On Entropy Martingale Optimal Transport Theory

In this paper, we give an overview of (nonlinear) pricing-hedging duality and of its connection with the theory of entropy martingale optimal transport (EMOT), recently developed, and that of convex risk measures. Similarly to Doldi and Frittelli (2023), we here establish a duality result between a convex optimal transport and a utility maximization problem. Dif-ferently from Doldi and Frittelli (2023), we provide here an alternative proof that is based on a compactness assumption. Subhedging and superhedging can be obtained as applications of the duality discussed above. Furthermore, we provide a dual representation of the generalized Optimized Certainty Equivalent associated to indirect utility.


Introduction
Two topics widely exploited in Mathematical Finance are nonlinear option pricing and risk measures.Here below we give an overview of nonlinear pricing-hedging duality and of its connection with entropy martingale optimal transport (EMOT) and risk measures.We proceed step-by-step, starting from the classical pricing theory and its relation with coherent/convex risk measures via the subhedging pricing, going through the recent theory of model uncertainty and pathwise finance and its links with martingale optimal transport.At the end of this introduction, we then discuss the duality between the EMOT problem and subhedging, already established in [35] and here proved under stronger assumptions but with an alternative proof.

Risk measures and the pricing-hedging duality: the classical setup
The notion of subhedging price is one of the most analyzed concepts in financial mathematics.Although specular considerations can be done for the superhedging price, in this introduction we focus on the subhedging price.We are assuming a discrete time market model with zero interest rate.It may be convenient for the reader to have at hand the summary described in Table 1 on page 10.In the classical setup of stochastic securities market models, one considers an adapted stochastic process X = (X t ) t , t = 0, ..., T, defined on a filtered probability space (Ω, F, (F t ) t , P ), representing the price of some underlying asset.Let P(P ) be the set of all probability measures on Ω that are absolutely continuous with respect to P , Mart(Ω) be the set of all probability measures on Ω under which X is a martingale and M(P ) = P(P ) ∩ Mart(Ω).We also let H be the class of admissible integrands and I ∆ := I ∆ (X) be the stochastic integral of X with respect to ∆ ∈ H.Under reasonable assumptions on H, the equality holds for all Q ∈ M(P ) and, as well known, all linear pricing functionals compatible with no arbitrage are expectations E Q [•] under some probability Q ∈ M(P ) such that Q ∼ P .We denote with p the subhedging price of a contingent claim Z := c(X T ) written on the payoff X T of the underlying asset.If we let L(P ) ⊆ L 0 (Ω, F T , P ) be the space of random payoffs, then p : L(P ) → R is defined by The subhedging price is independent from the preferences of the agents, but it depends on the reference probability measure via the class of P -null events.It satisfies the following two key properties: (CA) Cash Additivity on L(P ): p(Z + k) = p(Z) + k, for all k ∈ R, Z ∈ L(P ).
When a functional p satisfies (CA), then Z, k and p(Z) must be expressed in the same monetary unit and this allows for the monetary interpretation of p, as the price of the contingent claim.This will be one of the key features that we will require also in the novel definition of the nonlinear subhedging value.The (IA) property and p(0) = 0 imply that the p price of any stochastic integral I ∆ (X) is equal to zero, as in (1).
Since the seminal works of El Karoui and Quenez [38], Karatzas [55], Delbaen and Schachermayer [33], it was discovered that, under the no arbitrage assumption, the dual representation of the subhedging price p is p(Z) = inf More or less in the same period, the concept of a coherent risk measure was introduced in the pioneering work by Artzner et al. [3].A Coherent Risk Measure ρ : L(P ) → R determines the minimal capital required to make acceptable a financial position and its dual formulation is assigned by −ρ(Y ) = inf where Y is a random variable representing future profit-and-loss and Q ⊆ P(P ).Coherent Risk Measures ρ are convex, cash additive, monotone and positively homogeneous.We take the liberty to label both the representations in (3) and in (4) as the "sublinear case".
In the study of incomplete markets the concept of the (buyer) indifference price p b , originally introduced by Hodges and Neuberger [53], received, in the early 2000, increasing consideration (see Frittelli [41], Rouge and El Karoui [63], Delbaen et al. [32], Bellini and Frittelli [9]) as a tool to assess, consistently with the no arbitrage principle, the value of non replicable contingent claims, and not just to determine an upper bound (the superhedging price) or a lower bound (the subhedging price) for the price of the claim.Differently from the notion of subhedging, p b is based on some concave increasing utility function u : R → [−∞, +∞) of the agent.By defining the indirect utility function U (w 0 ) := sup where w 0 ∈ R is the initial wealth, the indifference price p b is defined as Under suitable assumptions, the dual formulation of p b is and the penalty term α u : M(P ) → [0, +∞] is associated to the particular utility function u appearing in the definition of p b via the Fenchel conjugate of u.We observe that in case of the exponential utility function u(x) = 1 − exp(−x), the penalty is α exp (Q) := H(Q, P ) − min Q∈M(P ) H(Q, P ), where H(Q, P ) := F dQ dP dP , if Q ≪ P and F (y) = y ln(y), is the relative entropy.In this case, the penalty α exp is a divergence functional, similarly e.g. to those considered in (11).Observe that the functional p b is concave, monotone increasing and satisfies both properties (CA) and (IA), but it is not necessarily linear on the space of all contingent claims.As recalled in the conclusion of Frittelli [41], "there is no reason why a price functional defined on the whole space of bundles and consistent with no arbitrage should be linear also outside the space of marketed bundles".It was exactly the particular form (5) of the indifference price that suggested to Frittelli and Rosazza Gianin [42] to introduce the concept of Convex Risk Measure (also independently introduced by Follmer and Schied [39]), as a map ρ : L(P ) → R that is convex, cash additive and monotone decreasing.Under good continuity properties, the Fenchel Moreau Theorem shows that any convex risk measure admits the following representation for some penalty α : P(P ) → [0, +∞].We will then label functional in the form ( 5) or ( 6) as the "convex case".As a consequence of the cash additivity property, in the dual representations ( 5) or ( 6) the infimum is taken with respect to probability measures, namely with respect to normalized non negative elements in the dual space, which in this case can be taken as L 1 (P ).Differently from the indifference price p b , convex risk measures do not necessarily take into account the presence of the stochastic security market, as reflected by the absence of any reference to martingale measures in the dual formulation ( 6) and ( 4), in contrast to ( 5) and (3).

Pathwise finance
In the classical setting of nonlinear pricing recalled before, it was implicitly assumed that a reference probability P was fixed and known a priori.The financial crises in 2008, however, somehow inspired and motivated an increasing interest in the case where uncertainty in the selection of a reference probability occurs.The classical notions of arbitrage and pricing-hedging duality have been therefore investigated in this new framework.Two main approaches have been adopted to deal with uncertainty in P .One approach consisted in replacing the single reference probability P with a family of -a priori non dominated -probability measures, leading to the theory of Quasi-Sure Stochastic Analysis (see Bayraktar and Zhang [6], Bayraktar and Zhou [7], Bouchard and Nutz [16], Cohen [25], Denis and Martini [34], Peng [59], Soner et al. [69]).An alternative approach, even more radical, developed a probability free, pathwise, theory of financial markets, see Acciaio et al. [1], Burzoni et al. [20], Burzoni et al. [21], Burzoni et al. [19], Riedel [62].In such framework, Optimal Transport theory became a very powerful tool to prove pathwise pricing-hedging duality results with relevant contributions by many authors (Beiglböck et al. [8], Davis et al. [30], Dolinsky and Soner [36] and [37], Galichon et al. [44], Henry-Labordère [47], Henry-Labordère et al. [49]; Hou and Ob lój [54], Tan and Touzi [70], Bartl et al. [5], Cheridito et al. [23] and [24], Guo and Ob lój [46], Backhoff-Veraguas and Pammer [4], Neufeld and Sester [57], Sester [67] and [66]).These contributions mainly deal with what we labeled above as the sublinear case, while our main interest in this paper is to develop the convex case theory, as explained below.
From now on, we will abandon the classical setup described above and work without a reference probability measure.We consider a finite horizon T ∈ N, T ≥ 1, and for K 0 , . . ., K T subsets of R and assume that K 0 is a singleton, that is: K 0 = {x 0 }, x 0 ∈ R. We let X 0 , . . ., X T be the canonical projections X t : Ω → K t , for t = 0, 1, ..., T .We denote Mart(Ω) := {Martingale probability measures for the canonical process of Ω} , and when µ is a measure defined on the Borel σ-algebra of (K 0 × • • • × K T ), its marginals will be denoted with µ 0 , . . ., µ T .We consider a contingent claim c : Ω → (−∞, +∞] which is now allowed to depend on the whole path and, for hedging purposes, we will adopt semistatic trading strategies.In other words, in addition to dynamic trading in X via the admissible integrands ∆ ∈ H, we may invest in "vanilla" options φ t : K t → R. For modeling purposes we take vector subspaces E t ⊆ C b (K t ) for t = 0, . . ., T , where C b (K t ) is the space of real-valued, continuous, bounded functions on K t .For each t, E t represents the set of options to be used for hedging, say affine combinations of options with same maturity t but different strikes.The key assumption in the robust, Optimal Transport based, formulation is that the marginals ( Q 0 , Q 1 , ..., Q T ) of the underlying price process X are known.This assumption can be justified (see the seminal papers by Breeden and Litzenberger [17] and Hobson [52], as well as the many contributions by Hobson [50], Cox and Ob lój [26], [27], Cox and Wang [28], Labordère et al. [49], Brown et al. [18], Hobson and Klimmerk [51]) by assuming the knowledge of a sufficiently large number of plain vanilla options maturing at each intermediate date, implying then the possibility of calibration.Thus, represents the set of arbitrage-free pricing measures that are compatible with the observed prices of the options.In this framework, the set of admissible trading strategies and of the corresponding stochastic integrals are, respectively, given by and the sub-hedging duality, obtained in [8] Th. 1.1, takes the form: where the RHS of ( 9) is known as the robust subhedging price of c. Comparing (9) with the duality between (2) and (3), we observe that: (i) the P −a.s.inequality in (2) has been replaced by an inequality that holds for all x ∈ Ω; (ii) in (9) the infimum of the price of the contingent claim c is taken under all martingale measure compatible with the option prices, with no reference to the probability P ; (iii) in (9) static hedging with options is allowed.As can be seen from the LHS of (9), this case falls into the category labeled above as the sublinear case, and the purpose of this paper (as well as of [35]) is to investigate the convex case, in the robust setting, using the tools from Entropy Optimal Transport (EOT) recently developed in Liero et al. [56].Let us first describe the financial interpretation of the problems that we are going to study.

The dual problem
Differently from the pricing theory in finance where the problem inf Q∈M( Q0, Q1,... Q T ) E Q [c] in the LHS of ( 9) is a dual problem, in Martingale Optimal Transport (MOT) it represents the primal problem (called henceforth sublinear case of MOT).In [56], the primal Entropy Optimal Transport (EOT) problem takes the form inf µ∈Meas(Ω) where Meas(Ω) is the set of all positive finite measures µ on Ω, and D Ft, Qt (µ t ) is a divergence in the form: otherwise D Ft, Qt (µ t ) := +∞.We label with F := (F t ) t=0,...,T the family of divergence functions F t : R → R∪ {+∞} appearing in (11).Problem (10) represents the convex case of OT theory.Notice that in the EOT primal problem (10) the typical constraint that µ has prescribed marginals ( Q 0 , Q 1 , ... Q T ) has been relaxed thanks to the introduction of the divergence functional D Ft, Qt (µ t ), which penalizes those measures µ that are "far" from some reference marginals ( Q 0 , Q 1 , ... Q T ).We are then naturally let to the study of the convex case of MOT, i.e. to the Entropy Martingale Optimal Transport (EMOT) problem inf having also a clear financial interpretation.The marginals are not any more fixed a priori, to capture the fact that the available information might not be enough to detect them with satisfactory precision.So the infimum is taken over all martingale probability measures, but those that are far from some estimate ( Q 0 , Q 1 , ... Q T ) are appropriately penalized through D Ft, Qt .Of course, when D Ft, Qt (•) = δ Qt (•), we recover the sublinear MOT problem, where only martingale probability measures with fixed marginals are allowed.Observe that in addition to the martingale property, the elements Q ∈ Mart(Ω) in ( 12) are required to be probability measures, while in the EOT (10) theory all positive finite measure are allowed.As it was recalled after equation (6), this normalization feature of the dual elements (µ(Ω) = 1) is not surprising when one deals with dual problems of primal problems with a cash additive objective functional as, for example, in the theory of coherent and convex risk measures.Potentially, we could push our smoothing argument above even further: in place of the functionals D Ft, Qt (µ t ), t = 0, ..., T , we might as well consider more general marginal penalizations, not necessairly in the divergence form (11), yielding the problem These penalizations D 0 , . . ., D T will be better specified later.We point out that an additional entropic term has been added to optimal transport problems since the seminal work of Cuturi [29] (see also the survey/monograph Peyré and Cuturi [61]).On this topic, we also cite Nutz and Wiesel [58], Bernton et al. [12], Ghosal et al. [45], De March and Henry-Labordère [31] Henry-Labordère [48], Blanchet et al. [15].We also point out that in all the works stemming from Cuturi [29], the exact matching of the marginals is still required.
In this paper such constraint is absent to take into account uncertainty regarding the marginals themselves.

The primal problem: the nonlinear subhedging value
We provide the financial interpretation of the primal problem which will yield the EMOT problem in (12) as its dual.It is convenient to reformulate the robust subhedging price in the RHS of (9) in a more general setting.
Definition 1.1.Consider a measurable function c : Ω → R representing a (possibly path dependent) option, the set V of hedging instruments and a suitable pricing functional π : V → R. Then the robust Subhedging Value of c is defined by In the classical setting, functionals of this form (and even with a more general formulation) are known as general capital requirement, see for example Frittelli and Scandolo [43].We stress however that in Definition 1.1 the inequality v ≤ c holds for all elements in Ω with no reference to a probability measure whatsoever.The novelty in this definition is that a priori π may not be linear and it is crucial to understand which evaluating functional π we may use.For our discussion, we assume that the vector subspaces E t ⊆ C b (K t ) satisfies E t + R =E t , for t = 0, . . ., T .We let Suppose we took a linear pricing rule π : where we used (1) and the fact that Q t is the marginal of Q.In this case, we would trivially obtain for the robust subhedging value of c = sup where in the last equality we replaced φ t with (E Qt [φ t ] − φ t ) ∈ E t , which satisfies: Interpretation: Π π,V (c) is the supremum amount m ∈ R for which we may buy options φ t and dynamic strategies ∆ ∈ H such that m + T t=0 φ t + I ∆ ≤ c , where the value of both the options and the stochastic integrals are computed as the expectation under the same martingale measure ( Q for the integral I ∆ ; its marginals Q t for each option φ t ).
However, as mentioned above when presenting the indifferent price p b , there is a priori no reason why one has to allow only linear functional in the evaluation of v ∈ V. We thus generalize the expression for Π π,V (c) by considering valuation functionals S : V → R and S t : Nonetheless, in order to be able to repeat the same key steps we used in ( 15)-( 16) and therefore to keep the same interpretation, we shall impose that such functionals S and S t satisfy the property in ( 17) and the two properties (i) and (ii) in equation ( 14), that is: We immediately recognize that (a) is the Cash Additivity (CA) property on C b (K t ) of the functional S t and (b) implies the Integral Additivity (IA) property on V.As a consequence, repeating the same steps in ( 15)-( 16), we will obtain as primal problem the nonlinear subhedging value of c : to be compared with (16).
Interpretation: P(c) is the supremum amount m ∈ R for which we may buy zero value options φ t and dynamic strategies ∆ ∈ H such that m + T t=0 φ t + I ∆ ≤ c, where the value of both the options and the stochastic integrals are computed with the same functional S.
Stock Additivity Before further elaborating on these issues, let us introduce the concept of Stock Additivity, which is the natural counterpart of properties (IA) and (CA) when we are evaluating hedging instruments depending solely on the value of the underlying stock X at some fixed date t ∈ {0, . . ., T }.Let Id t be the identity function on K t We recall that the set of hedging instruments is denoted by E t ⊆ C b (K t ) and we will suppose that Id t ∈ E t (that is, we can use units of stock at time t for hedging) and that E t + R = E t (that is, deterministic amounts of cash can be used for hedging as well).
We now clarify the role of stock additive functionals in our setup.Suppose that S t : E t → R are stock additive on E t , t = 0, . . ., T .It can be shown (see Lemma A.7) that if there exist φ, ψ ∈ E 0 × ... × E T and ∆ ∈ H such that This allows us to define a functional S : Then S is a well defined, integral additive functional on V, and S, S 0 , . . ., S T satisfy the properties (a), (b), (c).There is a natural way to produce a variety of Stock Additive functionals, as explained in Example 1.3 below.
Example 1.3.Consider a Martingale measure Q ∈ Mart(Ω), a concave non decreasing utility function u t : R → [−∞, +∞), satisfying u t (0) = 0 and u t (x t ) ≤ x t ∀x t ∈ R, and define If we take S t (φ t ) = U Qt (φ t ) then, as shown in Lemma 4.2, the stock additivity property is satisfied for these functionals.Two relevant examples of S t = U Qt are those corresponding to linear or exponential utility functions (see Section 4.1).For a linear utility u t (x) = x, we get For the exponential utility u t (x) = 1 − e −x , we obtain where α ∈ R satisfies the martingale condition: for using the notation

and 4.5 for details).
When we consider stock additive functionals S 0 , . . ., S T that induce the functional S as explained in (20), we can focus our attention to the optimization problem ( 18) or ( 19), that will be referred to as our primal problem.We mention at this point that different formulations of nonlinear subhedging prices can be already found in the literature, see Föllmer and Schied [40], Cheridito et al. [24], Pennanen and Perkkiö [60].We refer to [35] Section 2.3 for further discussion of this related literature.

Entropy Martingale Optimal Transport Duality
It was proved in [35] Theorem 3.4 that under fairly general assumptions if In the particular case of S 0 , . . ., S T induced by utility functions, as explained in Example 1.3, this yields the duality inf The functions F t appearing in D Ft, Qt , defined in (11), are associated to the utility functions u t appearing in U Qt via the coniugacy relation: where v(y) := −u(−y).Thus, depending on which utility function u is selected in the primal problem in the RHS of (26) to evaluate the options through U Qt , the penalization term D Ft, Qt in (26) has a particular form induced by F t = v * t .In the special case of linear utility functions u t (x t ) = x t , we recover the sublinear MOT theory.Indeed, in this case, v * t (y) = +∞, for all y ̸ = 1 and v * t (1) = 0, so that and thus we obtain the robust pricing-hedging duality (9) of the classical MOT.While for the exponential utility u t (x) = 1 − e −x one can verify that D Ft, Qt (Q t ) = H(Q t , Q t ), (see (58)).
In this work, we focus on the case of compact underlying space which allows us to provide an alternative, simpler proof of the duality (25) .
To achieve this, we first need to present a more general setting (which will lead to the duality (28) below), and then we show how to recover ( 25) from (28).Following [35], we start by introducing two general functionals U and D U that are associated through a Fenchel Moreau type relation (see (31)).The functional U : E → [−∞, +∞) is defined on the vector space E ⊆ C b (Ω; R T +1 ) consisting of continuous and bounded functions defined on some Polish space Ω and with values in R T +1 , while D U : ca(Ω) → (−∞, +∞] with ca(Ω) being the set of all finite signed Borel measures on Ω.As better discussed later, we can think at E as the set of financial instruments that can be used for hedging, while at U as the evaluation functional of the hedging instruments.The map U is not necessarily cash additive.In order to turn U in a cash additive functional, we then rely on the notion of the Optimized Certainty Equivalent (OCE), that was introduced in Ben Tal and Teboulle [10] and further analyzed in Ben Tal and Teboulle [11].We introduce the Generalized Optimized Certainty Equivalent associated to U as the functional As it is easily recognized, any OCE is, except for the sign, a particular convex risk measure and so it is cash additive.The cash additivity S U (φ + ξ) = S U (φ) + T t=0 ξ t of the map S U will ensure that in the problem (10) the elements µ ∈Meas(Ω) are normalized, i.e. are probability measures.A family of examples of S U can be built by considering U (φ) = T t=0 U Qt (φ t ), with Q t being the marginal of Q ∈ Mart(Ω) and U Qt as in Example 1.3.In particular, for a linear utility function since U already satisfies cash additivity.In Theorem 2.4 below we then prove the following duality inf where Observe that sup which is equal to the RHS of (42).Referring to the class of examples above and for a fixed Q, we have: for a linear utility function Note that the aforementioned Theorem 2.4, which in principle could be considered as a corollary of [35] Theorem 2.4 (see [35] Corollary 2.5), is obtained by a different technique, which allows for avoiding much of the technicalities involved in the noncompact case.Indeed, we first show a Kantorovich-type duality result for a generalization of the EOT problem (10) (see Theorem 2.3).We then deduce (28) following closely [8] by means of a minimax argument.The duality (25) is then deduced in Section 3, Theorem 3.1.In Section 4.1 we provide an example of application, obtaining a nonlinear pricing-hedging duality for the stock-additive pricing functionals of the type in Example 1.3.Finally, in Section 4.2 we show the flexibility of our previous result beyond subhedging and superhedging dualities.Indeed, we prove a dual robust representation of the generalized Optimized Certainty Equivalent associated to the indirect utility function.
We summarize the preceding discussion in the following Table and we point out that in this paper (as well as in [35]) we develop the duality theory sketched in the last line of the Table and provide its financial interpretation.Differently from rows 1, 2, 5, 6, in rows 3, 4, 7, 8, the financial market is present and martingale measures are involved in the dual formulation.In rows 1, 2, 3, 4 we illustrate the classical setting, where the conditions in the functional form hold P -a.s., while in the last four rows Optimal Transport is applied to treat the robust versions, where the inequalities holds for all elements of Ω.
with given marginals}; Meas(Ω) is the set of all positive finite measures on Ω; Sub(c) is the set of static parts of semistatic subhedging strategios for c; U is a concave proper utility functional and S U is the associated generalized Optimized Certainty Equivalent.

A generalized optimal transport duality
The main duality in Theorem 2.4 is obtained applying a preparatory result stated in Theorem 2.3 that we now illustrate.
We introduce some notations used in the sequel.For unexplained concepts on Measure Theory we refer to the Appendix A.1.We let Ω be a Polish Space, B(Ω) its Borel sigma-algebra and define the following sets: We define D : ca(Ω) → (−∞, +∞] by D is a convex functional and is σ(ca(Ω), E)-lower semicontinuous, even if we do not require that U is σ(E, ca(Ω))-upper semicontinuous.
The following Assumption will hold throughout all the paper without further mention.
Remark 2.2.So far, we have introduced the functionals U and D by fixing U and then defining D by duality.As already discussed in [35], an alternative approach could be to do the converse.Let D : ca(Ω) → (−∞, +∞] be a proper convex functional that is σ(ca(Ω), E) -lower semicontinuous for some vector subspace E ⊆ C b (Ω, R M +1 ).Set now V the Fenchel-Moreau (convex) conjugate of D, i.e.
V (φ) := sup and By the Fenchel-Moreau theorem, it holds that D if of the form (31).
To this end, without loss of generality we can assume α := sup φ∈C inf µ∈Meas(Ω) L(µ, φ) < +∞ and we have to find φ ∈ C and C > α such that the sublevel set µ C := {µ ∈ Meas(Ω) | L(µ, φ) ≤ C} is weakly compact.The functional c is proper, lower continuous and has compact sublevel sets, hence it attains a minimum on Ω.Therefore, for any ε > 0 we can choose, by Assumption (34), a deterministic vector φ ∈ C having all components φ m equal to some constant −k n m < 0, such that φ ∈ dom(U ) and For such choice of φ and for a sufficiently big constant C > α there exists another constant Consequently, the set µ C is: 1. Nonempty, as the measure µ ≡ 0 belongs to µ C .φ m dµ is the pointwise supremum of narrowly lower semicontinuous functions, and is lower semicontinuous with respect to the narrow topology itself.We conclude that for each φ ∈ C the functional L( • , φ) is narrowly lower semicontinuous, and has closed sublevel sets.This implies that in particular µ C is narrowly closed, using the central expression in (37).
Observing that the sublevels of f are compact, by lower semicontinuity of c and compactness of its sublevel sets, we see that {f > α} are complementaries of compact subsets of Ω and can be taken with arbitrarily small measure, just by increasing α, uniformly in µ ∈ µ C .Thus tightness follows.

A subset of M.
These properties in turns yield narrow compactness of µ C in Meas(Ω), by Theorem A.6, and therefore σ(Meas(Ω), C b (Ω))-compactness (recalling that weak and narrow topology coincide in our setup).As a consequence, by Item 5 , µ C is compact in the relative topology σ(Meas(Ω), C b (Ω))| M .We now may apply Theorem A.9. Indeed, L is real valued on M × C. Items 1 and 2 of Theorem A.9 are fulfilled for: A = M endowed with the topology σ(Meas(Ω), C b (Ω))| M ; B = C; and C taken as above.We only justify explicitly lower semicontinuity σ(Meas(Ω), C b (Ω))| M for Item 1, which can be obtained arguing as in Item 4 above and observing that narrow topology and weak topology coincide in our setup (see Proposition A.4). Hence we may interchange sup and inf in RHS of (36), obtaining where the last equality follows from the fact that L(µ, φ) = +∞ on the complementary of M in Meas(Ω) for every φ ∈ C. It is now easy to check that for every φ ∈ C which concludes the proof, given (36) and (38).

The Entropy Martingale Optimal Transport Duality
In order to describe a suitable theory to develop the entropy optimal transport duality in a dynamic setting, in this section we will adopt a particular product structure of the set Ω.
To this end, in addition to the notations already introduced at the beginning of Section 2, we consider a finite horizon T ∈ N, T ≥ 1, and for K 0 , . . ., K T ⊆ R, with K 0 = {x 0 }, x 0 ∈ R. We denote with X 0 , . . ., X T the canonical projections X t : Ω → K t and we set X = [X 0 , . . ., X T ] : Ω → R T +1 , to be considered as discrete-time stochastic process X representing the price of an underlying asset.We denote with: Mart(Ω) := {Martingale probability measures for the canonical process of Ω}. ( 40) , its marginals will be denoted with: µ 0 , . . ., µ T .We recall, respectively from ( 7) and ( 8), that H is the set of admissible trading strategies and I is the set of elementary stochastic integral.We take is a vector subspace, for every t = 0, . . ., T .Then E is clearly a vector subspace of C b (Ω; R T +1 ), and in the stochastic processes interpretation its elements are processes adapted to the natural filtration of the process X.
We suppose that U : E → [−∞, +∞) is proper and concave, D : Meas(Ω) → (−∞, +∞] is defined in (31) and, as in (27), The following result establishes the duality in (42).The result here presented can be obtained as a corollary of Theorem 2.4 of Doldi and Frittelli [35] where, differently from below, there is no assumption on the compactness of the sets K 0 , . . ., K T .We provide, however, a proof that is different from [35] and simpler, yet holding under the compactness assumption.
Theorem 2.4.Assume that and proper.Suppose also U satisfies (34), and that Then the following holds: where for each ∆ ∈ H Proof.The first part of the proof in inspired by [8] = inf = inf = inf = inf The equality chain above is justified as follows: (44)=( 45) is trivial; (45)=( 46) follows using the same argument as in [8] Lemma 2.3, which yields that the inner supremum explodes to +∞ unless Q is a martingale measure on Ω; (46)=( 47) and ( 47 From (41), we observe that K is real valued on N × (H × R T +1 ) and that K(µ, ∆, λ) = +∞ if µ ∈ Meas(Ω) \ N , for all (∆, λ) ∈ H × R T +1 .This, together with our previous computations, provides inf As in the proof of Theorem 2.3, we wish to apply the Minimax Theorem A.9 in order to interchange inf and sup in RHS of ( 51) and without loss of generality we can assume that α := sup ∆∈H λ∈R T +1 inf µ∈N K(µ, ∆, λ) < +∞.The functional K is real valued on N ×(H×R T +1 ) and convexity in Item 1, concavity in 2 of Theorem A.9 are clearly satisfied.We have to find ∆ ∈ H, λ ∈ R T +1 and C > α such that the sublevel set As the functional c is lower semicontinuous on the compact Ω, it is lower bounded on Ω and we can take ∆ = 0 and λ sufficiently big in such a way that inf x∈Ω (c(x) + T t=0 λ t ) > ε.For such a choice of (∆, λ) we have that M C satisfies | N -lower semicontinuous, while D is by definition σ(ca(Ω), E)| N lower semicontinuous (being supremum of linear functionals each continuous in such a topology).Since sum of lower semicontinuous functions is lower semicontinuous, the desired lower semicontinuity of K(•, ∆, λ) follows.All the hypotheses of Theorem A.9 are now verified, and we may then interchange sup and inf in RHS of ( 51) and obtain where in (⋆) we used the fact that K(µ, ∆, λ) = +∞ on the complementary of N in Meas(Ω), for every (∆, λ) ∈ H × R T +1 .We apply now Theorem 2.3 to the inner infimum with the cost functional c − I ∆ + T t=0 λ t , observing that, since we are assuming dom(U ) + R T +1 = dom(U ) (see (41)), the condition ( 34) is satisfied.We get that where Φ ∆,λ (c), which depends on ∆, λ ∈ H × R T +1 , is defined according to (35) by From ( 41), (φ t − λ t ) t ∈ dom(U ) and we can absorb λ in φ obtaining Φ ∆,λ (c) = Φ ∆ (c) + λ, ∀ λ ∈ R T +1 , ∆ ∈ H , with Φ ∆ (c) given in (43), so that We now recognize the expression in (27) and we conclude that inf and consequently, recalling our minimax argument, inf Q∈M art Eq.( 51) Remark 2.5.(i) The assumptions of Theorem 2.4 are reasonably weak and are satisfied, for example, if: dom(U ) = E, there exists a µ ∈ Meas(Ω) ∩ ∂U (0) such that c ∈ L 1 ( µ), and c is lower semicontinuous.Indeed, for all µ ∈ Meas(Ω), (ii) The step (45)=( 46) is the crucial point where compactness of the sets K 0 , . . ., K T ⊆ R is necessary for a smooth argument, since integrability of the underlying stock process is in this case automatically satisfied for all Q ∈ Prob(Ω), not only for Q ∈ Mart(Ω).Also, compactness is key in guaranteeing that the cost functional c − I ∆ + t λ t is bounded from below, in order to apply Theorem 2.3.
The following result guarantees that, under suitable assumptions, the infimum in ( 42) is attained.A similar result can be also found in Corollary 2.5 of Doldi and Frittelli [35].

Duality in an additive setting
Differently from Section 2, we will now assume an additive structure of U and D. In the whole Section 3 we consider for each t = 0, . . ., T a vector subspace Note that this automatically implies that E + R T +1 = E. Furthermore, E can be seen as a subspace of C b (Ω, R T +1 ) once E 0 , . . ., E T can be interpreted as subspaces of C b (Ω).
The following result provides the form of the penalization in an additive setup and the duality P(c) = D(c).
Theorem 3.1.Suppose for each t = 0, . . ., T E t ⊆ C b (K t ) is a vector subspace satisfying Id t ∈ E t and E t + R = E t and that S t : E t → R is a concave, cash additive functional null in 0. Consider for every t = 0, . . ., T the penalizations and set be lower semicontinuous and let D(c) and P(c) be defined respectively in (13) and (18).
), for φ ∈ E, and let D defined as in (31) for M = T .For any µ ∈ Meas(Ω) we have D(µ) ≥ T t=0 S t (0) − 0 = 0 hence D is lower bounded on Meas(Ω).Observe that dom(U ) = E, which implies dom(U ) + R T +1 = dom(U ), and that we are in Setup 3.2 of [35].By [35] T t=0 S t (φ t ), since S 0 , . . ., S T are Cash Additive, and D coincides on Mart(Ω) with the penalization term Q → T t=0 D t (Q t ), as provided in the statement of this Theorem.Since all the assumptions of Theorem 2.4 are fulfilled, we can apply Corollary 2.7, which yields exactly D(c) = P(c).Assumption 3.2.We consider concave, upper semicontinuous nondecreasing functions u 0 , . . ., u T : R → For each t = 0, . . ., T we define v t (x) := −u t (−x), x ∈ R, and its convex conjugate By Fenchel-Moreau Theorem, we recall that v t (y) = v * * t (y) = sup x∈R (xy − v * t (y)) for all y ∈ R and that v * t is convex, lower semicontinuous and lower bounded on R. Example 3.3.Assumption 3.2 is satisfied by a wide range of functions.Just to mention a few with various peculiar features, we might take u t of the following forms: Proof.Since V Qt (φ t ) = −U Qt (−φ t ), w.l.o.g.we prove the claims only for U Qt .Clearly U Qt (φ t ) > −∞, as we may choose λ t ∈ R so that (φ t + 0Id t + λ t ) ∈ dom(u) ⊇ [0, +∞).Furthermore, Item 2: trivial from the definitions.Item 3: we see that in which we recognize the definition of U Qt (φ t ) + α t x 0 + λ t .
As in [8], in the next two Corollaries we suppose that the elements in E t represent portfolios obtained combining deterministic amounts, units of the underlying stock at time t (x t ), and call options with maturity t, that is E t consists of all the functions in C b (K t ) with the following form: As shown in the proof, one could as well take preserving validity of ( 61), ( 62), ( 67) and (68).As for Lemma 4.2, the following result extends Corollary 4.3 [35] to cover the stock additive case: indeed, the definition of U Qt (φ t ) used in the next Corollary is different from the one in Corollary 4.3 [35].inf sup Proof.We prove (61), since (62) can be obtained in a similar fashion.Set U (φ) = T t=0 U Qt (φ t ) for φ ∈ E. We observe that E t consists of all piecewise linear functions on K t , which are norm dense in C b (K t ).By Lemma 4.2 for each t = 0, . . ., T the monotone concave functional φ t → U Qt (φ t ) is actually well defined, finite valued, concave and nondecreasing on the whole C b (K t ).Hence, by the Extended Namioka-Klee Theorem (see [13]) it is norm continuous on C b (K t ) and we can take (61) and prove equality to LHS in this more comfortable case (notice that S sub (c) depends on E).We also observe that in this case we are in Setup of Section 3. Define D as in (31) with M = T .Using the facts that if φ t ∈ E t , α, λ ∈ R then (φ t + αId t + λ) ∈ E t , that Q ∈Mart(Ω) and that v t (•) := −u t (−•) one may easily check that where the last equality follows from Proposition 3.4 Equation ( 59).The Standing Assumption 2.1 is satisfied.Indeed, from Assumption 3.2 we have v * 0 (1), . . ., v * T (1) < +∞, hence | Ω c dµ < +∞ .Moreover, by Lemma 4.2 Item 1, dom(U ) = E, and for every µ ∈ Meas(Ω) D(µ) ≥ U (0) − 0 = 0, hence D is lower bounded on the whole Meas(Ω).We conclude that U and D satisfy the assumptions of Theorem 2.4.Using Lemma 3.3 of Doldi and Frittelli [35] and the fact that U Q0 , . . ., U Q T are cash additive we get S U (φ) = T t=0 S U Q t (φ t ) = T t=0 U Qt (φ t ) = U (φ), and by Corollary 2.7 Equation ( 55) we obtain inf We stress the fact that in Corollary 4.3 we assume that all the functions u 0 , . . ., u T are real valued on the whole R.
Proposition 4.4.The following dual representations hold: where, with a slight abuse of notation and consistently with (40), Mart(K Proof.We prove the dual representation for V Qt as the other one follows by a change of signs.By Extended Namioka Klee Theorem [13], together with the compactness of the underlying canonical space we have that where the conjugate (V Qt ) * (γ) is given as usual by (V Qt ) * (γ) = sup ψt∈C b (Kt) Kt ψ t dγ − V Qt (ψ t ) .
We show that (V Qt ) * (γ) < +∞ implies γ ∈ Mart(K 0 × K t ).By standard monotonicity and cash additivity arguments (see e.g.[40] Remark 4.18) it can be seen that (V Qt ) * (γ) implies that γ is a non-negative normalized element of the dual space (C b (K t )) * .Since K t is compact, γ is then identified with an element of Prob(K t ).We show that the martingale property must hold: for any Now, the last term in RHS is finite if and only if Kt Id t dγ = x 0 .To conclude, observe that arguing as we did to obtain (63), we can also show here that for any Q ∈ Mart(K 0 × K t ) we have by Proposition 3.4 Equation ( 59).Proof.One can verify that (58)).Hence, from (64), The first order conditions for U Qt (φ t ) := sup α,λ f (α, λ) are: Assuming there exists α ∈ R satisfying (23), and taking λ = log E(exp(−φ t − αId t )) we easily get that ( 65) and ( 66) are satisfied, and we compute From the definition of dQ α d Qt in (24) we obtain: Since U Qt (φ t ) = f ( α, λ) ∈ R, we conclude: .
We are only left with proving that such an α ∈ R exists.Since x 0 and we must have is continuous, and the existence of a solution α for (23) follows.
We now take u t (x) = x for each t = 0, . . ., T , and get Hence with an easy computation we have

Beyond uniperiodal semistatic hedging
We now explore the versatility of Corollary 2.7, which can be used beyond the semistatic subhedging and superhedging problems in Section 4.1.Note that in Section 4.1 we chose for static hedging portfolios the sets E t , t = 0, . . ., T consisting of deterministic amounts, units of underlying stock at time t and call options with the same maturity t but different strike prices.This affected the primal problem in the fact that the penalty D turned out to depend solely on the (one dimensional) marginals of Q. Nonetheless, Theorem 2.4 allows to choose for each t = 0, . . ., T a subspace , potentially allowing to consider also Asian and path dependent options in the sets E t .We expect that this would translate in the penalty D depending no more only on the one dimesnional marginals of Q.The study of these less restrictive, yet technically more complex cases is left for future research.
In the following we will treat a slightly different problem, which however helps understanding how also the extreme case

Dual representation for generalized OCE associated to the indirect utility function
Theorem 2.4 yields the following dual robust representation of the generalized Optimized Certainty Equivalent associated to the indirect utility function.We stress here the fact that, again, Q ∈ Mart(Ω) is a fixed martingale measure, but we will not focus anymore on its marginals only, as will become clear in the following.and S U be the associated Optimized Certainty Equivalent defined according to (27), namely Then for every c ∈ C b (Ω) where for µ ∈ Meas(Ω) x ∈ R we also have U (ψ) ≤ T t=0 ∥φ t ∥ ∞ < +∞.Moreover, it is easy to verify that defining D as in (31) for any Q ∈ Mart(Ω) we have and arguing as in Proposition 3.4 we get D(Q) = D Q (Q).From the fact that u(x) ≤ x for every x ∈ R we have v * (1) < +∞, hence from Assumption 4.1 Q ∈ dom(D).This and c ∈ L 1 ( Q) in turns yields Q ∈ N (see (41)).Moreover dom(U ) = E and by definition of D for any µ ∈ Meas(Ω) we have D(µ) ≥ U (0) − 0 = 0, hence D is lower bounded on the whole Meas(Ω).We conclude that U and D satisfy the assumptions of Theorem 2.4.We then get inf Observe now that S U satisfies

A Appendix
A.1 Setting

A.1.1 Measures
We start fixing our setup and some notation.Let Ω be a Polish space and endow it with the Borel sigma algebra B(Ω) generated by its open sets.A set function µ : B(Ω) → R is a finite signed measure if µ(∅) = 0 and µ is σ-additive.A finite measure µ is a finite signed measure such that µ(B) ≥ 0 for all B ∈ B(Ω).A finite measure µ such that µ(Ω) = 1 will be called a probability measure.Recall from Section 2 the notations for ca(Ω), Meas(Ω), Prob(Ω).The following result is well known, see e.g.[14] Theorem 1.1 and 1.3.
Proposition A.1.Every finite measure µ on B(Ω) is a Radon Measure, that is for every B ∈ B(Ω) and every ε > 0 there exists a compact Proposition A.4.When Ω is a Polish Space, the weak and narrow topologies coincide.
Remark A.5.Even though the two topologies coincide in our setting, because of their different definitions we will find more convenient to exploit the one or the other topology in our proofs.

A.2 Auxiliary results and proofs
Lemma A.7.Take compact K 1 , . . ., K T ⊆ R, and suppose that K 0 = {x 0 } and card(K t+1 ) ≥ card(K t ) for every t = 0, . . ., T − 1.Take E = E 0 × • • • × E T for vector subspaces E t ⊆ C b (K t ) such that Id t ∈ E t and E t + R = E t , for t = 0, . . ., T .Suppose there exist φ, ψ ∈ E and ∆ ∈ H, where H is defined in (7), such that T t=0 φ t = T t=0 ψ t + I ∆ .Then there exist constants k 0 , . . ., k T , h 0 , . . ., h T ∈ R such that for each t = 0, . . ., T ψ t (x t ) = φ t (x t ) + k t x t + h t , ∀x t ∈ K t .In particular for S t : E t → R, t = 0, . . ., T Stock Additive functionals we have ∆ t (x 0 , . . ., x t )(x t+1 − x t ) + ∆ T −1 (x 0 , . . ., x T −1 )(x T − x T −1 ) = f (x 0 , . . ., x T −1 ) + ∆ T −1 (x 0 , . . ., x T −1 )x T for some function f .If ∆ T −1 were not constant, on two points it would assume values a ̸ = b, with corresponding values of f that we call f a , f b .Then f a + ax T = f b + bx T has a unique solution, contradicting the fact that all the equalities need to hold on the whole K 0 , . . ., K T and in particular for two different values of x T .We proceed one step backward.If card(K T −1 ) = 1, the claim trivially follows, given our previous step.If card(K T −1 ) ≥ 2, similarly to the previous computation An argument similar to the one we used in the previous time step shows that ∆ T −2 (x 0 , . . ., x T −2 )− ∆ T −1 is constant, hence so is ∆ T −2 .Our argument can be clearly be iterated up to ∆ 0 .STEP 2: we prove existence of the vectors k, h ∈ R T +1 , as stated in the Lemma.From Step 1 it is clear that there exist constants k 0 , . . ., k T such that I ∆ (x) = T t=0 k t x t .Hence T t=0 φ t (x t ) = T t=0 (ψ t (x t ) + k t x t ) for all x ∈ Ω, which yields for each t = 0, . . ., T that φ t (x t ) − (ψ t (x t ) + k t x t ) does not depend on x t , hence is constant, call it −h t .Then k 0 , . . ., k T , h 0 , . . ., h T ∈ R satisfy our requirements.The last claim T t=0 S t (φ t ) = T t=0 S t (ψ t ) is then an easy consequence of stock additivity.STEP 3: well posedness and properties of S. Observe that whenever φ, ψ ∈ E, ∆, H ∈ H are given with T t=0 φ t + I ∆ = T t=0 ψ t + I H we have by Steps 1-2 that T t=0 S t (φ t ) = T t=0 S t (ψ t ) .As a consequence, S is well defined.Cash Additivity is inherited from S 0 , . . ., S T while Integral Additivity is trivial from the definition.

2 .
Narrowly closed.Indeed, for each φ ∈ C the function c − φ is lower semicontinuous on Ω, and so it is the pointwise supremum of bounded continuous functions (c n ) n ⊆ C b (Ω).For each n, µ → Ω c n dµ is narrowly lower semicontinuous on Meas(Ω), by definition.Hence by Monotone Convergence Theorem the map µ → Ω c − M m=0

D
(µ) =: D where D ∈ R since D(•) is lower bounded by hypothesis.By (41) and for large enough C, the set M C is nonempty, and the same arguments in Items 2, 3 and 4 of the proof of Theorem 2.3 can be applied to conclude that the set M C is narrowly closed, bounded and tight, hence narrowly and σ(Meas(Ω), C b (Ω))-compact.Moreover we see that M C ⊆ N , hence it is also compact in the topology σ(Meas(Ω), C b (Ω))| N .We finally verify σ(Meas(Ω), C b (Ω))| N -lower semicontinuity of K(•, ∆, λ) on N for every (∆, λ) ∈ (H × R T +1 ).To see this, observe that arguing as in Item 2 of the proof of Theorem 2.3 we get that µ → Ω c − I ∆ + T t=0 λ t dµ − T t=0 λ t is σ(Meas(Ω), C b (Ω))

Table 1 :
Π(Ω) is the set of all probabilities on Ω; P then we must have Q t = δ {x0} (the latter being a Diract delta at the point x 0 ) by the martingale property and a solution to (23) exists trivially.Otherwise, assume x 0 ∈ (ess inf Q (Id t ), ess sup Q (Id t )).By Lemma A.8 we have that lim α→+∞