see Proposition 3.4. As opposed to this, D in (1.5), when viewed as afunctional, does not possess the desired upward continuity. This prevents a direct application ofChoquet’s capacitability theorem to the unconstrained duality D (Φ) = P (Φ) in Proposition 1.1;see Remark 3.3 for details.By taking N → ∞ in the constrained duality D N (Φ) = P N (Φ), we obtain a new characteriza-tion of D (Φ), for upper semi-analytic Φ; see Theorem 2.1, the main result of this paper. This newcharacterization asserts a generalized version of risk-neutral pricing. To find the model-free super-hedging price D (Φ), we need to compute expected values of Φ, but not directly under risk-neutral(martingale) measures Q ∈ M . As prescribed by Theorem 2.1, we should consider sequences ofmeasures { Q n } n ∈ N that converge to M appropriately, and compute the limiting expected values, i.e.lim sup n →∞ E Q n [Φ]. The supremum of these limiting expected values then characterizes D (Φ). Forthe special case where Φ is upper semicontinuous, these limiting expected values can be attained bymeasures Q ∈ M , as shown in Proposition 2.1. The generalized duality in Theorem 2.1 thus reducesto one that involves solely measures in M , recovering the classical duality in Proposition 1.1.In deriving the generalized duality in Theorem 2.1 from the constrained one D N (Φ) = P N (Φ),one needs the relation lim N →∞ D N (Φ) = D (Φ). This is equivalent to D ∞ (Φ) = D (Φ), where D ∞ (Φ) is defined as in (1.5), with H therein replaced by H ∞ := { ∆ ∈ H : ∆ t is bound , ∀ t = 1 , · · · , T − } . (1.9)This turns out to be highly nontrivial, and is established through delicate probabilistic estimations;see Proposition 4.2 for details. Such a relation is economically intriguing in itself: it states thatrestricting to bounded trading strategies does not increase the cost of model-free superhedging. Tothe best of our knowledge, this harmless restriction to bounded strategies has not been identifiedin the literature under such generality.The rest of the paper is organized as follows. Section 2 introduces the main result of this paper,a generalized duality that characterizes D (Φ), for upper semi-analytic Φ. Section 3 establishes aportfolio-constrained duality for upper semi-analytic contingent claims, by using Choquet’s capacitytheory. Section 4 derives an unconstrained duality for upper semi-analytic contingent claims, asthe limiting case of the constrained one in Section 3; this completes the proof of the main result. Let Y = R t + for some t = 1 , , ..., T . We denote by G ( Y ) the set of all functions from Ω to R .Moreover, let USA( Y ), B ( Y ), and USC( Y ) be the sets of functions in G ( Y ) that are upper semi-analytic, Borel measurable, and upper semicontinuous, respectively. Throughout this paper, forany Φ ∈ G (Ω) and Q ∈ Π, we will interpret E Q [Φ] as the outer expectation of Φ. When Φ is actuallyBorel measurable, it reduces to the standard expectation of Φ.For any u ∈ L ( µ ), we will write ⊕ u ( x ) := P Tt =1 u t ( x t ) for x = ( x , ..., x T ) ∈ Ω and µ ( u ) := P Tt =1 R R + u t dµ t , as specified below (1.5). 4 The Main Result
Given N ∈ N , recall H N defined in (1.8). For each Q ∈ Π, we introduce A NT ( Q ) := sup ∆ ∈H N E Q [(∆ · S ) T ] = sup ∆ ∈H Nc E Q [(∆ · S ) T ] , (2.1)where H Nc := { ∆ ∈ H N : ∆ t is continuous , ∀ t = 1 , · · · , T − } . Note that the reduction to continuous trading strategies in (2.1) is justified by [11, Lemma 3.3].The set M in (1.3) can be fully characterized by A NT ( Q ) as follows. Lemma 2.1. Q ∈ M ⇐⇒ A T ( Q ) = 0 ⇐⇒ A NT ( Q ) = 0 for all N ∈ N .Proof. By definition, A NT ( Q ) = N A T ( Q ). Thus, A T ( Q ) = 0 if and only if A NT ( Q ) = 0 for all N ∈ N . Now, by (2.1), “ A NT ( Q ) = 0 for all N ∈ N ” is equivalent to “ E Q [(∆ · S ) T ] = 0 for all∆ ∈ H Nc , for any N ∈ N ”. The latter condition holds if and only if Q ∈ M , by [2, Lemma 2.3].Lemma 2.1 indicates that a pseudometric on Π can be defined by d ( Q , Q ) := (cid:12)(cid:12) A T ( Q ) − A T ( Q ) (cid:12)(cid:12) , ∀ Q , Q ∈ Π . (2.2)It is only a pseudometric, but not a metric, because d ( Q , Q ) = 0 does not necessarily imply Q = Q . We can turn it into a metric by considering equivalent classes induced by d . Specifically,we say Q , Q ∈ Π are equivalent (denoted by Q ∼ Q ) if d ( Q , Q ) = 0, or A T ( Q ) = A T ( Q ).Equivalent classes are then defined by [ Q ] := { Q ′ ∈ Π : d ( Q ′ , Q ) = 0 } for all Q ∈ Π. On thequotient space Π ∗ := Π / ∼ = { [ Q ] : Q ∈ Π } , ρ ([ Q ] , [ Q ]) := d ( Q , Q ) (2.3)defines a metric. Remark 2.1.
In view of Lemma 2.1, M = [ Q ] for any Q ∈ M . Remark 2.2.
Instead of the pseudometric on Π in (2.2) , one can consider the semi-norm k Q k := sup ∆ ∈H Z Ω (∆ · S ) T dQ defined on the vector space K := { Q : Q is a signed measure on Ω } . When we restrict the semi-norm to Π ⊂ K , we have k Q k = 0 if and only Q ∈ M (thanks to Lemma 2.1). This can be used todefine a metric equivalent to (2.3) . To state the main result of this paper, Theorem 2.1 below, we need to consider a sequence { Q N } N ∈ N in Π that converge to M under the metric ρ ; that is, by Remark 2.1, ρ ([ Q N ] , M ) = ρ ([ Q N ] , [ Q ]) → N → ∞ , ∀ Q ∈ M . For simplicity, this will be denoted by Q N ρ → M . As Q N ρ → M is equivalent to A T ( Q N ) →
0, by(2.3) and (2.2), they will be used interchangeably throughout the paper.Crucially, Q N ρ → M entails weak convergence to M (up to a subsequence).5 emma 2.2. Consider { Q N } N ∈ N in Π such that Q N ρ → M . For any subsequence { Q N k } k ∈ N thatconverges weakly, it must converge weakly to some Q ∗ ∈ M .Proof. Let Q ∗ ∈ Π denote the probability measure to which Q N k converges weakly. First, recallthat Q N ρ → M is equivalent to A T ( Q N ) →
0, which in turn implies A T ( Q N k ) →
0. Next, for any∆ ∈ H c , since | (∆ · x ) T | ≤ h ( x ) := x + 2( x + ... + x T − ) + x T , we deduce from [21, Lemma 4.3]that Q E Q [(∆ · S ) T ] is continuous under the topology of weak convergence. It follows that Q A T ( Q ) = sup ∆ ∈H c E Q [(∆ · S ) T ]is lower semicontinuous under the topology of weak convergence. Hence, A T ( Q ∗ ) ≤ lim inf k →∞ A T ( Q N k ) = 0 . We then conclude A T ( Q ∗ ) = 0, which implies Q ∗ ∈ M thanks to Lemma 2.1. Now, we are ready to present the main result of this paper.
Theorem 2.1.
For any Φ ∈ USA(Ω) for which there exists
K > such that | Φ( x ) | ≤ K (1 + x + · · · + x T ) ∀ x = ( x , · · · , x T ) ∈ Ω , (2.4) we have D (Φ) = e P (Φ) := sup (cid:26) lim sup N →∞ E Q N [Φ] : Q N ρ → M (cid:27) . (2.5)When Φ is additionally upper semicontinuous, Theorem 2.1 recovers the classical duality inProposition 1.1, as the next result demonstrates. Proposition 2.1.
For any Φ ∈ USC(Ω) that satisfies (2.4) , e P (Φ) reduces to P (Φ) in (1.6) .Proof. For any Q ∈ M , by taking Q N := Q for all N ∈ N , the definition of e P (Φ) in (2.5) directlyimplies e P (Φ) ≥ E Q [Φ]. Taking supremum over Q ∈ M yields e P (Φ) ≥ P (Φ).On the other hand, take an arbitrary { Q N } N ∈ N in Π such that Q N ρ → M . For any ε >
0, thereexists a subsequence { Q N k } k ∈ N such thatlim k →∞ E Q Nk [Φ] ≥ lim sup N →∞ E Q N [Φ] − ε. (2.6)As Π is compact (recall the explanation below (1.2)), there is a further subsequence, which willstill be denoted by { Q N k } k ∈ N , that converges weakly to some Q ∗ ∈ Π. By Lemma 2.2, Q ∗ mustbelong to M . Now, as Φ is upper semicontinuous and satisfies (2.4), we deduce from [21, Lemma4.3] and { Q N k } converging weakly to Q ∗ ∈ M thatlim k →∞ E Q Nk [Φ] ≤ E Q ∗ [Φ] ≤ P (Φ) . This, together with (2.6) and the arbitrariness of ε >
0, shows that lim sup N →∞ E Q N [Φ] ≤ P (Φ).As { Q N } N ∈ N such that Q N ρ → M is arbitrarily chosen, we conclude that e P (Φ) ≤ P (Φ).6heorem 2.1 extends the standard wisdom for risk-neutral pricing. To find the model-freesuperhedging price D (Φ), one needs to compute expected values of Φ, but not directly under risk-neutral (martingale) measures Q ∈ M . Instead, one should consider, more generally, sequences ofmeasures { Q N } N ∈ N in Π that converge appropriately to M , and compute the limiting expectedvalues of Φ. Only when Φ is continuous enough (i.e. upper semicontinuous) can we restrict ourattention to solely martingale measures in M , as Proposition 2.1 indicates.The next example demonstrates explicitly that despite D (Φ) > P (Φ), the generalized duality D (Φ) = e P (Φ) holds. Example 2.1.
Let T = 2 and µ = µ be the Lebesgue measure on [0 , . Then M contains onesingle measure P , under which ( S , S ) is uniformly distributed on { ( x, y ) ∈ [0 , : x = y } . Forthe lower semicontinuous Φ( x , x ) := 1 { x = x } , it is shown in [3, Example 8.1] that P (Φ) Given Φ ∈ USC(Ω) that satisfies (1.7) , D N (Φ) = P N (Φ) for all N ∈ N . Section 3 focuses on extending this portfolio-constrained duality to one that allows for uppersemi-analytic Φ. Intriguingly, by using Choquet’s capacity theory, we will show that the sameduality D N (Φ) = P N (Φ) simply holds for upper semi-analytic Φ; there is no need to adjust P N (Φ).By taking N → ∞ , Section 4 elaborates how D N (Φ) = P N (Φ) turns into the desired duality (2.5). Given N ∈ N , the goal of this section is to establish the complete duality D N (Φ) = P N (Φ) for uppersemi-analytic Φ. As such a duality is known to hold for upper semicontinuous Φ (Proposition 2.2),our strategy is to treat P N and D N as functionals, and exploit their continuity properties.Let us first recall the notion of a Choquet capacity. Recall also the notation in Section 1.1. Definition 3.1. A functional C : G (Ω) → R is called a Choquet capacity associated with USC(Ω) (or simply capacity) if it satisfies(i) C ( φ ) ≤ C ( ψ ) if φ ≤ ψ ;(ii) if φ i ↑ φ , then sup i ∈ N C (Φ i ) = C (Φ) ;(iii) for any sequence { φ i } in USC(Ω) such that φ i ↓ φ , inf i ∈ N C (Φ i ) = C (Φ) . Choquet’s capacitability theorem (see [15, Proposition 2.11] or [7, Section 3]) asserts a desirablecontinuity property of a capacity. Lemma 3.1. Let C : G (Ω) → R be a Choquet capacity associated with USC(Ω) . Then, for any Φ ∈ USA(Ω) , C (Φ) = sup { C ( φ ) : φ ≤ Φ with φ ∈ USC(Ω) } . Hence, if two capacities C and C coincide on USC(Ω) , they coincide on USA(Ω) . Remark 3.1. The original Choquet’s capacitability theorem gives a more general result: if C and C are two Choquet capacities associated with a set of functions A and they coincide on functionsin A , then they coincide on A -Suslin functions. Here, we take A = USC(Ω) in Definition 3.1 andLemma 3.1, and note that “ USC(Ω) -Suslin functions” are simply “upper semi-analytic functions( USA(Ω) )”; see [15, Proposition 2.13] and [4, Definition 7.21]. .1 Continuity of P N Proposition 3.1. Consider { Φ i } i ∈ N in G (Ω) for which there exists K > such that for each i ∈ N , Φ i ( x ) ≥ − K (1 + x + · · · + x T ) , ∀ x = ( x , · · · , x T ) ∈ Ω . (3.1) If Φ i ↑ Φ , then sup i ∈ N P N (Φ i ) = P N (Φ) , ∀ N ∈ N . Proof. Since Φ satisfies (3.1), the monotone convergence theorem for outer expectation gives E Q [Φ i ] ↑ E Q [Φ], for all Q ∈ Π. By changing the order of two supremums, we getsup i ∈ N P N (Φ i ) = sup Q ∈ Π sup i ∈ N (cid:0) E Q [Φ i ] − A NT ( Q ) (cid:1) = sup Q ∈ Π (cid:0) E Q [Φ] − A NT ( Q ) (cid:1) = P N (Φ) , for each N ∈ N . Proposition 3.2. Consider { Φ i } i ∈ N in USC(Ω) for which there exists K > such that (1.7) issatisfied for each Φ i . If Φ i ↓ Φ , then inf i ∈ N P N (Φ i ) = P N (Φ) , ∀ N ∈ N . Proof. Fix N ∈ N . As Φ i ↓ Φ clearly implies inf i ∈ N P N (Φ i ) ≥ P N (Φ), we focus on provingthe “ ≤ ” relation. Assume inf i ∈ N P N (Φ i ) > −∞ , otherwise the proof would be trivial. For any δ < inf i ∈ N P N (Φ i ), define M N (Φ i , δ ) := { Q ∈ Π : E Q [Φ i ] − A NT ( Q ) ≥ δ } for all N ∈ N . We intend to show that M N (Φ i , δ ) is compact under the topology of weak convergence. As Π iscompact (recall the explanation below (1.2)), it suffices to prove that M N (Φ i , δ ) is closed. SinceΦ i is upper semicontinuous and satisfies (1.7), we deduce from [21, Lemma 4.3] that Q E Q [Φ i ]is upper semicontinuous under the topology of weak convergence. On the other hand, by the sameargument in the proof of Lemma 2.2, Q A NT ( Q ) is lower semicontinuous under the topology ofweak convergence. As a result, Q E Q [Φ i ] − A NT ( Q ) is upper semicontinuous, which gives thedesired closedness of M N (Φ i , δ ).Now, since {M N (Φ i , δ ) } i ∈ N is a nonincreasing sequence of compact sets, T ∞ i =1 M N (Φ i , δ ) = ∅ .Take ˜ Q ∈ T ∞ i =1 M N (Φ i , δ ), and observe that P N (Φ) ≥ E ˜ Q [Φ] − A NT ( ˜ Q ) = lim i →∞ E ˜ Q [Φ i ] − A NT ( ˜ Q ) ≥ δ, where the equality follows from the reverse monotone convergence theorem, applicable here as(1.7) is satisfied for each Φ i , and the last inequality results from the definition of M N (Φ i , δ ). With δ < inf i ∈ N P N (Φ i ) arbitrarily chosen, we conclude inf i ∈ N P N (Φ i ) ≤ P N (Φ). D N The downward continuity of D N is a consequence of Propositions 2.2 and 3.2. Proposition 3.3. Consider { Φ i } i ∈ N in USC(Ω) for which there exists K > such that (1.7) issatisfied for each Φ i . If Φ i ↓ Φ , then inf i ∈ N D N (Φ i ) = D N (Φ) , ∀ N ∈ N . roof. As the infimum of a sequence of upper semicontinuous functions satisfying (1.7), Φ is againupper semicontinuous and satisfies (1.7). It then follows from Proposition 2.2 thatinf i ∈ N D N (Φ i ) = inf i ∈ N P N (Φ i ) = P N (Φ) = D N (Φ) , where the second equality is due to Proposition 3.2.The upward continuity of D N , by contrast, is much more obscure. We need the followingtechnical result, Lemma 3.2, to construct certain compactness for the space of semi-static strategies( u, ∆), which will facilitate the derivation of the upward continuity of D N in Proposition 3.4 below.This lemma can be viewed as a generalization of [15, Lemma 1.27] to the case of martingale optimaltransport. The main idea involved is to extract additional compactness from the portfolio bound N > B ( R t + ) be equipped with the topology of pointwise convergence. Inaddition, consider the product measure ν := µ ⊗ · · · ⊗ µ T on Ω, and denote by L ( µ t ) (resp. L ( ν ))the set of µ t -integrable (resp. ν -integrable) functions. Also recall m ( µ t ), t = 1 , ..., T , from (1.1). Lemma 3.2. Fix N ∈ N and Φ ∈ G (Ω) that satisfies (3.1) and D N (Φ) < ∞ . For any δ > D N (Φ) ,define L (Φ , δ, N ) as the collection of all pairs (Θ , ∆) , with Θ := n ( u k , ..., u kT , W k ) o k ∈ N ∈ (cid:0) Π Tt =1 L ( µ t ) × L ( ν ) (cid:1) N and ∆ ∈ H N , (3.2) satisfying (i) For each k ∈ N , ≤ u kt ≤ k , ∀ t = 1 , · · · , T ; (ii) u t ≤ u t ≤ · · · , ∀ t = 1 , · · · , T ; (iii) For each k ∈ N , µ ( u k ) ≤ δ + ( K + 2 N )(1 + m ( µ ) + · · · + m ( µ T )) ; (iv) For each k ∈ N , W k ∈ L ( ν ) with ≤ W k ≤ Λ , where Λ ∈ L ( ν ) is defined by Λ( x ) := 2 N ( x + · · · + x T ); moreover, W k = 0 on the set { x : Λ( x ) < k } ; (v) For each k ∈ N , ⊕ u k ≥ (Φ + Γ) ∧ k + (∆ · x ) T − W k , where Γ ∈ L ( ν ) is defined by Γ( x ) := ( K + 2 N )(1 + x + · · · + x T ) . Here, the constant K > in (iii) and (v) comes from (3.1) .The set L (Φ , δ, N ) is a nonempty compact subset of (cid:0) Π Tt =1 L ( µ t ) × L ( ν ) (cid:1) N × Π T − t =1 B ( R t + ) , underthe product of the weak topologies of the spaces L ( µ t ) , L ( ν ) , and B ( R t + ) .Proof. Step 1: We show that L (Φ , δ, N ) is nonempty . As δ > D N (Φ), there exist u = ( u , ..., u T ) ∈ L ( µ ) and ∆ ∈ H N such that µ ( u ) ≤ δ and ⊕ u + (∆ · x ) T ≥ Φ . As Φ satisfies (3.1) and | (∆ · x ) T | ≤ Λ( x ), we have ⊕ u ( x ) ≥ Φ( x ) − (∆ · x ) T ≥ − Γ( x ). This impliesthat we can find constants a , a , ..., a T such that P Tt =1 a t = 0 and a t + u t ≥ − ( K + 2 N )(1 /T + x t )10or all t = 1 , ..., T . Now, define ¯ u t := a t + u t + ( K + 2 N )(1 /T + x t ) ≥ t = 1 , ..., T . Then,one can write ⊕ ¯ u ≥ Φ + Γ + ( ¯∆ · x ) T , with ¯∆ := − ∆ ∈ H N . On the other hand, by the concavity of x x ∧ (2 k ),( ⊕ ¯ u ) ∧ (2 k ) ≥ (cid:0) (Φ + Γ) + ( ¯∆ · x ) T (cid:1) ∧ (2 k ) ≥ (Φ + Γ) ∧ k + ( ¯∆ · x ) T ∧ k. Since ¯ u t ≥ t = 1 , ..., T , it can be checked that ⊕ (¯ u ∧ (2 k )) ≥ ( ⊕ ¯ u ) ∧ (2 k ). This, togetherwith the previous inequality, gives ⊕ (¯ u ∧ (2 k )) ≥ (Φ + Γ) ∧ k + ( ¯∆ · x ) T ∧ k. (3.3)We claim that u kt := ¯ u t ∧ (2 k ), W k := ( ¯∆ · x ) T − ( ¯∆ · x ) T ∧ k , and ¯∆ form an element of L (Φ , δ, N ). Byconstruction, it is straightforward to verify conditions (i), (ii), and (v). Since ⊕ ¯ u k ≤ ⊕ ¯ u = ⊕ u + Γ,we have µ (¯ u k ) ≤ δ + ( K + 2 N ) (1 + m ( µ ) + · · · + m ( µ T )), i.e. condition (iii) is satisfied. For each k ∈ N , observe that 0 ≤ W k ≤ | ( ¯∆ · x ) T | ≤ Λ( x ). In particular, if Λ( x ) ≤ k , then | ( ¯∆ · x ) T | ≤ k and thus W k = 0 by definition. This shows that condition (iv) is satisfied. Step 2: We prove that L (Φ , δ, N ) is contained in a weakly compact space of functions. Observethat the following collections of functions U ( t, k ) := { u ∈ L ( µ t ) : 0 ≤ u ≤ k } t = 1 , ..., T and k ∈ N ,V := { W ∈ L ( ν ) : 0 ≤ W ≤ Λ } are all uniformly integrable, and thus relatively weakly compact thanks to the Dunford-Pettistheorem. It follows that the countable product (Π t,k U ( t, k )) × V N is also relatively weakly compact.On the other hand, for each t = 1 , · · · , T − F t := { f : R t + → R : | f | ≤ N } = Π x ∈ R t + [ − N, N ] x is compact under the topology of pointwise convergence, as a consequence of Tychonoff’s theorem.The space F t is therefore weakly compact, and this carries over to the product space H N = Π t F t .We then conclude that Π t,k U ( t, k ) × V N × H N is a weakly compact set containing L (Φ , δ, N ). Step 3: We prove that L (Φ , δ, N ) is strongly closed . Take a sequence n { ( u k,m , · · · , u k,mT , W k,m ) } k ∈ N , ∆ m o m ∈ N in L (Φ , δ, N ) such that it converges to ( { ( u k , · · · , u kT , W k ) } k ∈ N , ∆) in the strong sense. That is, u k,mt → u kt in L ( µ t ), W k,m → W k in L ( ν ), and ∆ m → ∆ pointwise in H N . We intend to showthat ( { ( u k , · · · , u kT , W k ) } k ∈ N , ∆) also lies in L (Φ , δ, N ).The convergence in L ( µ t ) (resp. L ( ν )) implies the existence of a subsequence that converges µ t -a.e (resp. ν -a.e.). Then, as m → ∞ , we conclude from ⊕ u k,m ≥ (Φ + Γ) ∧ k + (∆ m · x ) T − W k,m that ⊕ u k ≥ (Φ + Γ) ∧ k + (∆ · x ) T − W k (3.4)holds outside a ν -null set N . We can then modify ( u kt ) Tt =1 and W k on N such that (3.4) holdseverywhere, i.e. condition (vi) is satisfied. Also, we see from the convergence u k,mt → u kt and∆ m → ∆ that conditions (i), (ii), and (v) are satisfied, and Fatou’s lemma implies the validityof (iii). From the convergence W k,m → W k , we have 0 ≤ W k ≤ Λ. Moreover, W k = 0 on { x : Λ( x ) < k } because W k,m = 0 on { x : Λ( x ) < k } for all m ∈ N . This shows that condition11iv) is satisfied. We therefore conclude that ( { ( u k , · · · , u kT , W k ) } k ∈ N , ∆) ∈ L (Φ , δ, N ), and thus L (Φ , δ, N ) is closed under the strong topology. Step 4: We prove the desired compactness of L (Φ , δ, N ). Observe that L (Φ , δ, N ) is convex.Since a strongly closed convex set is also weakly closed, and the weak topology of a product spacecoincides with the product of the weak topologies, we conclude that L (Φ , δ, N ) is closed under theproduct of the weak topologies in the spaces L ( µ t ), L ( ν ), and B ( R t + ). It is therefore weaklycompact in view of Step 2. Remark 3.2. While the motivation of Lemma 3.2 is to construct some compactness for the spaceof semi-static strategies ( u, ∆) , we have to introduce the auxiliary random variable W k in (3.2) toensure the convexity of L (Φ , δ, N ) , needed in the last step of the proof. Proposition 3.4. Consider { Φ i } i ∈ N in G (Ω) for which there exists K > such that (3.1) issatisfied for all i ∈ N . If Φ i ↑ Φ , then sup i ∈ N D N (Φ i ) = D N (Φ) , ∀ N ∈ N . Proof. Fix N ∈ N . As Φ i ↑ Φ clearly implies sup i ∈ N D N (Φ i ) ≤ D N (Φ), we focus on proving the“ ≥ ” relation. Assume sup i ∈ N D N (Φ) < ∞ , otherwise the proof would be trivial. Pick an arbitrary δ > sup i ∈ N D N (Φ i ). By Lemma 3.2, {L (Φ i , δ, N ) } i ∈ N is a nonincreasing sequence of nonemptycompact sets. We can therefore choose some (cid:0) { ( u k , ..., u kT , W k ) } k ∈ N , ∆ (cid:1) ∈ T i ∈ N L (Φ i , δ, N i ). Inview of conditions (i), (ii), and (iii) in Lemma 3.2, u t := lim k →∞ ↑ u kt ∈ L ( µ t ) is well-defined, and u = ( u , ..., u T ) satisfies µ ( u ) ≤ δ + ( K + 2 N )(1 + m ( µ ) + · · · + m ( µ T )) . (3.5)Moreover, condition (v) in Lemma 3.2 implies that for each k and i , ⊕ u k ≥ (Φ i + Γ) ∧ k + (∆ · x ) T − W k . Recall from condition (iv) in Lemma 3.2 that W k = 0 on { x : Λ( x ) < k } . This in particular implies W k ( x ) → x ∈ Ω as k → ∞ . Therefore, by taking k → ∞ in the previous inequality, weget ⊕ u ≥ Φ i + Γ + (∆ · x ) T . As i → ∞ , this yields ⊕ u ≥ Φ + Γ + (∆ · x ) T . (3.6)Now, define ¯ u t := u t − ( K + 2 N )(1 /T + x t ) for all t = 1 , · · · , T . By (3.5) and (3.6), µ (¯ u ) = µ ( u ) − ( K + 2 N )(1 + m ( µ ) + · · · + m ( µ T )) ≤ δ, ⊕ ¯ u = ⊕ u − Γ ≥ Φ + (∆ · x ) T . This readily implies D N (Φ) ≤ δ . With δ > sup i ∈ N D N (Φ i ) arbitrarily chosen, we concludesup i ∈ N D N (Φ i ) ≥ D N (Φ). Theorem 3.1. For any Φ ∈ USA(Ω) that satisfies (2.4) , D N (Φ) = P N (Φ) , ∀ N ∈ N . (3.7) Moreover, there exists an optimizer ( u, ∆) ∈ L ( µ ) × H N for D N (Φ) whenever D N (Φ) < ∞ . roof. Fix N ∈ N . Define ζ K ( x ) := K (1 + x + ... + x T ), with K > P N and ¯ D N defined by¯ P N ( ϕ ) := P N ( − ζ K ∨ ( ϕ ∧ ζ K )) and ¯ D N ( ϕ ) := D N ( − ζ K ∨ ( ϕ ∧ ζ K )) , for ϕ ∈ G (Ω) . In view of Propositions 3.1 and 3.2 (resp. Propositions 3.3 and 3.4), ¯ P N (resp. ¯ D N ) is a Cho-quet capacity associated with USC(Ω); recall Definition 3.1. Moreover, thanks to Proposition 2.2,¯ D N ( ϕ ) = ¯ P N ( ϕ ) for all ϕ ∈ USC(Ω). We then conclude from Lemma 3.1 that ¯ D N ( ϕ ) = ¯ P N ( ϕ ) forall ϕ ∈ USA(Ω). That is to say, D N ( ϕ ) = P N ( ϕ ) for all ϕ ∈ USA(Ω) satisfying | ϕ | ≤ ζ K , or (2.4).It remains to prove the existence of an optimizer for D N (Φ). If D N (Φ) < ∞ , take a realsequence { δ i } such that δ i ↓ D N (Φ). By Lemma 3.2, {L (Φ , δ i , N ) } i ∈ N is a nonincreasing se-quence of nonempty compact sets. We can therefore choose some (cid:0) { ( u k , ..., u kT , W k ) } k ∈ N , ∆ (cid:1) ∈ T i ∈ N L (Φ , δ i , N ). In view of conditions (i), (ii), and (iii) in Lemma 3.2, u t := lim k →∞ ↑ u kt ∈ L ( µ t )is well-defined, and u = ( u , ..., u T ) satisfies µ ( u ) ≤ D N (Φ) + ( K + 2 N )(1 + m ( µ ) + · · · + m ( µ T )) . (3.8)Moreover, condition (v) in Lemma 3.2 implies that for each k and i , ⊕ u k ≥ (Φ + Γ) ∧ k + (∆ · x ) T − W k . As shown in the proof of Proposition 3.4, W k ( x ) → x ∈ Ω as k → ∞ . Thus, by taking k →∞ in the previous inequality, we get ⊕ u ≥ Φ+Γ+(∆ · x ) T . Now, define ¯ u t := u t − ( K +2 N )(1 /T + x t )for all t = 1 , · · · , T . Then, ⊕ ¯ u = ⊕ u − Γ ≥ Φ + (∆ · x ) T . Moreover, by (3.8), µ (¯ u ) = µ ( u ) − ( K + 2 N )(1 + m ( µ ) + · · · + m ( µ T )) ≤ D N (Φ) . This implies that, (¯ u, − ∆) ∈ L ( µ ) × H N is an optimizer of D N (Φ). Remark 3.3. When we view D and P , defined in (1.5) and (1.6) , as functionals, arguments similarto (and simpler than) those in Sections 3.1 and 3.2 yield the upward and downward continuity of P ,as well as the downward continuity of D . However, the upward continuity of D is obscure. Withoutthe portfolio bound N > , it is unclear how the space of semi-static strategies ( u, ∆) ∈ L ( µ ) × H can be made compact under any topology, so that the upward continuity does not follow from thearguments in Proposition 3.4.In fact, since D (Φ) = P (Φ) for some Borel measurable Φ (as shown in [3, Example 3.1]),the upward continuity of D must not hold. Otherwise, we could apply Choquet’s capacitabilitytheorem directly to the classical duality D (Φ) = P (Φ) in Proposition 1.1, extending it from uppersemicontinuous Φ to upper semi-analytic ones (which include Borel measurable ones). Remark 3.4. Recall Example 2.1, where P (Φ) < D (Φ) = 1 . We will show that P N (Φ) = D N (Φ) for all N ∈ N . Fix N ∈ N . Recall that ( u ∗ , u ∗ , ∆ ∗ ) ≡ (1 , , is an optimizer of D (Φ) . As ∆ ∗ ∈ H N , ( u ∗ , u ∗ , ∆ ∗ ) is also an optimizer of D N (Φ) , and thus D N (Φ) = D (Φ) = 1 . On the otherhand, consider { Q M } M ∈ N in Π constructed in (2.7) . By (2.8) , A N ( Q M ) = N A ( Q M ) = N M . Itfollows that P N (Φ) = sup Q ∈ Π (cid:8) E Q [Φ] − A N ( Q ) (cid:9) ≥ lim M →∞ (cid:8) E Q M [Φ] − A N ( Q M ) (cid:9) = 1 . As Φ ≤ already implies P N (Φ) ≤ , we conclude P N (Φ) = 1 = D N (Φ) . Remark 3.5. P N (Φ) in general does not admit an optimizer, unless Φ is upper semicontinuous.To illustrate, in Example 2.1, suppose that there exists Q ∗ ∈ Π such that E Q ∗ [Φ] − A N ( Q ∗ ) = P N (Φ) = 1 for some N ∈ N . Then, ≤ A N ( Q ∗ ) = E Q ∗ [Φ] − ≤ , which yields A N ( Q ∗ ) = 0 .By Proposition 2.1, Q ∗ must belong to M and thus coincide with P . This, however, entails E Q ∗ [Φ] − A N ( Q ∗ ) = 0 , a contradiction. Derivation of Theorem 2.1 This section is devoted to proving Theorem 2.1. To connect the portfolio-constrained duality (3.7)to the desired (unconstrained) duality (2.5), it is natural to relax the constraint N > N → ∞ , leading to the next result. Recall D ∞ (Φ) defined above (1.9) and e P (Φ) defined in (2.5). Proposition 4.1. For any Φ ∈ USA(Ω) that satisfies (2.4) , D ∞ (Φ) = e P (Φ) .Proof. First, we show that D ∞ (Φ) ≥ e P (Φ). Fix { Q N } N ∈ N in Π such that Q N ρ → M (or equiv-alently, A T ( Q N ) → h such that h ( N ) → ∞ and h ( N ) A T ( Q N ) → h ( N ) := 1 / q A T ( Q N )). For each N ∈ N , there exist( u, ∆) ∈ L ( µ ) × H h ( N ) with µ ( u ) < D h ( N ) (Φ) + 1 /N such that ⊕ u + (∆ · S ) T ≥ Φ. If follows that D h ( N ) (Φ) + 1 /N + A h ( N ) T ( Q N ) ≥ µ ( u ) + E Q N [(∆ · S ) T ] ≥ E Q N [Φ] , where the first inequality follows from the definition of A h ( N ) T ( Q N ) in (2.1). As N → ∞ in theabove inequality, since A h ( N ) T ( Q N ) = h ( N ) A T ( Q N ) → D h ( N ) (Φ) → D ∞ (Φ) by definition, weget D ∞ (Φ) ≥ lim sup N →∞ E Q N [Φ]. With { Q N } N ∈ N arbitrarily chosen, we obtain D ∞ (Φ) ≥ e P (Φ).On the other hand, for any N ∈ N , by the definition of P N (Φ), we can take Q N ∈ Π such that P N (Φ) ≥ E Q N [Φ] − A NT ( Q N ) > P N (Φ) − /N. (4.1)This, together with A NT ( Q N ) = N A T ( Q N ), shows that A T ( Q N ) < E Q N [Φ] − P N (Φ) + 1 /NN ≤ CN , ∀ N ∈ N . (4.2)Here, the constant C > N , thanks to (2.4) and (1.1). This inparticular implies A T ( Q N ) → 0. In view of (4.1), this yieldslim N →∞ P N (Φ) = lim N →∞ { E Q N [Φ] − A NT ( Q N ) } ≤ lim sup N →∞ E Q N [Φ] ≤ e P (Φ) . (4.3)Finally, by taking N → ∞ in the constrained duality (3.7) and using the above inequality, weobtain D ∞ (Φ) = lim N →∞ P N (Φ) ≤ e P (Φ).In view of Proposition 4.1, to obtain the desired duality (2.5), it remains to show D ∞ (Φ) = D (Φ)for all Φ ∈ USA(Ω) satisfying (2.4). That is, restricting to bounded trading strategies does notincrease the cost of model-free superhedging. To this end, we need the following technical result. Lemma 4.1. Given Φ ∈ G (Ω) that satisfies (2.4) , we define Φ n ∈ G (Ω) , for each n ∈ N , by Φ n ( x , ..., x T ) := Φ( x , ..., x T )1 { x ≤ n,...,x T ≤ n } ( x , ..., x T ) , ∀ x = ( x , ..., x T ) ∈ Ω . (4.4) For any ε > , there exists n ∈ N large enough such that | E Q [Φ] − E Q [Φ n ] | < ε, ∀ Q ∈ Π . (4.5) Proof. Fix ε > 0. Let δ := εK ( T + T ) . Thanks to (1.1), we can take n ∈ N large enough such that µ t (( n, ∞ )) < δ and Z { y>n } ydµ t ( y ) < δ, ∀ t = 1 , ..., T. (4.6)14or simplicity, we will write A = { x ∈ Ω : x ≤ n, ..., x T ≤ n } . Observe that A c ⊆ [ t ∈{ ,...,T } { x ∈ Ω : x t > n } . (4.7)Moreover, for each fixed t = 1 , ..., T , A c = { x ∈ Ω : x t > n } ∪ [ i ∈{ ,...,T }\{ t } { x ∈ Ω : x t ≤ n and x i > n } . (4.8)Now, for any Q ∈ Π, by (2.4), | E Q [Φ] − E Q [Φ n ] | ≤ E Q [ | Φ | A c ] ≤ K (cid:18) E Q [1 A c ] + X t =1 ,...,T E Q [ x t A c ( x )] (cid:19) . (4.9)The first inequality above requires the linearity of outer expectations; recall from Section 1.1 that E Q [ · ] denotes an outer expectation if the integrad need not be Borel measurable. While the linearityof outer expectations does not hold in general, it holds specifically here thanks to the definition ofΦ n . Indeed, by [16, Lemma 6.3], there exists Φ ∗ ∈ B (Ω), a minimal Borel measurable majorant ofΦ, such that E Q [Φ ∗ ] = E Q [Φ] and E Q [Φ ∗ B ] = E Q [Φ1 B ] for any Borel subset B of Ω. It follows that E Q [Φ] − E Q [Φ n ] = E Q [Φ ∗ ] − E Q [Φ ∗ A ] = E Q [Φ ∗ A c ] = E Q [Φ1 A c ] , where the second equality follows from the linearity of standard expectations, as Φ ∗ and Φ ∗ A areboth Borel measurable.Thanks to (4.7), E Q [1 A c ( x )] ≤ X t =1 ,...,T E Q [1 { x t >n } ( x )] = X t =1 ,...,T µ t (( n, ∞ )) < T δ, where the last inequality follows from (4.6). On the other hand, for any t = 1 , ..., T , (4.8) implies E Q [ x t A c ( x )] = E Q [ x t { x t >n } ( x )] + X i ∈{ ,...,T }\{ t } E Q [ x t { x t ≤ n, x i >n } ( x )] ≤ E Q [ x t { x t >n } ( x )] + X i ∈{ ,...,T }\{ t } E Q [ x i { x i >n } ( x )]= X i =1 ,...,T E Q [ x i { x i >n } ( x )] = X i =1 ,...,T Z { y>n } ydµ i ( y ) < T δ, where the last inequality follows from (4.6). Hence, we conclude from (4.9) that | E Q [Φ( x )] − E Q [Φ n ( x )] | ≤ K ( T + T ) δ = ε , as desired. Corollary 4.1. If D ∞ (Φ) = D (Φ) for all bounded Φ ∈ USA(Ω) , then the same equality holds forall Φ ∈ USA(Ω) satisfying (2.4) .Proof. First, we show that D ∞ (Φ) = D (Φ) for all nonnegative Φ ∈ USA(Ω) satisfying (2.4). GivenΦ ∈ USA(Ω) that is nonnegative and satisfies (2.4), consider Φ n , n ∈ N , defined in (4.4). As aproduct of Φ ∈ USA(Ω) and a nonnegative Borel measurable function, Φ n also belongs to USA(Ω),thanks to [4, Lemma 7.30]. In view of the estimate (4.5) and the definition of e P in (2.5), we deducefrom Proposition 4.1 that D ∞ (Φ n ) = e P (Φ n ) → e P (Φ) = D ∞ (Φ). Now, note that every Φ n is15ounded, thanks to the fact that Φ satisfies (2.4). As the boundedness of Φ n ∈ USA(Ω) implies D ∞ (Φ n ) = D (Φ n ) for all n ∈ N , we have D ∞ (Φ) = lim n →∞ D ∞ (Φ n ) = lim n →∞ D (Φ n ) ≤ D (Φ) , where the inequality stems from Φ n ↑ Φ, thanks to the fact that Φ is nonnegative. Since D ∞ (Φ) ≥ D (Φ) by definition, we conclude that D ∞ (Φ) = D (Φ).Now, take an arbitrary Φ ∈ USA(Ω) that satisfies (2.4) (which need not be nonnegative).Consider v = ( v , ..., v T ) ∈ L ( µ ) defined by v t ( y ) := K ( T + y ) for t = 1 , ..., T , where K > ⊕ v is nonnegative. Indeed, (Φ + ⊕ v )( x ) ≥ − K (1 + x + ... + x T ) + P Tt =1 K ( T + x t ) = 0, for all x ∈ Ω. Moreover, Φ + ⊕ v again satisfies (2.4), with apossibly larger K > 0. Hence, we have D ∞ (Φ + ⊕ v ) = D (Φ + ⊕ v ) . (4.10)Note that E Q [Φ + ⊕ v ] = E Q [Φ] + µ ( v ) for all Q ∈ Π. This, together with Proposition 4.1, implies D ∞ (Φ + ⊕ v ) = e P (Φ + ⊕ v ) = e P (Φ) + µ ( v ) = D ∞ (Φ) + µ ( v ) . (4.11)On the other hand, by definition D (Φ + ⊕ v ) = inf { µ ( u ) : u ∈ L ( µ ) satisfying ∃ ∆ ∈ H s.t. ⊕ u + (∆ · S ) T ≥ Φ + ⊕ v on Ω } = inf { µ ( u ) : u ∈ L ( µ ) satisfying ∃ ∆ ∈ H s.t. ⊕ ( u − v ) + (∆ · S ) T ≥ Φ on Ω } = inf { µ (˜ u ) + µ ( v ) : ˜ u ∈ L ( µ ) satisfying ∃ ∆ ∈ H s.t. ⊕ ˜ u + (∆ · S ) T ≥ Φ on Ω } = D (Φ) + µ ( v ) . (4.12)On the strength of (4.11) and (4.12), (4.10) yields D ∞ (Φ) = D (Φ).Now, we are ready to establish D ∞ (Φ) = D (Φ) for all upper semi-analytic Φ satisfying (2.4). Proposition 4.2. For any Φ ∈ USA(Ω) that satisfies (2.4) , D ∞ (Φ) = D (Φ) .Proof. First, by Corollary 4.1, we can assume without loss of generality that Φ ∈ USA(Ω) isbounded. We take C > | Φ | ≤ C on Ω.As D ∞ (Φ) ≥ D (Φ) by definition, we focus on proving the opposite inequality. Fix δ > 0. Thereexist u = ( u , ..., u T ) ∈ L ( µ ) and ∆ ∈ H such that µ ( u ) < D (Φ) + δ/ ⊕ u ( x ) + (∆ · x ) T ≥ Φ( x ) ∀ x ∈ Ω . (4.13) Step 1: We replace u ∈ L ( µ ) by nonnegative functions. By the Vitali-Carath´eodory theorem,there exists v = ( v , ..., v T ) ∈ L ( µ ), with u t ≤ v t and v t bounded from below for all t = 1 , ..., T ,such that µ ( u ) ≤ µ ( v ) ≤ µ ( u ) + δ/ 2. Take ℓ > v t ≥ − ℓ for all t = 1 , ..., T .By setting ¯ v t := v t + ℓ ≥ 0, we deduce from (4.13) that µ ( v ) < D (Φ) + δ and ⊕ ¯ v ( x ) + (∆ · x ) T ≥ Φ( x ) + T ℓ ∀ x ∈ Ω . (4.14) Step 2: We construct a bounded trading strategy ¯∆ ∈ H ∞ and replace (4.14) by a superhedgingrelation involving ¯∆. Fix arbitrary ε , ε , ..., ε T − > v is µ -integrable, by [19, Problem 14, p.63], there exists M ∈ B ( R + ) such that µ ( R + \ M ) < ε and¯ v is bounded on M . We can assume without loss of generality that M contains { } . Indeed, if16 ( { } ) = 0, adding { } to M does not change the above statement; if µ ( { } ) > 0, then M hasto contain { } as long as ε < µ ( { } ). For any m > 1, define f M := M ∩ ( { } ∪ (1 /m , m )) . Note that µ ( f M ) ↑ µ ( M ) as m → ∞ . Now, we claim that∆ is bounded on f M , ∀ m > . By contradiction, suppose that there exist { x n } n ∈ N in f M such that ∆ ( x n ) → ∞ or −∞ . Bytaking x = x n and x = x = ... = x T ∈ R + in the second part of (4.14) and using the fact | Φ | ≤ C , we get ¯ v ( x n ) + ¯ v ( x ) + . . . + ¯ v T ( x ) + ∆ ( x n )( x − x n ) ≥ − C + T ℓ. (4.15)For the case ∆ ( x n ) → ∞ (resp. ∆ ( x n ) → −∞ ), we take x = m (resp. x = m + 1) in(4.15). As n → ∞ , by the boundedness of ¯ v on f M , the left hand side of (4.15) tends to −∞ ,a contradiction. Similarly to the above, by [19, Problem 14, p.63], there exists M ∈ B ( R + ),containing { } , such that µ ( R + \ M ) < ε and ¯ v is bounded on M . For any m > 1, define f M := M ∩ ( { } ∪ (1 /m , m )) , and note that µ ( f M ) ↑ µ ( M ) as m → ∞ . We claim that∆ is bounded on f M × f M , ∀ m , m > . By contradiction, suppose that there exist { ( x n , x n ) } n ∈ N in f M × f M such that ∆ ( x n , x n ) → ∞ or −∞ . By taking ( x , x ) = ( x n , x n ) and x = x = ... = x T ∈ R + in the second part of (4.14)and using the fact | Φ | ≤ C , we get¯ v ( x n ) + ¯ v ( x n ) + ¯ v ( x ) + ... + ¯ v T ( x ) + ∆ ( x n )( x n − x n ) + ∆ ( x n , x n )( x − x n ) ≥ − C + T ℓ. (4.16)For the case ∆ ( x n , x n ) → ∞ (resp. ∆ ( x n , x n ) → −∞ ), we take x = m (resp. x = m + 1)in (4.16). As n → ∞ , by the boundedness of ¯ v (on f M ), ¯ v (on f M ), and ∆ (on f M ), theleft hand side of (4.16) tends to −∞ , a contradiction. By repeating the same argument for all t = 3 , , ..., T − 1, we obtain { M t } T − t =1 in B ( R + ) such that for each t = 1 , ..., T − µ t ( M ct ) = µ t ( R + \ M t ) < ε t ;(ii) µ t ( f M t ) ↑ µ t ( M t ) as m t → ∞ , where f M t := M t ∩ ( { } ∪ (1 /m t , m t )) for m t > t ( x , x , . . . , x t ) is bounded on f M × f M × ... × f M t .We also consider a t := sup f M × f M × ... × f M t | ∆ t ( x , x , . . . , x t ) | < ∞ , (4.17)for all t = 1 , ..., T − 1, which will be used in Step 3 of the proof.Now, let us define the bounded strategy ¯∆ = { ¯∆ t } T − t =1 ∈ H ∞ by¯∆ t ( x , x , . . . , x t ) := ∆ t ( x , x , . . . , x t )1 f M × ... × f M t ( x , x , ..., x t ) , ∀ t = 1 , ..., T − . x = ( x , ..., x T ) ∈ Ω, we introduce¯Φ( x ) := (Φ( x ) + T ℓ )1 f M × ... × f M T − ( x ) + T − X t =2 (∆ · S ) t f M × ... × f M t − × f M ct ( x , x , ..., x t ) . (4.18)We claim that ⊕ ¯ v ( x ) + ( ¯∆ · x ) T ≥ ¯Φ( x ) , ∀ x ∈ Ω . (4.19)Indeed, for any x ∈ Ω such that x t ∈ f M t for all t = 1 , ..., T − 1, the above inequality simply reducesto the second part of (4.14). For any x ∈ Ω such that x t / ∈ f M t for some t = 1 , ..., T − 1, consider t ∗ := inf { t ∈ { , , ..., T − } : x t / ∈ f M t } . Observe that ⊕ ¯ v ( x ) + ( ¯∆ · x ) T = ⊕ ¯ v ( x ) + ∆ ( x )( x − x ) + · · · + ∆ t ∗ − ( x , x , . . . , x t ∗ − )( x t ∗ − x t ∗ − )= ⊕ ¯ v ( x ) + (∆ · x ) t ∗ ≥ (∆ · x ) t ∗ = ¯Φ( x ) , where the inequality follows from ¯ v t ≥ 0, and the last equality is deduced from the definitions of ¯Φand t ∗ . We therefore conclude that (4.19) holds. Step 3: We show that for any ε > { f M t } T − t =1 can be constructed appropriately so that E Q [ ¯Φ] ≥ E Q [Φ + T ℓ ] − ε for all Q ∈ Π. For any Q ∈ Π, by (4.18), E Q [ ¯Φ] = E Q h (Φ + T ℓ )1 f M × ... × f M T − i + T − X t =2 E Q h (∆ · S ) t f M × ... × f M t − × f M ct i . (4.20)Note that E Q h (Φ + T ℓ ) (cid:16) − f M × ... × f M T − (cid:17)i ≤ ( C + T ℓ ) E Q h f M c ( x ) + 1 f M c ( x ) + ... + 1 f M cT − ( x T − ) i = ( C + T ℓ ) (cid:16) µ ( f M c ) + µ ( f M c ) + · · · + µ T − ( f M cT − ) (cid:17) . (4.21)On the other hand, for any t = 2 , ..., T − E Q [(∆ · S ) t f M × ... × f M t − × f M ct ( x , x , ..., x t )]= E Q h { ∆ ( x − x ) + ... + ∆ t − ( x t − x t − ) } f M ( x ) ... f M t − ( x t − )1 f M ci ( x t ) i ≥ − E Q h {| ∆ | ( x + x ) + · · · + | ∆ t − | ( x t + x t − ) } f M ( x ) ... f M t − ( x t − )1 f M ci ( x t ) i ≥ − E Q h { a ( m + m ) + ... + a t − ( x t + m t − ) } f M ( x ) ... f M t − ( x t − )1 f M ct ( x t ) i ≥ − E Q h { a ( m + m ) + ... + a t − ( x t + m t − ) } f M ct ( x t ) i = − [ a ( m + m ) + ... + a t − ( m t − + m t − ) + a t − m t − ] µ i ( f M ct ) − a t − Z f M ct ydµ t ( y ) , where the first inequality follows from x i ≥ y < m i for all y ∈ f M i and | ∆ i | ≤ a i on f M i , for all i = 1 , ..., T − 1. We then deduce from (4.20), (4.21), and the18revious inequality that E Q [ ¯Φ] ≥ E Q [Φ + T ℓ ] − ( C + T ℓ ) (cid:16) µ ( f M c ) + µ ( f M c ) + µ T − ( f M cT − ) (cid:17) − T − X t =2 (cid:18) [ a ( m + m ) + ... + a t − ( m t − + m t − ) + a t − m t − ] µ t ( f M ct )+ a t − Z f M ct ydµ t ( y ) (cid:19) . (4.22)The above inequality particularly requires the linearity of outer expectations, which holds here forΦ + T ℓ and (Φ + T ℓ )1 f M × ... × f M T − . This can be proved as in the discussion below (4.9). We willshow that every term on the right hand side of (4.22), except E Q [Φ + T ℓ ], can be made arbitrarilysmall, by choosing m t and a t appropriately for all t = 1 , ..., T − ε > 0, and define η := ε ( C + T ℓ )( T − T − > 0. Taking ε = η in Step 2 gives µ ( M c ) < η .Since µ ( f M ) ↑ µ ( M ) as m → ∞ , we can pick m > µ ( f M c ) < η . With m chosen, a ≥ m and a , we can take ε ∈ (0 , ε )small enough such that a m ε + a R A ydµ ( y ) < η for all A ∈ B ( R + ) with µ ( A ) < ε . Using this ε > µ ( M c ) < ε . Since µ ( f M ) ↑ µ ( M ) as m → ∞ , we can pick m > η , i.e. a m µ ( f M c ) + a Z f M c ydµ ( y ) < η. With m , m chosen, a ≥ m t and a t for t = 1 , ε ∈ (0 , ε ) small enough such that [ a ( m + m ) + a m ] ε + a R A ydµ ( y ) < η for all A ∈ B ( R + ) with µ ( A ) < ε . Using this ε > µ ( M c ) < ε . Since µ ( f M ) ↑ µ ( M )as m → ∞ , we can pick m > η , i.e.[ a ( m + m ) + a m ] µ ( f M c ) + a Z f M c ydµ ( y ) < η. By repeating the same argument for all t = 4 , ..., T − 1, we have µ t ( f M ct ), t = 1 , ..., T − 1, and everyterm in summation of (4.22) less than η . We then conclude from (4.22) that E Q [ ¯Φ] ≥ E Q [Φ + T ℓ ] − (( C + T ℓ )( T − 1) + ( T − η = E Q [Φ + T ℓ ] − ε. (4.23) Step 4: We establish D (Φ) ≥ D ∞ (Φ). 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