Entropy Martingale Optimal Transport and Nonlinear Pricing-Hedging Duality
aa r X i v : . [ q -f i n . M F ] J u l Entropy Martingale Optimal Transport andNonlinear Pricing-Hedging Duality
Alessandro Doldi ∗ Marco Frittelli † July 30, 2020
Abstract
The objective of this paper is to develop a duality between a novel Entropy Martingale Op-timal Transport problem ( A ) and an associated optimization problem ( B ). In ( A ) we followthe approach taken in the Entropy Optimal Transport (EOT) primal problem by Liero et al.“Optimal entropy-transport problems and a new Hellinger-Kantorovic distance between posi-tive measures”, Invent. math. 2018, but we add the constraint, typical of Martingale OptimalTransport (MOT) theory, that the infimum of the cost functional is taken over martingaleprobability measures, instead of finite positive measures, as in Liero et al. The Problem ( A )differs from the corresponding problem in Liero et al. not only by the martingale constraint,but also because we admit less restrictive relaxation terms D , which may not have a diver-gence formulation. In Problem (B) the objective functional, associated via Fenchel coniugacyto the terms D , is not any more linear, as in OT or in MOT. This leads to a novel optimiza-tion problem which also has a clear financial interpretation as a non linear subhedging value.Our results allow us to establish a novel nonlinear robust pricing-hedging duality in financialmathematics, which covers a wide range of known robust results in its generality. Keywords : Martingale Optimal Transport problem, Entropy Optimal Transport problem, Pricing-hedging duality, Robust finance, Pathwise finance.
Mathematics Subject Classification (2020) :49Q25, 49J45, 60G46, 91G80, 90C46
JEL Classification : C61, G13
In this research we exploit optimal transport theory to develop the duality A := inf Q ∈ Mart(Ω) ( E Q [ c ] + D U ( Q )) = sup ∆ ∈H sup ϕ ∈ Φ ∆ ( c ) S U ( ϕ ) := B. (1)In ( A ) we recognize the approach taken in the Entropy Optimal Transport primal problem (Lieroet al. [42]) with the additional constraints, typical of Martingale Optimal Transport (MOT), ∗ Dipartimento di Matematica, Universit`a degli Studi di Milano, Via Saldini 50, 20133 Milano, Italy, [email protected] . † Dipartimento di Matematica, Universit`a degli Studi di Milano, Via Saldini 50, 20133 Milano, Italy, [email protected] . c is taken over martingale probability measures, insteadof finite positive measures , as in [42]. This is a consequence of the additional supremum over theintegrands ∆ ∈ H in problem ( B ), and of the cash additivity of the functional S U . The functional S U is associated to a, typically non linear, utility functional U and represents the pricing ruleover continuous bounded functions ϕ defined on Ω. We observe that the marginal constraints,typical of Optimal Transport (OT) problems, in ( A ) are relaxed by introducing the functional D U ,also associated to the map U , which may have a divergence formulation. The counterpart of thisin Problem ( B ) is that the functional S U , associated via Fenchel coniugacy to the penalizationfunctional D U is not necessarily linear , as in OT or in MOT. Both S U and D U may also dependon some martingale measure b Q which in the classical OT and MOT theories, plays the role offixing the marginals. The duality (1) generalizes the well known robust pricing hedging duality infinancial mathematics.We provide a clear financial interpretation of both problems and observe that the novel concept ofa non linear subhedging value expressed by ( B ) was not previously considered in the literature. The notion of subhedging price is one of the most analyzed concepts in financial mathematics. Inthis introduction we will take the point of view of the subhedging price, but obviously an analogoustheory for the superhedging price can be developed as well. We are assuming a discrete time marketmodel with zero interest rate. It may be convenient for the reader to have at hand the summarydescribed in Table 1 on page 11.
The classical setup
In the classical setup of stochastic securities market models, one consid-ers 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 prob-ability measures on Ω that are absolutely continuous with respect to P , Mart(Ω) be the set of allprobability measures on Ω under which X is a martingale and M ( P ) = P ( P ) ∩ Mart(Ω) . We alsolet H be the class of admissible integrands and I ∆ := I ∆ ( X ) be the stochastic integral of X withrespect to ∆ ∈ H . Under reasonable assumptions on H , the equality E Q (cid:2) I ∆ ( X ) (cid:3) = 0 (2)holds for all Q ∈ M ( P ) and, as well known, all linear pricing functionals compatible with noarbitrage 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 c : R → R written on the payoff X T of the underlying asset. If we let L ( P ) ⊆ L ((Ω , F T , P )) be the space of random payoff andlet Z := c ( X T ) ∈ L ( P ), then p : L ( P ) → R is defined by p ( Z ) := sup (cid:8) m ∈ R | ∃ ∆ ∈ H s.t. m + I ∆ ( X ) ≤ Z , P − a.s. (cid:9) . (3)The subhedging price is independent from the preferences of the agents, but it depends on thereference probability measure via the class of P -null events. It satisfies the following two keyproperties: 2CA) Cash Additivity on L ( P ): p ( Z + k ) = p ( Z ) + k, for all k ∈ R , Z ∈ L ( P ) . (IA) Integral Additivity on L ( P ): p ( Z + I ∆ ) = p ( Z ) , for all ∆ ∈ H , Z ∈ L ( P ) . When a functional p satisfies (CA), then Z, k and p ( Z ) must be expressed in the same monetaryunit and this allows for the monetary interpretation of p , as the price of the contingent claim. Thiswill be one of the key features that we will require also in the novel definition of the nonlinearsubhedging 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 (2).Since the seminal works of El Karoui and Quenez [28], Karatzas [41], Delbaen and Schachermayer[24], it was discovered that, under the no arbitrage assumption, the dual representation of thesubhedging price p is p ( Z ) = inf Q ∈M ( P ) E Q [ Z ] . (4)More or less in the same period, the concept of coherent risk measure was introduced in thepioneering work by Artzner et al. [3]. A Coherent Risk Measure ρ : L ( P ) → R determinesthe minimal capital required to make acceptable a financial position and its dual formulation isassigned by − ρ ( Y ) = inf Q ∈Q⊆P ( P ) E Q [ Y ] , (5)where Y is a random variable representing future profit-and-loss and Q ⊆ P ( P ). Coherent RiskMeasures ρ are convex, cash additive, monotone and positively homogeneous. We take the libertyto label both the representations in (4) and in (5) as the “ sublinear case ”.In the study of incomplete markets the concept of the (buyer) indifference price p b , originallyintroduced by Hodges and Neuberger [39], received, in the early 2000, increasing consideration(see Frittelli [30], Rouge and El Karoui [45], Delbaen et al. [23], Bellini and Frittelli [7]) as atool to assess, consistently with the no arbitrage principle , the value of non replicable contingentclaims, and not just to determine an upper bound (the superhedging price) or a lower bound (thesubhedging price) for the price of the claim. Differently from the notion of subhedging, p b is basedon some concave increasing utility function u : R → [ −∞ , + ∞ ) of the agent. By defining theindirect utility function U ( w ) := sup ∆ ∈H E P [ u ( w + I ∆ ( X ))] , where w ∈ R is the initial wealth, the indifference price p b is defined as p b ( Z ) := sup { m ∈ R | U ( Z − m ) ≥ U (0) } . Under suitable assumptions, the dual formulation of p b is p b ( Z ) = inf Q ∈M ( P ) { E Q [ Z ] + α u ( Q ) } , (6)and the penalty term α u : M ( P ) → [0 , + ∞ ] is associated to the particular utility function u appear-ing in the definition of p b via the Fenchel conjugate of u . We observe that in case of the exponentialutility 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 ) := Z F (cid:18) d Q d P (cid:19) d P , if Q ≪ P and F ( y ) = y ln( y ) ,
3s the relative entropy. In this case, the penalty α exp is a divergence functional, as those that willbe considered below in Section 3.4. Observe that the functional p b is concave, monotone increasingand satisfies both properties (CA) and (IA), but it is not necessarily linear on the space of allcontingent claims. As recalled in the conclusion of Frittelli [30], “there is no reason why a pricefunctional defined on the whole space of bundles and consistent with no arbitrage should be linearalso outside the space of marketed bundles”.It was exactly the particular form (6) of the indifference price that suggested to Frittelli andRosazza Gianin [31] to introduce the concept of Convex Risk Measure (also independentlyintroduced by Follmer and Schied [29]), as a map ρ : L ( P ) → R that is convex, cash additive andmonotone decreasing. Under good continuity properties, the Fenchel Moreau Theorem shows thatany convex risk measure admits the following representation − ρ ( Y ) = inf Q ∈P ( P ) { E Q [ Y ] + α ( Q ) } (7)for some penalty α : P ( P ) → [0 , + ∞ ]. We will then label functional in the form (6) or (7) as the“ convex case ”. As a consequence of the cash additivity property, in the dual representations (6) or(7) the infimum is taken with respect to probability measures , namely with respect to normalizednon negative elements in the dual space, which in this case can be taken as L ( P ). Differently fromthe indifference price p b , convex risk measures do not necessarily take into account the presence ofthe stochastic security market, as reflected by the absence of any reference to martingale measuresin the dual formulation (7) and (5), in contrast to (6) and (4). Pathwise finance
As a consequence of the financial crises in 2008, the uncertainty in the se-lection of a reference probability P gained increasing attention and led to the investigation of thenotions of arbitrage and the pricing hedging duality in different settings. On the one hand, thesingle reference probability P was replaced with a family of - a priori non dominated - probabilitymeasures, leading to the theory of Quasi-Sure Stochastic Analysis (see Bayraktar and Zhang [4],Bayraktarand Zhou [5], Bouchard and Nutz [12], Cohen [18], Denis and Martini [25], Peng [43],Soner et al.[48]). On the other hand, taking a even more radical approach, a probability free,pathwise, theory of financial markets was developed, as in Acciaio et al. [1], Burzoni et al. [16],Burzoni et al. [17], Burzoni et al. [15], Riedel [44]. In such framework, Optimal Transport theorybecame a very powerful tool to prove pathwise pricing-hedging duality results with very relevantcontributions by many authors (Beiglb¨ock et al. [6], Davis et al. [22], Dolinksi and Soner [26],Dolinsky and Soner [27]; Galichon et al. [33], Henry-Labord`ere [34], Henry-Labord`ere et al. [35];Hou and Ob l´oj [40], Tan and Touzi [49]). These contributions mainly deal with what we labeledabove as the sublinear case, while our main interest in this paper is to develop the convex casetheory, as explained below.From now on, we will abandon the classical setup described above and work without a referenceprobability measure. We consider T ∈ N , T ≥
1, andΩ := K × · · · × K T for K , . . . , K T subsets of R and denote with X , . . . , X T the canonical projections X t : Ω → K t ,4or t = 0 , , ..., T . We denoteMart(Ω) := { Martingale probability measures for the canonical process of Ω } , and when µ is a measure defined on the Borel σ -algebra of ( K × · · · × K T ), its marginals willbe denoted with µ , . . . , µ T . We consider a contingent claim c : Ω → ( −∞ , + ∞ ] which is nowallowed to depend on the whole path and we will adopt semistatic trading strategies for hedging.This means that in addition to dynamic trading in X via the admissible integrands ∆ ∈ H , wemay 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, boundedfunctions on K t . For each t, E t is the set of static options that can be used for hedging, sayaffine combinations of options with different strikes and same maturity t . The key assumptionin the robust, Optimal Transport based, formulation is that the marginals ( b Q , b Q , ..., b Q T ) of theunderlying price process X are known. This assumption can be justified (see the seminal papers byBreeden and Litzenberger [13] and Hobson [38], as well as the many contributions by Hobson [36],Cox and Ob l´oj [19], [20], Cox and Wang [21], Labord`ere et al. [35], Brown et al. [14], Hobson andKlimmerk [37]) by assuming the knowledge of a sufficiently large number of plain vanilla optionsmaturing at each intermediate date.Thus the class of arbitrage-free pricing measures that are compatible with the observed prices ofthe options is given by M ( b Q , b Q , ... b Q T ) := n Q ∈ Mart(Ω) | X t ∼ Q b Q t for each t = 0 , . . . , T o . In this framework, H := { ∆ = [∆ , . . . , ∆ T − ] | ∆ t ∈ C b ( K × · · · × K t ; R ) } (8) I := ( I ∆ ( x ) = T − X t =0 ∆ t ( x , . . . , x t )( x t +1 − x t ) | ∆ ∈ H ) (9)and the sub-hedging duality, obtained in [6] Th. 1.1, takes the form:inf Q ∈M ( b Q , b Q ,... b Q T ) E Q [ c ] = sup ( T X t =0 E b Q t [ ϕ t ] | ∃ ∆ ∈ H s.t. T X t =0 ϕ t ( x t ) + I ∆ ( x ) ≤ c ( x ) ∀ x ∈ Ω ) , (10)where the RHS of (10) is known as the robust subhedging price of c . Comparing (10) with theduality between (3) and (4), we observe that: (i) the P − a.s. inequality in (3) has been replacedby an inequality that holds for all x ∈ Ω; (ii) in (10) the infimum of the price of the contingentclaim c is taken under all martingale measure compatible with the option prices, with no referenceto the probability P ; (iii) static hedging with options is allowed.As can be seen from the LHS of (10), this case falls into the category labeled above as the sublinearcase , and the purpose of this paper is to investigate the convex case , in the robust setting, usingthe tools from Entropy Optimal Transport (EOT) recently developed in Liero et al. [42].Let us first describe the financial interpretation of the problems that we are going to study. The dual problem
The LHS of (10), namely inf Q ∈M ( b Q , b Q ,... b Q T ) E Q [ c ], represents the dualproblem in the financial application, but is typically the primal problem in Martingale OptimalTransport (MOT) 5e label this case as the sublinear case of MOT. In [42], the primal
Entropy Optimal Transport (EOT) problem takes the form inf µ ∈ Meas (Ω) E Q [ c ] + T X t =0 D F t , b Q t ( µ t ) ! , (11)where Meas(Ω) is the set of all positive finite measures µ on Ω , and D F t , b Q t ( µ t ) is a divergence inthe form: D F t , b Q t ( µ t ) := Z K t F t (cid:18) d µ t d b Q t (cid:19) d b Q t , if µ t ≪ b Q t , (12)otherwise D F t , b Q t ( µ t ) := + ∞ . We label with F := ( F t ) t =0 ,...,T the family of divergence functions F t : R → R ∪ { + ∞} appearing in (12). Problem (11) represents the convex case of OT theory.Notice that in the EOT primal problem (11) the typical constraint that µ has prescribed marginals( b Q , b Q , ... b Q T ) is relaxed (as the infimum is taken with respect to all positive finite measures) byintroducing the divergence functional D F t , b Q t ( µ t ), which penalizes those measures µ that are “far”from some reference marginals ( b Q , b Q , ... b Q T ) . We are then naturally let to the study of the convexcase of MOT, i.e. to the Entropy Martingale Optimal Transport (EMOT) problem D F, b Q ( c ) := inf Q ∈ Mart(Ω) E Q [ c ] + T X t =0 D F t , b Q t ( Q t ) ! (13)having also a clear financial interpretation. The marginals are not any more fixed a priori, as in(10), because we may not have sufficient information to detect them with enough accuracy. Sothe infimum is taken over all martingale probability measures , but those that are far from someestimate ( b Q , b Q , ... b Q T ) are appropriately penalized through D F t , b Q t . Of course, when D F t , b Q t ( · ) = δ b Q t ( · ), we recover the sublinear MOT problem, where only martingale probability measures withfixed marginals are allowed. Observe that in addition to the martingale property, the elements Q ∈ Mart(Ω) in (13) are required to be probability measures, while in the EOT (11) theory allpositive finite measure are allowed. As it was recalled after equation (7), this normalization featureof the dual elements ( µ (Ω) = 1) is not surprising when one deals with dual problems of primalproblems with a cash additive objective functional as, for example, in the theory of coherent andconvex risk measures.Potentially, we could push our smoothing argument above even further: in place of the function-als D F t , b Q t ( µ t ), t = 0 , ..., T , we might as well consider more general marginal penalizations, notnecessairly in the divergence form (12), yielding the problem D ( c ) := inf Q ∈ Mart(Ω) E Q [ c ] + T X t =0 D t ( Q t ) ! . (14)These penalizations D , . . . , D T will be better specified later. The primal problem: the Nonlinear Subhedging Value
We provide the financial interpre-tation of the primal problem which will yield the EMOT problem D F, b Q as its dual. It is convenientto reformulate the robust subhedging price in the RHS of (10) in a more general setting.6 efinition 1.1. Consider a measurable function c : Ω → R representing a (possibly path depen-dent) option, the set V of hedging instruments and a suitable pricing functional π : V → R . Thenthe robust Subhedging Value of c is defined by Π π, V ( c ) = sup { π ( v ) | v ∈ V s.t. v ≤ c } . In the classical setting, functionals of this form (and even with a more general formulation) areknown as general capital requirement, see for example Frittelli and Scandolo [32]. We stresshowever that in Definition 1.1 the inequality v ≤ c holds for all elements in Ω with no referenceto a probability measure whatsoever. The novelty in this definition is that a priori π may not belinear 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 E := E × · · · × E T , and V := E + · · · + E T + I . Suppose we took a linear pricing rule π : V → R defined via a b Q ∈ Mart(Ω) by π ( v ) := E b Q " T X t =0 ϕ t + I ∆ ( i ) = E b Q " T X t =0 ϕ t ( ii ) = T X t =0 E b Q t [ ϕ t ] , (15)where we used (2) and the fact that b Q t is the marginal of b Q . In this case, we would trivially obtainfor the robust subhedging value of c Π π, V ( c ) = sup { π ( v ) | v ∈ V s.t. v ≤ c } (16)= sup ( T X t =0 E b Q t [ ϕ t ] | ∃ ∆ ∈ H s.t. T X t =0 ϕ t ( x t ) + I ∆ ( x ) ≤ c ( x ) ∀ x ∈ Ω ) = sup ( m ∈ R | ∃ ∆ ∈ H , ϕ ∈ E , s.t. m − T X t =0 E b Q t [ ϕ t ] + T X t =0 ϕ t + I ∆ ≤ c ) = sup ( m ∈ R | ∃ ∆ ∈ H , ϕ ∈ E , with E b Q t [ ϕ t ] = 0 s.t. m + T X t =0 ϕ t + I ∆ ≤ c ) , (17)where in the last equality we replaced ϕ t with ( E b Q t [ ϕ t ] − ϕ t ) ∈ E t , which satisfies: E b Q t h E b Q t [ ϕ t ] − ϕ t i = 0 . (18) Interpretation : Π π, V ( c ) is the supremum amount m ∈ R for which we may buy options ϕ t anddynamic strategies ∆ ∈ H such that m + P Tt =0 ϕ t + I ∆ ≤ c , where the value of both the optionsand the stochastic integrals are computed as the expectation under the same martingale measure( b Q for the integral I ∆ ; its marginals b Q t for each option ϕ t ).However, as mentioned above when presenting the indifferent price p b , there is a priori no reasonwhy 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 : E t → R more general than E b Q [ · ] and E b Q t [ · ] . Nonetheless, in order to be able to repeat the same key steps we used in (16)-(17) and therefore tokeep the same interpretation, we shall impose that such functionals S and S t satisfy the propertyin (18) and the two properties (i) and (ii) in equation (15), that is:7a) S t [ ϕ t + k ] = S t [ ϕ t ] + k and S t [0] = 0 , for all ϕ t ∈ C b ( K t ), k ∈ R , t = 0 , . . . , T .(b) S (cid:20)(cid:18) T P t =0 ϕ t (cid:19) + I ∆ ( x ) (cid:21) = S (cid:20) T P t =0 ϕ t (cid:21) for all ∆ ∈ H and ϕ ∈ E . (c) S (cid:20) T P t =0 ϕ t (cid:21) = T P t =0 S t [ ϕ t ] for all ϕ ∈ E .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 thesame steps in (16)-(17), we will obtain as primal problem the nonlinear subhedging value of c : P ( c ) = sup { S ( v ) | v ∈ V : v ≤ c } = sup ( T X t =0 S t ( ϕ t ) | ∃ ∆ ∈ H s.t. T X t =0 ϕ t ( x t ) + I ∆ ( x ) ≤ c ( x ) ∀ x ∈ Ω ) (19)= sup ( m ∈ R | ∃ ∆ ∈ H , ϕ ∈ E , with S t ( ϕ t ) = 0 s.t. m + T X t =0 ϕ t + I ∆ ≤ c ) , (20)to be compared with (17). 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 + P Tt =0 ϕ t + I ∆ ≤ c , where the value of both theoptions and the stochastic integrals are computed with the same functional S. Before further elaborating on these issues, let us introduce the concept of Stock Additivity, whichis the natural counterpart of properties (IA) and (CA) when we are evaluating hedging instrumentsdepending solely on the value of the underlying stock X at some fixed date t ∈ { , . . . , T } . LetId t be the identity function x t x t on K t . As before, the set of such 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 forhedging) and that E t + R = E t (that is, deterministic amounts of cash can be used for hedging aswell). Definition 1.2.
A functional p t : E t → R is stock additive on E t if p t (0) = 0 and p t ( ϕ t + α t Id t + λ t ) = p t ( ϕ t ) + α t x + λ t ∀ ϕ t ∈ E t , λ t ∈ R , α t ∈ R , 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 × ... × E T and ∆ ∈ H such that P Tt =0 ϕ t = P Tt =0 ψ t + I ∆ then T X t =0 S t ( ϕ t ) = T X t =0 S t ( ψ t ) . This allows us to define a functional S : V = E + · · · + E T + I → R by S ( υ ) := T X t =0 S t ( ϕ t ) , for υ = T X t =0 ϕ t + I ∆ . (21)Then S is a well defined, integral additive functional on V , and S, S , . . . , S T satisfy the properties(a), (b), (c). There is a natural way to produce a variety of Stock Additive functionals, as explainedin Example 1.3 below. 8 xample . Consider a Martinagle measure b Q ∈ Mart(Ω) and a concave non decreasing utilityfunction u t : R → [ −∞ , + ∞ ) , satisfying u (0) = 0 and u t ( x t ) ≤ x t ∀ x t ∈ R . We can then take S t ( ϕ t ) = U b Q t ( ϕ t ) := sup α ∈ R , λ ∈ R (cid:18)Z Ω u t ( ϕ t ( x t ) + αx t + λ ) d b Q t ( x t ) − ( αx + λ ) (cid:19) . As shown in Lemma 4.2 the stock additivity property is then satisfied for these functionals.When we consider stock additive functionals S , . . . , S T that induce the functional S as explainedin (21), we can focus our attention to the optimization problem (19) or (20), that will be referredto as our primal problem. The Duality
As a consequence of our main results we prove the following duality (see Theorem3.3). If D t ( Q t ) := sup ϕ t ∈E t (cid:18) S t ( ϕ t ) − Z K t ϕ t d Q t (cid:19) for Q t ∈ Prob( K t ), t = 0 , . . . , T, and D ( c ) and P ( c ) are defined respectively in (14) and (19), then D ( c ) = P ( c ) . In the particular case of S , . . . , S T induced by utility functions, as explained in Example 1.3, theproblem corresponding to (19) will be denoted by P U, b Q ( c ), that is P U, b Q ( c ) := sup ( T X t =0 U b Q t ( ϕ t ) | ∃ ∆ ∈ H s.t. T X t =0 ϕ t ( x t ) + I ∆ ( x ) ≤ c ( x ) ∀ x ∈ Ω ) . (22)We also show the duality between (13) and (22), namely we prove in Section 4.1 D F, b Q ( c ) := inf Q ∈ Mart(Ω) E Q [ c ] + T X t =0 D F t , b Q t ( Q t ) ! = P U, b Q ( c ) . (23)The divergence functions F t appearing in D F, b Q (via D F t , b Q t ) are associated to the utility functions u t appearing in U b Q t and in P U, b Q via the coniugacy relation: F t ( y ) := v ∗ t ( y ) = sup x t ∈ R { x t y − v ( y ) } = sup x t ∈ R { u t ( x t ) − x t y } , where v ( y ) := − u ( − y ). Thus, depending on which utility function u is selected in the primalproblem P U, b Q ( c ) to evaluate the options through U b Q t , the penalization term D F t , b Q t in the the dualformulation D F, b Q ( c ) has a particular form induced by F t = v ∗ t . In the special case of linear utilityfunctions u t ( x t ) = x t , we recover the sublinear MOT theory. Indeed, in this case, v ∗ t ( y ) = + ∞ , forall y = 1 and v ∗ t (1) = 0, so that D F t , b Q t ( · ) = δ b Q t ( · ) and thus we obtain the robust pricing-hedgingduality (10) of the classical MOT. In the previous section we described and provided the financial interpretation of the new duality(23). This will be a particular case of a more general duality established in Theorem 2.3 andTheorem 2.4 and announced in the first section of this Introduction.9n our main result (Theorem 2.4) we start by introducing two general functionals, U : E → [ −∞ , + ∞ ) and D U : ca(Ω) → ( −∞ , + ∞ ] , that are associated through a Fenchel Moreau typerelation (see 26). The vector space E ⊆ C b (Ω; R T +1 ) consists of continuous and bounded functionsdefined on some Polish space Ω and with values in R T +1 . The map U is not necessarily cashadditive. We then rely on the notion of the Optimized Certainty Equivalent (OCE), that wasintroduced in Ben Tal and Teboulle [8] and further analyzed in Ben Tal and Teboulle [9]. As it iseasily recognized, any OCE is, except for the sign, a particular convex risk measure and so it iscash additive. We introduce the Generalized Optimized Certainty Equivalent associated to U asthe functional S U : E → [ −∞ , + ∞ ] defined by S U ( ϕ ) := sup ξ ∈ R T +1 U ( ϕ + ξ ) − T X t =0 ξ t ! , ϕ ∈ E . (24)Thus we obtain a cash additive map S U ( ϕ + ξ ) = S U ( ϕ ) + P Tt =0 ξ t , which will guarantee that inthe problem (11) the elements µ ∈ Meas(Ω) are normalized, i.e. are probability measures. Thenthe duality will take the forminf Q ∈ Mart(Ω) ( E Q [ c ( X )] + D U ( Q )) = sup ∆ ∈H sup ϕ ∈ Φ ∆ ( c ) S U ( ϕ ) , (25)where Φ ∆ ( c ) := ( ϕ ∈ dom( U ) , T X t =0 ϕ t ( x t ) + I ∆ ( x ) ≤ c ( x ) ∀ x ∈ Ω ) , and we also prove the existence of the optimizer for the problem in the LHS of (25), see Proposition2.6.The penalization term D := D U associated to U does not necessarily have an additive structure( D ( Q ) = P Tt =0 D t ( Q t )) nor needs to have the divergence formulation, as described in (12), and soit does not necessarily depend on a given martingale measure b Q . As explained in Sections 4.1.2 and4.2 this additional flexibility in choosing D allows several different interpretation and constitutesone generalization of the Entropy Optimal Transport theory of [42]. Of course, the other additionaldifference with EOT is the presence in (25) of the additional supremum with respect to admissibleintegrand ∆ ∈ H . As a consequence, in the LHS of (25) the infimum is now taken with respectto martingale measures. We also point out that in [42], the cost functional c is required to belower semicontinuous and nonnegative and that the theory is developed only for the bivariate case( t = 0 , c lower semicontinuous and with compact level sets, hencebounded from below, and consider the multivariate case ( t = 0 , ..., T ).In [42] the authors work with a Hausdorf topological space Ω , while we request Ω to be a Polishspace and for some of the results even a compact subset of R N . This stronger assumption however istotally reasonable for the applications we deal with (see Remark 4.4) and, in case of the Polish spaceassumption, it would be also compatible with a theory for stochastic processes X in continuoustime, a topic left for future investigation.To conclude, our framework allows to establish and comprehend several different duality results,even if under different type of assumptions:1. The new non linear robust pricing-hedging duality with options described in (23) and provedin Section 4.1, Corollary 4.3. 10. The new non linear robust pricing-hedging duality with options and singular components(see Corollary 4.5.3. The linear robust pricing-hedging duality with options (see [6] Th. 1.1, or Acciaio et al. [1]Th 1.4) described in (10) and proved in Corollary 4.7.4. The linear robust pricing-hedging duality without options (see for example Burzoni et al.[17] Th. 1.1) proved in Corollary 4.8.5. A new robust pricing-hedging duality with penalization function based on market data (seeSection 4.1.2).6. A new dual robust representation for the Optimized Certainty Equivalent functional (seeSection 4.2.1).We will explain in Section 4.2 why the degree of freedom in the choice of E may be relevant alsofor the application in the financial context.We summarize the preceding discussion in the following Table and we point out that in thispaper we develop the duality theory sketched in the last line of the Table and provide its financialinterpretation . Differently from rows 1, 2, 5, 6, in rows 3, 4, 7, 8, the financial market is presentand martingale measures are involved in the dual formulation. In rows 1, 2, 3, 4 we illustrate theclassical setting, where the conditions in the functional form hold P -a.s., while in the last fourrows Optimal Transport is applied to treat the robust versions, where the inequalities holds for allelements of Ω. Table 1:
Π(Ω) is the set of all probabilities on Ω; P ( P ) = { Q ∈ Π(Ω) | Q ≪ P } ; Mart(Ω) is the set of all martingaleprobabilities on Ω; M ( P ) = Mart(Ω) ∩ P ( P ); Π( Q , Q ) = { Q ∈ Π(Ω) with given marginals } ; Mart( Q , Q ) = { Q ∈ Mart(Ω) with given marginals } ; Meas(Ω) is the set of all positive finite measures on Ω; Sub( c ) is the set ofstatic parts of semistatic subhedging strategios for c ; U is a concave proper utility functional and S U is the associatedgeneralized Optimized Certainty Equivalent. ;FUNCTIONAL FORM SUBLINEAR CONVEX1 - Coherent R.M. − inf { m | c + m ∈ A} , A cone inf Q ∈Q⊆P ( P ) E Q [ c ]2 - Convex R.M. − inf { m | c + m ∈ A} , A convex inf Q ∈P ( P ) ( E Q [ c ] + α A ( Q ))3 Subreplic. price sup n m | ∃ ∆ : m + I ∆ ( X ) ≤ c o inf Q ∈M ( P ) E Q [ c ]4 Indiff. price sup { m | U ( c − m ) ≥ U (0) } inf Q ∈M ( P ) ( E Q [ c ] + α U ( Q ))5 O.T. sup ϕ + ψ ≤ c (cid:0) E Q [ ϕ ] + E Q [ ψ ] (cid:1) inf Q ∈ Π( Q ,Q E Q [ c ]6 E.O.T. sup ϕ + ψ ≤ c U ( ϕ, ψ ) inf Q ∈ Meas(Ω) ( E Q [ c ] + D U ( Q ))7 M.O.T. sup [ ϕ,ψ ] ∈ Sub( c ) (cid:0) E Q [ ϕ ] + E Q [ ψ ] (cid:1) inf Q ∈ Mart( Q ,Q E Q [ c ]8 E.M.O.T. sup [ ϕ,ψ ] ∈ Sub( c ) S U ( ϕ, ψ ) inf Q ∈ Mart(Ω) ( E Q [ c ] + D U ( Q )) A Generalized Optimal Transport Duality
For unexplained concepts on Measure Theory we refer to the Appendix A.1. We let Ω be a PolishSpace and define the following sets:ca(Ω) := { γ : B (Ω) → ( −∞ , + ∞ ) | γ is finite signed Borel measures on Ω } , Meas(Ω) := { µ : B (Ω) → [0 , + ∞ ) | µ is a non negative finite Borel measures on Ω } , Prob(Ω) := { Q : B (Ω) → [0 , | Q is a probability Borel measures on Ω } . C b (Ω , R M ) := ( C b (Ω)) M = { ϕ : Ω → R M | ϕ is bounded and continuous on Ω } . We let
E ⊆ C b (Ω; R M +1 ) be a vector subspace, U : E → [ −∞ , + ∞ ) be a proper concave functionaland set V ( ϕ ) := − U ( − ϕ ) . We define D : ca(Ω) → ( −∞ , + ∞ ] by D ( γ ) := sup ϕ ∈E U ( ϕ ) − M X m =0 Z Ω ϕ m d γ ! = sup ϕ ∈E M X m =0 Z Ω ϕ m d γ − V ( ϕ ) ! , γ ∈ ca(Ω) . (26) 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. Standing Assumption 2.1. D is proper, i.e. dom( D ) = { γ ∈ ca(Ω) | D ( γ ) < + ∞} 6 = ∅ . Remark . Another way to introduce our setting, that will be used in Subsection 4.1.2, is tostart initially with a proper convex functional D : ca(Ω) → ( −∞ , + ∞ ] which is σ (ca(Ω) , E )-lowersemicontinuous for some vector subspace E ⊆ C b (Ω , R M +1 ). By Fenchel-Moreau Theorem we thenhave the representaion D ( γ ) = sup ϕ ∈E M X m =0 Z Ω ϕ m d γ − V ( ϕ ) ! , where now V is the Fenchel-Moreau (convex) conjugate of D , namely V ( ϕ ) := sup γ ∈ ca(Ω) M X m =0 Z Ω ϕ m d γ − D ( γ ) ! . (27)Setting U ( ϕ ) := − V ( − ϕ ), ϕ ∈ E , (28)we get back that D satisfies (26) and additionally that U is σ ( E , ca(Ω))-upper semicontinuous.In conclusion, a pair ( U, D ) satisfying (26) might be defined either providing a proper concave U : E → [ −∞ , + ∞ ), as described at the beginning of this section, or assigning a proper convex and σ ( E , ca(Ω))-lower semicontinuous functional D : ca(Ω) → ( −∞ , + ∞ ] as explained in this Remark.We set dom( U ) := { ϕ ∈ E | U ( ϕ ) > −∞} . (29)12 heorem 2.3. Let c : Ω → ( −∞ , + ∞ ] be proper lower semicontinuous with compact sublevel setsand assume the following holds:There exists a sequence ( k n ) n ⊆ R M +1 with lim sup n M X m =0 k nm = + ∞ such that U ( − k n ) > −∞ ∀ n . (30) Then inf µ ∈ Meas(Ω) (cid:18)Z Ω c d µ + D ( µ ) (cid:19) = sup ϕ ∈ Φ ( c ) U ( ϕ ) . where Φ ( c ) := ( ϕ ∈ dom( U ) | M X m =0 ϕ m ( x ) ≤ c ( x ) ∀ x ∈ Ω ) . (31) Proof.
We start applying (26) to get that Z Ω c d µ + D ( µ ) = Z Ω c d µ + sup ϕ ∈E U ( ϕ ) − M X m =0 Z Ω ϕ m d µ ! . We then consider L : Meas(Ω) × dom( U ) → ( −∞ , + ∞ ] defined by L ( µ, ϕ ) := Z Ω c − M X m =0 ϕ m ! d µ + U ( ϕ )and we set M := { µ ∈ Meas(Ω) | R Ω c d µ < + ∞} . We observe that L is real valued on M × dom( U )and for any µ ∈ Meas(Ω) \ M we have L ( µ, ϕ ) = + ∞ for all ϕ ∈ dom( U ) (since c is bounded frombelow). We also see that setting C := dom( U )inf µ ∈ Meas(Ω) (cid:18)Z Ω c d µ + D ( µ ) (cid:19) = inf µ ∈ Meas(Ω) sup ϕ ∈C L ( µ, ϕ ) = inf µ ∈ M sup ϕ ∈C L ( µ, ϕ ) . (32)The aim is now to interchange sup and inf in RHS of (32), using Theroem A.8.To this end, without loss of generality we can assume α := sup ϕ ∈C inf µ ∈ Meas(Ω) L ( µ, ϕ ) < + ∞ andwe 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 > ϕ ∈ C having all components ϕ m equal to some constant − k nm <
0, such that ϕ ∈ dom( U ) and inf x ∈ Ω c ( x ) − M X m =0 ϕ m ( x ) ! > ε > . For such choice of ϕ and for a sufficiently big constant C > α there exists another constant D := C − U ( ϕ ) ≥
0, independent of µ , such that µ C = ( µ ∈ Meas(Ω) | Z Ω c − M X m =0 ϕ m ! d µ ≤ D ) = ( µ ∈ Meas(Ω) | Z Ω c − M X m =0 ϕ m − ε ! d µ + εµ (Ω) ≤ D ) . (33)Consequently, the set µ C is: 13. Nonempty, as the measure µ ≡ µ C .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). Foreach n, µ R Ω c n d µ is narrowly lower semicontinuous on Meas(Ω) , by definition. Henceby Monotone Convergence Theorem the map µ R Ω (cid:16) c − P Mm =0 ϕ m (cid:17) d µ is the pointwisesupremum of narrowly lower semicontinuous functions, and is lower semicontinuous withrespect 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 (33).3. Bounded: having a sequence of measures in µ C with unbounded total mass would resultin a contradiction with the constraint in the last item of (33), taking into account that c − P Mm =0 ϕ m − ε ≥ ε > ≤ f := c − P Mm =0 ϕ m − ε. Since εµ (Ω) ≥ µ ∈ Meas(Ω), by the centralexpression of (33) the inclusion µ C ⊆ { µ ∈ M eas | R Ω f d µ ≤ D } holds. Now it is easy tocheck that for all µ ∈ µ C and α > D ≥ Z Ω f d µ ≥ Z f>α f d µ ≥ αµ ( { f > α } ) . Observing that the sublevels of f are compact, by lower semicontinuity of c and compactnessof its sublevel sets, we see that { f > α } are complementaries of compact subsets of Ω andcan be taken with arbitrarily small measure, just by increasing α , uniformly in µ ∈ µ C . Thustightness follows.5. A subset of M. These properties in turns yield narrow compactness of µ C in Meas(Ω), by Theorem A.6, andtherefore σ (Meas(Ω) , C b (Ω)) compactness (recalling that weak and narrow topology coincide in oursetup). 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.8. Indeed, L is real valued on M × C . Items 1 and 2 of TheoremA.8 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 weaktopology coincide in our setup (see Proposition A.4). Hence we may interchange sup and inf inRHS of (32), obtaininginf µ ∈ M sup ϕ ∈C L ( µ, ϕ ) = sup ϕ ∈C inf µ ∈ M L ( µ, ϕ ) = sup ϕ ∈C inf µ ∈ Meas(Ω) L ( µ, ϕ ) (34)where the last equality follows from the fact that L ( µ, ϕ ) = + ∞ on the complementary of M inMeas(Ω) for every ϕ ∈ C . It is now easy to check that for every ϕ ∈ C inf µ ∈ Meas L ( µ, ϕ ) = U ( ϕ ) if P Mm =0 ϕ m ( x ) ≤ c ( x ) ∀ x ∈ Ω −∞ otherwise14hus sup ϕ ∈C inf µ ∈ Meas(Ω) L ( µ, ϕ ) = sup ϕ ∈ Φ ( c ) U ( ϕ )which concludes the proof, given (32) and (34). In order to describe a suitable theory to develop the entropy optimal transport duality in a dynamicsetting, in this section we will adopt a particular product structure of the set Ω.To this end, in addition to the notations already introduced in Section 2, we consider T ∈ N , T ≥
1, and Ω := K × · · · × K T (35)for K , . . . , K T ⊆ R . We denote with X , . . . , X T the canonical projections X t : Ω → K t and weset X = [ X , . . . , X T ] : Ω → R T +1 , to be considered as discrete-time stochastic process. We denotewith: Mart(Ω) := { Martingale measures for the canonical process of Ω } . When µ ∈ Meas( K × · · · × K T ), its marginals will be denoted with: µ , . . . , µ T .We recall, respectively from (8) and (9), that H is the set of admissible trading strategies and I isthe set of elementary stochastic integral. We take E = E × · · · × E T where E t ⊆ C b ( K × · · · × K t )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 naturalfiltration of the process ( X t ) t .We suppose that U : E → [ −∞ , + ∞ ) is proper and concave, D : Meas(Ω) → ( −∞ , + ∞ ] is definedin (26) and, as in (24), S U ( ϕ ) := sup ξ ∈ R T +1 U ( ϕ + ξ ) − T X t =0 ξ t ! , ϕ ∈ E . Theorem 2.4.
Assume that
Ω := K × · · · × K T for compact sets K , . . . , K T ⊆ R , that c : Ω → ( −∞ , + ∞ ] is lower semicontinuous, that D : Meas(Ω) → ( −∞ , + ∞ ] is lower bounded on Meas(Ω) and proper. Suppose also U satisfies (30) , and that N := (cid:26) µ ∈ Meas(Ω) ∩ dom( D ) | Z Ω c d µ < + ∞ (cid:27) = ∅ and dom( U ) + R T +1 ⊆ dom( U ) . (36) Then the following holds: inf Q ∈ Mart(Ω) ( E Q [ c ( X )] + D ( Q )) = sup ∆ ∈H sup ϕ ∈ Φ ∆ ( c ) S U ( ϕ ) (37) where for each ∆ ∈ H Φ ∆ ( c ) := ( ϕ ∈ dom( U ) , T X t =0 ϕ t ( x t ) + T − X t =0 ∆ t ( x , . . . , x t )( x t +1 − x t ) ≤ c ( x ) ∀ x ∈ Ω ) . (38) Proof.
The first part of the proof in inspired by [6] Equations (3.4)-(3.3)-(3.2)-(3.1).inf Q ∈ Mart(Ω) ( E Q [ c ( X )] + D ( Q )) (39)15 inf Q ∈ Mart(Ω) sup ∆ ∈H E Q " c ( X ) − T − X t =0 ∆ t ( X , . . . , X t )( X t +1 − X t ) + D ( Q ) ! (40)= inf Q ∈ Prob(Ω) sup ∆ ∈H E Q " c ( X ) − T − X t =0 ∆ t ( X , . . . , X t )( X t +1 − X t ) + D ( Q ) ! (41)= inf Q ∈ Prob(Ω) sup ∆ ∈H λ ∈ R E Q " c ( X ) − T − X t =0 ∆ t ( X , . . . , X t )( X t +1 − X t ) + λ − λ + D ( Q ) ! (42)= inf µ ∈ Prob(Ω) sup ∆ ∈H λ ∈ R (cid:18)Z Ω (cid:2) c ( x ) − I ∆ ( x ) + λ (cid:3) d µ ( x ) − λ + D ( µ ) (cid:19) (43)= inf µ ∈ Meas(Ω) sup ∆ ∈H λ ∈ R (cid:18)Z Ω (cid:2) c − I ∆ + λ (cid:3) d µ − λ + D ( µ ) (cid:19) (44)= inf µ ∈ Meas(Ω) sup ∆ ∈H λ ∈ R T +1 Z Ω " c − I ∆ + T X t =0 λ t d µ − T X t =0 λ t + D ( µ ) ! . (45)The equality chain above is justified as follows: (39)=(40) is trivial; (40)=(41) follows using thesame argument as in [6] Lemma 2.3, which yields that the inner supremum explodes to + ∞ unless Q is a martingale measure on Ω; (41)=(42) and (42)=(43) are trivial; (43)=(44) follows observingthat the inner supremum over λ ∈ R explodes to + ∞ unless µ (Ω) = 1; (44)=(45) is trivial.We define now K : Meas(Ω) × ( H × R M ) → ( −∞ , + ∞ ] as K ( µ, ∆ , λ ) := Z Ω " c − I ∆ + T X t =0 λ t d µ − T X t =0 λ t + D ( µ ) . From (36), 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,providesinf Q ∈ Mart(Ω) ( E Q [ c ( X )] + D ( Q )) = inf µ ∈ Meas(Ω) sup ∆ ∈H λ ∈ R T +1 K ( µ, ∆ , λ ) = inf µ ∈N sup ∆ ∈H λ ∈ R T +1 K ( µ, ∆ , λ ) . (46)As in the proof of Theorem 2.3, we wish to apply the Minimax Theorem A.8 in order to in-terchange inf and sup in RHS of (46) 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 convex-ity in Item 1, concavity in 2 of Theorem A.8 are clearly satisfied. We have to find ∆ ∈ H , λ ∈ R T +1 and C > α such that the sublevel set M C := { µ ∈ Meas(Ω) | K ( µ, ∆ , λ ) ≤ C } is weakly compact.Fix a ε >
0. As the functional c is lower semicontinuous on the compact Ω, it is lower boundedon Ω and we can take ∆ = 0 and λ sufficiently big in such a way that inf x ∈ Ω ( c ( x ) + P Tt =0 λ t ) > ε .For such a choice of (∆ , λ ) we have that M C satisfies M C ⊆ ( µ ∈ Meas(Ω) | Z Ω " c + T X t =0 λ t − ε d µ ( x ) + εµ (Ω) ≤ C + λ − inf µ ∈ Meas(Ω) D ( µ ) =: D ) where D ∈ R since D ( · ) is lower bounded by hypothesis. By (36) 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 be16pplied 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 topol-ogy 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 ofTheorem 2.3 we get that µ R Ω h c − I ∆ + P Tt =0 λ t i d µ − P Tt =0 λ t is σ (Meas(Ω) , C b (Ω)) | N -lowersemicontinuous, while D is by definition σ (ca(Ω) , E ) | N lower semicontinuous (being supremum oflinear functionals each continuous in such a topology). Since sum of lower semicontinuous functionsis lower semicontinuous, the desired lower semicontinuityof K ( · , ∆ , λ ) follows. All the hypothesesof Theorem A.8 are now verified, and we may then interchange sup and inf in RHS of (46) andobtain inf µ ∈N sup ∆ ∈H λ ∈ R T +1 K ( µ, ∆ , λ ) = sup ∆ ∈H λ ∈ R T +1 inf µ ∈N K ( µ, ∆ , λ ) ( ⋆ ) = sup ∆ ∈H λ ∈ R T +1 inf µ ∈ Meas(Ω) K ( µ, ∆ , λ )= sup ∆ ∈H λ ∈ R T +1 inf µ ∈ Meas(Ω) Z Ω " c − I ∆ + T X t =0 λ t d µ + D ( µ ) ! − T X t =0 λ t , (47)where in ( ⋆ ) we used the fact that K ( µ, ∆ , λ ) = + ∞ on the complementary of N in Meas(Ω), forevery (∆ , λ ) ∈ H × R T +1 .We apply now Theorem 2.3 to the inner infimum with the cost functional c − I ∆ + P Tt =0 λ t ,observing that, since we are assuming dom( U ) + R T +1 = dom( U ) (see (36)), the condition (30) issatisfied. We get that (47) = sup ∆ ∈H λ ∈ R T +1 sup ϕ ∈ Φ ∆ ,λ ( c ) U ( ϕ ) − T X t =0 λ t ! where Φ ∆ ,λ ( c ), which depends on ∆ , λ ∈ H × R T +1 , is defined according to (31) by Φ ∆ ,λ ( c ) = ( ϕ ∈ dom( U ) , T X t =0 ϕ t ( x ) ≤ c ( x ) − I ∆ ( x ) + T X t =0 λ t ∀ x ∈ Ω ) . From (36), ( 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 (38), so that(47) = sup ∆ ∈H λ ∈ R T +1 sup ϕ ∈ Φ ∆ ( c ) U ( ϕ + λ ) − T X t =0 λ t ! = sup ∆ ∈H sup ϕ ∈ Φ ∆ ( c ) sup λ ∈ R T +1 U ( ϕ + λ ) − T X t =0 λ t ! . We now recognize the expression in (24) and we conclude thatinf µ ∈ Meas(Ω) sup ∆ ∈H λ ∈ R T +1 K ( µ, ∆ , λ ) = (47) = sup ∆ ∈H sup ϕ ∈ Φ ∆ ( c ) S U ( ϕ ) , and consequently, recalling our minimax argument,inf Q ∈ Mart ( E Q [ c ( X )] + D ( Q )) Eq.(46) = (47) = sup ∆ ∈H sup ϕ ∈ Φ ∆ ( c ) S U ( ϕ ) . emark . (i) The assumptions of Theorem 2.4 are reasonably weak and are satisfied, for example,if: dom( U ) = E , there exists a b µ ∈ Meas(Ω) ∩ ∂U (0) such that c ∈ L ( b µ ) , and c is lower semicon-tinuous. Indeed, for all µ ∈ Meas(Ω), D ( µ ) ≥ U (0) − > −∞ . Clearly dom( U ) + R T +1 = dom( U ).Finally, b µ ∈ N , because c ∈ L ( b µ ) and −∞ < U (0) ≤ D ( b µ ) ≤ , by definition of D .(ii) The step (40)=(41) is the crucial point where compactness of the sets K , . . . , K T ⊆ R isnecessary for a smooth argument, since integrability of the underlying stock process is in this caseautomatically satisfied for all Q ∈ Prob(Ω), not only for Q ∈ Mart(Ω). Also, compactness is keyin guaranteeing that the cost functional c − I ∆ + P t λ t is bounded from below, in order to applyTheorem 2.3. Proposition 2.6.
Suppose that LHS of (37) is finite and that D| Meas(Ω) is σ (Meas(Ω) , C b (Ω)) -lower semicontinuous. Then, under the same the assumptions of Theorem 2.4, the problem in LHSof (37) admits an optimum.Proof. Similarly to what we argued in Item 2 of the proof of Theorem 2.3, the map µ R Ω c d µ is σ (Meas(Ω) , C b (Ω))-lower semicontinuous, and we deduce the lower semicontinuity of Q
7→ J ( Q ) := E Q [ c ] + T X t =0 D ( Q ) , Q ∈ Mart(Ω) . Moreover for C big enough the sublevel { Q ∈ Mart(Ω) | J ( Q ) ≤ C } is nonempty (since we areassuming LHS of (37) is finite), hence J is proper on Mart(Ω). Since K , ..., K T are compact,Prob(Ω) is σ (Meas(Ω) , C b (Ω)) compact (see [2] Theorem 15.11), and Mart(Ω) is σ (Meas(Ω) , C b (Ω))closed because, arguing as in [6] Lemma 2.3,Mart(Ω) = \ ∆ ∈H ( Q ∈ Prob(Ω) | Z Ω T − X t =0 ∆ t ( x , . . . , x t )( x t +1 − x t ) ! d Q ( x ) ≤ ) . We conclude that Mart(Ω) is σ (Meas(Ω) , C b (Ω))-compact, and J is lower semicontinuous andproper on it, hence it attains a minimum. Remark . The lower semicontinuity assumption in Proposition 2.6 is satisfied in many cases, asit will become clear in Section 3.
We now rephrase our findings in Theorem 2.4, with minor additions, to get the formulation inCorollary 2.8 which will simplify our discussion of Section 4. In particular, this reformulation willcome in handy when dealing with subhedging and superhedging dualitites in Corollaries 4.3-4.8and Proposition 4.9.For a given proper concave U : E → R , we recall the definition of S U in (24) and, for V ( · ) = − U ( −· ),we define dom( V ) := { ϕ ∈ E | V ( ϕ ) < + ∞} = − dom( U ) and S V ( ϕ ) := inf λ ∈ R T +1 V ( ϕ + λ ) − T X t =0 λ t ! = − S U ( − ϕ ) , ϕ ∈ dom( V ) . c : Ω → ( −∞ , + ∞ ], d : Ω → [ −∞ , + ∞ ) we introduce the sets S sub ( c ) := ( ϕ ∈ dom( U ) | ∃ ∆ ∈ H s.t. T X t =0 ϕ ( x t ) + I ∆ ( x ) ≤ c ( x ) ∀ x ∈ Ω ) (48) S sup ( d ) := ( ϕ ∈ dom( V ) | ∃ ∆ ∈ H s.t. T X t =0 ϕ ( x t ) + I ∆ ( x ) ≥ d ( x ) ∀ x ∈ Ω ) (49)and observe that S sup (Ψ) = −S sub ( − Ψ).
Corollary 2.8.
Suppose that the assumptions in Theorem 2.4 are satisfied, that d : Ω → [ −∞ , + ∞ ) is upper semicontinuous and that { µ ∈ Meas(Ω) ∩ dom( D ) | R Ω d d µ > −∞} 6 = ∅ . Then the followinghold inf Q ∈ Mart(Ω) ( E Q [ c ( X )] + D ( Q )) = sup ϕ ∈S sub ( c ) S U ( ϕ ) , (50)sup Q ∈ Mart(Ω) ( E Q [ d ( X )] − D ( Q )) = inf ϕ ∈S sup ( d ) S V ( ϕ ) . (51) Proof.
Equation (50) is an easy rephrasing of the corresponging (37). As to (51), we observe thatfor c := − d we get from (50)sup ϕ ∈S sub ( − d ) S U ( ϕ ) = inf Q ∈ Mart ( E Q [ − d ( X )] + D ( Q )) = − sup Q ∈ Mart ( E Q [ d ( X )] − D ( Q )) . Observing that S sup ( d ) = −S sub ( − d )and that S V ( · ) = − S U ( −· ) on dom( V ) we get sup ϕ ∈S sub ( − d ) S U ( ϕ ) = − inf ϕ ∈S sup ( d ) S V ( ϕ ). Thiscompletes the proof. In Section 2, we did not require any particular stuctural form of the functionals D , U . Here instead,we will assume in addition to (35) also an additive structure of U and, complementarily, an additivestructure of D . In the whole Section 3 we take for each t = 0 , . . . , T a vector subspace E t ⊆ C b ( K t )such that E t + R = E t and set E = E × · · ·× E T . Observe that we automatically have E + R T +1 = E .It is also clear that E is a subspace of C b (Ω , R T +1 ) , if we interpret E , . . . , E T as subspaces of C b (Ω). U Setup 3.1.
For every t = 0 , . . . , T we consider a proper concave functional U t : E t → [ −∞ , + ∞ ) .We define D t on ca( K t ) similarly to (26) as D t ( γ t ) := sup ϕ t ∈E t (cid:18) U t ( ϕ t ) − Z K t ϕ t d γ t (cid:19) γ t ∈ ca( K t ) and observe that D t can also be thought to be defined on ca(Ω) using for γ ∈ ca(Ω) the marginals γ , ..., γ T and setting D t ( γ ) := D t ( γ t ) . We may now define, for each ϕ ∈ E , U ( ϕ ) := P Tt =0 U t ( ϕ t ) and define D on ca(Ω) using (26) with M = T . Recall from (24) S U ( ϕ ) := sup ξ ∈ R T +1 U ( ϕ + ξ ) − T X t =0 ξ t ! , ϕ ∈ E , S U t ( ϕ t ) := sup α ∈ R ( U ( ϕ t + α ) − α ) , ϕ t ∈ E t . emma 3.2. In Setup 3.1 we have D ( γ ) = T X t =0 D t ( γ ) = T X t =0 D t ( γ t ) , ∀ γ ∈ ca(Ω) , S U ( ϕ ) = T X t =0 S U t ( ϕ t ) for all ϕ ∈ E , (52) and for all ϕ ∈ E S U ( ϕ + β ) = S U ( ϕ ) + T X t =0 β t , for β ∈ R T +1 , S U t ( ϕ t + d ) = S U t ( ϕ t ) + d, for d ∈ R Proof.
See Appendix A.2
Theorem 3.3.
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 . Considerfor every t = 0 , . . . , T the penalizations D t ( Q t ) := sup ϕ t ∈E t (cid:18) S t ( ϕ t ) − Z K t ϕ t d Q t (cid:19) for Q t ∈ Prob( K t ) , and set D ( Q ) := P Tt =0 D t ( Q t ) . Let c : Ω → ( −∞ , + ∞ ] be lower semicontinuous and let D ( c ) and P ( c ) be defined respectively in (14) and (19) . If N := (cid:8) µ ∈ Meas(Ω) ∩ dom( D ) | R Ω c d µ < + ∞ (cid:9) = ∅ then P ( c ) = D ( c ) .Proof. Set E = E × · · · × E T and U ( ϕ ) := P Tt =0 S t ( ϕ t ), for ϕ ∈ E , and let D defined as in (26) for M = T . For any µ ∈ Meas(Ω) we have D ( µ ) ≥ P Tt =0 S t (0) − D is lower bounded onMeas(Ω). Observe that dom( U ) = E , which implies dom( U ) + R T +1 = dom( U ), and that we arein Setup 3.1. Lemma 3.2 tells us that S U ( ϕ ) = P Tt =0 S U t t ( ϕ t ) = P Tt =0 S t ( ϕ t ), since S , . . . , S T areCash Additive, and that D coincides on Mart(Ω) with the penalization term Q P Tt =0 D t ( Q t ), asprovided in the statement of this Theorem. Since all the assumptions of Theorem 2.4 are fulfilled,we can apply Corollary 2.8, which yields exactly D ( c ) = P ( c ). D . The results of this subsection will be applied in Subsection 4.1.2. In the spirit of Remark 2.2, wemay now reverse the procedure taken in the previous subsection: we start from some functionals D t on ca( K t ) , for t = 0 , . . . , T , and build an additive functional D on ca(Ω) . Our aim is to find thecounterparts of the results in Section 3.1.
Setup 3.4.
For every t = 0 , . . . , T we consider a proper, convex, σ (ca( K t ) , E t ) -lower semicontin-uous functional D t : ca( K t ) → ( −∞ , + ∞ ] . We can then extend the functionals D t to ca(Ω) byusing, for any γ ∈ ca(Ω) , the marginals γ , . . . , γ T . If γ ∈ ca(Ω) , we set D t ( γ ) := D t ( γ t ) and D ( γ ) := T X t =0 D t ( γ ) = T X t =0 D t ( γ t ) . We define V ( ϕ ) for ϕ ∈ E and V t ( ϕ t ) for ϕ t ∈ E t , for t = 0 , . . . , T similarly to (27) , as V ( ϕ ) := sup γ ∈ ca(Ω) Z Ω T X t =0 ϕ t ! d γ − D ( γ ) ! and V t ( ϕ t ) := sup γ ∈ ca( K t ) (cid:18)Z K t ϕ t d γ − D t ( γ ) (cid:19) . e define on E the functional U ( · ) = − V ( −· ) , as in (28) , and similarly U t ( · ) = − V t ( −· ) on E t , for t = 0 , . . . , T . Finally, S U ( ϕ ) , S U ( ϕ ) , . . . , S U T ( ϕ T ) are defined as in Setup 3.1. Lemma 3.5.
In Setup 3.4 we have:1. D , . . . , D T , as well as D , are σ (ca(Ω) , E ) -lower semicontinuous.2. Under the additional assumption that dom( D t ) ⊆ Prob( K t ) for every t = 0 , . . . , T , for any ϕ = [ ϕ , . . . , ϕ T ] ∈ E × · · · × E T U ( ϕ ) = T X t =0 U t ( ϕ t ) = T X t =0 − V t ( − ϕ t ) , (53) S U ( ϕ ) = T X t =0 S U t ( ϕ t ) . (54) Proof.
See Appendix A.2
Assumption 3.6.
We consider concave, upper semicontinuous nondecreasing functions u , . . . , u T : R → [ −∞ , + ∞ ) with u (0) = · · · = u T (0) = 0 , u t ( x ) ≤ x ∀ x ∈ R (that is ∈ ∂u (0) ∩· · ·∩ ∂u T (0) ).For each t = 0 , . . . , T we define v t ( x ) := − u t ( − x ) , x ∈ R and v ∗ t ( y ) := sup x ∈ R ( xy − v t ( x )) = sup x ∈ R ( u t ( x ) − xy )) , y ∈ R . (55)We observe that v t ( y ) = v ∗∗ t ( y ) = sup x ∈ R ( xy − v ∗ t ( y )) for all y ∈ R by Fenchel-Moreau Theoremand that v ∗ t is convex, lower semicontinuous and lower bounded on R . Example . Assumption 3.6 is satisfied by a wide range of functions. Just to mention a few withvarius peculiar features, we might take u t of the following forms: u t ( x ) = 1 − exp( − x ), whoseconvex conjugate is given by v ∗ t ( y ) = −∞ for y < v ∗ t (0) = 0, v ∗ t ( y ) = ( y log( y ) − y + 1) for y > u t ( x ) = αx ( −∞ , ( x ) for α ≥
1, so that v ∗ t ( y ) = + ∞ for y < v ∗ t ( y ) = 0 for y ∈ [0 , α ], v ∗ t ( y ) = + ∞ for y > α ; u t ( x ) = log( x +1) for x > − u t ( x ) = −∞ for x ≤ −
1, so that v ∗ t ( y ) = + ∞ for y ≤ v ∗ t ( y ) = y − log( y ) − y > u t ( x ) = −∞ for x ≤ − u t ( x ) = xx +1 for x > − v ∗ t ( y ) = −∞ for y < v ∗ t ( y ) = y − √ y + 1 for y ≥ u t ( x ) = −∞ for x < u t ( x ) = 1 − exp( − x )for x ≥
0, so that v ∗ t ( y ) = + ∞ for y < v ∗ t ( y ) = y log( y ) − y + 1 for 0 ≤ y ≤ v ∗ t ( y ) = 0 for y > b µ t ∈ Meas( K t ). We pose for µ ∈ Meas( K t ) D v ∗ t , b µ t ( µ ) := R K t v ∗ t (cid:16) d µ d b µ t (cid:17) d b µ t if µ ≪ b µ t + ∞ otherwise , (56)In the next two propositions, whose proofs are postponed to the Appendix A.2, we provide thedual representation of the divergence terms. Proposition 3.8.
Take u , . . . , u T satisfying Assumption 3.6, and suppose dom( u ) = · · · =dom( u T ) = R . Let b µ t ∈ Meas( K t ) and v t ( · ) := − u t ( −· ) , t = 0 , . . . , T . Then v ∗ t , b µ t ( µ ) = sup ϕ t ∈C b ( K t ) (cid:18)Z K t ϕ t ( x t ) d µ ( x t ) − Z K t v t ( ϕ t ( x t )) d b µ t ( x t ) (cid:19) . (57)Set: ( v ∗ t ) ′∞ := lim y → + ∞ v ∗ t ( y ) y , t = 0 , . . . , T . As dom( u ) ⊇ [0 , + ∞ ) , ( v ∗ t ) ′∞ ∈ [0 , + ∞ ]. Let b Q t ∈ Prob( K t ) and, for µ ∈ Meas( K t ), let µ = µ a + µ s be the Lebesgue Decomposition of µ with respect to b Q t , where µ a ≪ b Q t and µ s ⊥ b Q t . Then wecan define for µ ∈ Meas( K t ) F t ( µ | b Q t ) := Z K t v ∗ t (cid:18) d µ a d b Q t (cid:19) d b Q t + ( v ∗ t ) ′∞ µ s ( K t )where we use the convention ∞ × v ∗ t ) ′∞ = + ∞ , µ s ( K t ) = 0. Observe that therestriction of F ( · | b Q t ) to Meas( K t ) coincides with the functional in [42] (2.35) with F = v ∗ t , andthat whenever dom( u t ) = R we have ( v ∗ t ) ′∞ = lim y → + ∞ v ∗ t ( y ) y = + ∞ and F t ( · | b Q t ) coincides with D v ∗ t , b Q t ( · ) (see (56)) on Meas( K t ). Proposition 3.9.
Suppose that u , . . . , u T : R → [ −∞ , + ∞ ) satisfy Assumption 3.6 and v t ( · ) := − u t ( −· ) . If b Q t ∈ Prob( K t ) , t ∈ { , . . . , T } , has full support then F t ( µ | b Q t ) = sup ϕ t ∈C b ( K t ) (cid:18)Z K t ϕ t ( x t ) d µ ( x t ) − Z K t v t ( ϕ t ( x t )) d b Q t ( x t ) (cid:19) . (58) Example . The requirement that b Q , . . . , b Q T have full support is crucial for the proof of Propo-sition 3.9. We provide a simple example to the fact that (58) does not hold in general when suchan assumption is not fulfilled. To this end, take K = {− , , } , b Q = δ {− } + δ { +2 } , µ = δ { } , u ( x ) := xx +1 for x ≥ − u ( x ) = −∞ for x < −
1. It is easy to see that the associated v ∗ via(55) is defined by v ∗ ( y ) = 1 + y − √ y for y ≥ v ∗ ( y ) = −∞ for y <
0, so that ( v ∗ t ) ′∞ = 1 . Itis also easy to see that µ ⊥ b Q , hence in the Lebesgue decomposition with respect to b Q, µ a = 0 and µ s = µ . Hence F ( µ | b Q ) = 1+1 µ ( K ) = 2. At the same time we see that taking ϕ N ∈ C b ( K ) definedvia ϕ N ( −
2) = ϕ N (2) = 0 , ϕ N (0) = − N (observe that for N sufficiently large u ( ϕ N ) / ∈ C b ( K )) wehave sup ϕ ∈C b ( K ) (cid:18)Z K ϕ d µ − Z K v ( ϕ ) d b Q (cid:19) = sup ϕ ∈C b ( K ) (cid:18)Z K u ( ϕ ) d b Q − Z K ϕ d µ (cid:19) ≥ sup N (cid:18)Z K u ( ϕ N ) d b Q − Z K ϕ N d µ (cid:19) ≥ sup N (cid:18) (0) 12 + (0) 12 − ( − N ) (cid:19) = + ∞ . In this section we suppose the following requirements are fulfilled:
Standing Assumption 4.1.
Ω := K × · · · × K T for compact sets K , . . . , K T ⊆ R and K = { x } ; the functional c : Ω → ( −∞ , + ∞ ] is lower semicontinuous and d : Ω → [ −∞ , + ∞ ) isupper semicontinuous; Mart(Ω) = ∅ ; b Q ∈ Mart(Ω) is a given probability measure with marginals b Q , . . . , b Q T ; c, d ∈ L ( b Q ) . .1 Subhedging and Superhedging As it will become clear from the proofs, in all the results in Section 4.1 the functional U is realvalued on the whole E , that is dom( U ) = E . Thus we will exploit Theorem 2.4 and Corollary 2.8,in particular (48) and (49), in the case dom( U ) = dom( V ) = E .We set for ϕ t ∈ C b ( K t ) U b Q t ( ϕ t ) = sup α,λ ∈ R (cid:18)Z K t u t ( ϕ t ( x t ) + α Id t ( x t ) + λ )d b Q t ( x t ) − ( αx + λ ) (cid:19) , (59) V b Q t ( ϕ t ) = − U b Q t ( − ϕ t ) = inf α,λ ∈ R (cid:18)Z K t v t ( ϕ t ( x t ) + α Id t ( x t ) + λ )d b Q t ( x t ) + ( αx + λ ) (cid:19) . We observe that Assumption 3.6 does not impose that the functions u t are real valued on thewhole R . Nevertheless, for the functionals U b Q t , V b Q t we have: Lemma 4.2.
Under Assumption 3.6, for each t = 0 , . . . , T U b Q t and V b Q t are real valued on C b ( K t ) and null in .2. U b Q t and V b Q t are concave and convex respectively, and both nondecreasing.3. U b Q t and V b Q t are stock additive on C b ( K t ) , namely for every α t , λ t ∈ R and ϕ t ∈ C b ( K t ) U b Q t ( ϕ t + α t Id t + λ t ) = U b Q t ( ϕ t ) + α t x + λ t , V b Q t ( ϕ t + α t Id t + λ t ) = V b Q t ( ϕ t ) + α t x + λ t . Proof.
Since V b Q t ( ϕ t ) = − U b Q t ( − ϕ t ) , w.l.o.g. we prove the claims only for U b Q t . Clearly U b Q t ( ϕ t ) > −∞ , as we may choose λ t ∈ R so that ( ϕ t + 0Id t + λ t ) ∈ dom( u ) ⊇ [0 , + ∞ ). Furthermore, U b Q t ( ϕ t ) ∈ ∂U t (0) ≤ sup α,λ ∈ R (cid:18)Z K t ( ϕ t + α Id t + λ ) d b Q t − ( αx + λ ) (cid:19) b Q ∈ Mart(Ω) = sup α,λ ∈ R (cid:18)Z K t ϕ t d b Q t + ( αx + λ − αx − λ ) (cid:19) ≤ k ϕ t k ∞ . Finally, 0 = R K t u (0) d b Q t ≤ U b Q t (0) ≤ k k ∞ .Item 2: trivial from the definitions. Item 3: we see that U b Q t ( ϕ t + α t Id t + λ t ) = sup α ∈ R λ ∈ R (cid:18)Z K t u t ( ϕ t ( x t ) + ( α + α t ) x t + ( λ + λ t )) d b Q t ( x t ) − ( αx + λ ) (cid:19) = sup α ∈ R λ ∈ R (cid:18)Z K t u t ( ϕ t ( x t ) + ( α + α t ) x t + ( λ + λ t )) d b Q t ( x t ) − (( α t + α ) x + ( λ t + λ )) (cid:19) + α t x + λ t , in which we recognize the definition of U b Q t ( ϕ t ) + α t x + λ t .As in [6], in the next two Corollaries we suppose that the elements in E t represent portfoliosobtained combining call options with maturity t , units of the underlying stock at time t ( x t ) anddeterministic amounts, that is E t consists of all the functions in C b ( K t ) with the following form: ϕ t ( x t ) = a + bx t + N X n =1 c n ( x t − K n ) + , for a, b, c n , k n ∈ R , x t ∈ K t and take E = E × · · ·× E T . As shown in the proof, one could as well take E = C b ( K ) × · · ·× C b ( K T )preserving validity of (60), (61), (62) and (63). 23 orollary 4.3. Take u , . . . , u T satisfying Assumption 3.6, and suppose dom( u ) = · · · = dom( u T ) = R . Then the following equalities hold: inf Q ∈ Mart(Ω) E Q [ c ( X )] + T X t =0 D v ∗ t , b Q t ( Q t ) ! = sup ( T X t =0 U b Q t ( ϕ t ) | ϕ ∈ S sub ( c ) ) (60)sup Q ∈ Mart(Ω) E Q [ d ( X )] − T X t =0 D v ∗ t , b Q t ( Q t ) ! = inf ( T X t =0 V b Q t ( ϕ t ) | ϕ ∈ S sup ( d ) ) (61) Proof.
We prove (60), since (61) can be obtained in a similar fashion. Set U ( ϕ ) = P Tt =0 U b Q t ( ϕ t )for ϕ ∈ E . We observe that E t consists of all piecewise linear functions on K t , which are norm densein C b ( K t ). By Lemma 4.2 for each t = 0 , . . . , T the monotone concave functional ϕ t U b Q t ( ϕ t )is actually well defined, finite valued, concave and nondecreasing on the whole C b ( K t ) . Hence, bythe Extended Namioka-Klee Theorem (see [10]) it is norm continuous on C b ( K t ) and we can take E = C b ( K t ) × · · · × C b ( K T ) in place of E × · · · × E T in the RHS of (60) and prove equality to LHSin this more comfortable case (notice that S sub ( c ) depends on E ). We also observe that in this casewe are in Setup 3.1. Define D as in (26) with M = T . Using the facts that if ϕ t ∈ E t , α, λ ∈ R then ( ϕ t + α Id t + λ ) ∈ E t , that Q ∈ Mart(Ω) and that v t ( · ) := − u t ( −· ) one may easily check that D ( Q ) := sup ϕ ∈E U ( ϕ ) − T X t =0 Z K t ϕ t d Q t ! = sup ϕ ∈E T X t =0 Z K t u t ( ϕ t ( x t )) d b Q t ( x t ) − T X t =0 Z K t ϕ t d Q t ! = T X t =0 sup ψ t ∈E t (cid:18)Z K t ψ t d Q t − Z K t v t ( ψ t ( x t )) d b Q t ( x t ) (cid:19) = T X t =0 D v ∗ t , b Q t ( Q t ) , ∀ Q ∈ Mart(Ω)where the last equality follows from Proposition 3.8 Equation (57). The Standing Assumption 2.1is satisfied. Indeed, from Assumption 3.6 we have v ∗ (1) , . . . , v ∗ T (1) < + ∞ , hence D v ∗ t , b Q t ( b Q t ) = R K t v ∗ t (cid:16) d b Q t d b Q t (cid:17) d b Q t < + ∞ and therefore b Q ∈ dom( D ). Recalling that c ∈ L ( b Q ) , this in turnsyields b Q ∈ N = (cid:8) µ ∈ Meas(Ω) ∩ dom( D ) | R Ω c d µ < + ∞ (cid:9) . Moreover, by Lemma 4.2 Item 1,dom( U ) = E , and for every µ ∈ Meas(Ω) D ( µ ) ≥ U (0) − D is lower bounded on thewhole Meas(Ω). We conclude that U and D satisfy the assumptions of Theorem 2.4.Using Lemma 3.2 and the fact that U b Q , . . . , U b Q T are cash additive we get S U ( ϕ ) = P Tt =0 S U b Qt ( ϕ t ) = P Tt =0 U b Q t ( ϕ t ) = U ( ϕ ), and by Corollary 2.8 Equation (50) we obtaininf Q ∈ Mart(Ω) E Q [ c ( X )] + T X t =0 D v ∗ t , b Q t ( Q t ) ! = sup ( T X t =0 U b Q t ( ϕ t ) | ϕ ∈ S sub ( c ) ) . Remark . At this point we can add some more discussion on the compactness condition of theunderlying sets K , . . . , K T in Assumption 4.1. In the classical non-robust setup the requirementof (essential) boundedness of the underlying stock is quite common. We observe that in ourcanonical setup for the underlying space, compactness is essentially tantamount to requiring thatthe stock (Id t ) t is bounded (everywhere). But we can actually say more: when D is taken as inCorollary 4.3, we automatically have that Q t ≪ b Q t , whenever D ( Q ) < + ∞ . If the marginals b Q t ,for every t = 0 , . . . , T , satisfy Id t ∈ L ∞ ( b Q t ) we then get automatically that Id t ∈ L ∞ ( Q t ), with k Id t k L ∞ ( Q t ) ≤ k Id t k L ∞ ( b Q t ) , for any Q ∈ dom( D ). Thus, it is possible to reformulate the hypotheses24n Corollary in 4.3 using K = · · · = K T = R but requesting that the marginals b Q , . . . , b Q T havecompact support. A version of Corollary 4.3 should hold even without the compactness requirementin Assumption 4.1, but it is a delicate issue. It would require a modification of the settings, as theset of continuous functions would not work well any more, and a generalization of [42] Theorem2.7, which is not trivial. We leave these interesting issues for future research.We stress the fact that in Corollary 4.3 we assume that all the functions u , . . . , u T are real valuedon the whole R . A more general result can be obtained when weakening this assumption, but itrequires an additional assumption on the marginals of b Q . Corollary 4.5.
Suppose Assumption 3.6 is fulfilled. Assume b Q , . . . , b Q T have full support on K , . . . , K T respectively. Then Equations (60) , (61) hold true replacing D v ∗ t , b Q t ( Q t ) with F t ( Q t | b Q t ) .Proof. The proof can be carried over almost literally as the proof of Corollary 4.3, with the excep-tion of replacing the reference to Proposition 3.8 with the reference to Proposition 3.9.
Remark . Observe that we are requesting the full support property on K , . . . , K T with respectto their induced (Euclidean) topology. In particular, this means that whenever k t ∈ K t is anisolated point, b Q t ( { k t } ) >
0. This is consistent with our assumption K = { x } , which implesProb( K ) reduces to the Dirac mesure, Prob( K ) = { δ { x } } ..We now take u t ( x ) = x for each t = 0 , . . . , T , and get U b Q t ( ϕ t ) = V b Q t ( ϕ t ) = E b Q t [ ϕ t ]. Hence withan easy computation we have D v ∗ t , b Q t ( Q t ) = Q t ≡ b Q t + ∞ otherwise. for all Q ∈ Mart(Ω) . Recalling that Mart( b Q , . . . , b Q T ) = { Q ∈ Mart(Ω) | Q t ≡ b Q t ∀ t = 0 , . . . , T } , from Corollary 4.3we can recover the following result of [6] (under more stringent assumptions on the underlyingspace). Corollary . The following equalities hold: inf Q ∈ Mart( b Q ,..., b Q T ) E Q [ c ] = sup ( T X t =0 E b Q t [ ϕ t ] | ϕ ∈ S sub ( c ) ) (62)sup Q ∈ Mart( b Q ,..., b Q T ) E Q [ d ] = inf ( T X t =0 E b Q t [ ϕ t ] | ϕ ∈ S sup ( d ) ) (63) Corollary . The following equalities hold: inf Q ∈ Mart(Ω) E Q [ c ] = sup { m ∈ R | m ∈ S sub ( c ) } := Π sub ( c ) , (64)sup Q ∈ Mart(Ω) E Q [ d ] = inf { m ∈ R | m ∈ S sup ( d ) } := Π sup ( d ) . (65) Proof.
We take E = · · · = E T = R and E = E × · · · × E T = R T +1 . We first focus on (64). Foreach ϕ ∈ E with ϕ = [ m , . . . , m T ], m ∈ R T +1 we select U ( ϕ ) := P Tt =0 m t (we notice that when25 = R T +1 , u t ( x t ) = x t , t = 0 , ..., T, and b Q ∈ Mart(Ω), the functional U b Q t defined in (59) is givenby U b Q t ( m t ) = m t and so U ( m ) = P Tt =0 U b Q t ( m t ) = P Tt =0 m t for all m ∈ E ). Then applying thedefinition of D in (26) we get D ( γ ) = γ ∈ ca(Ω) s.t. γ (Ω) = 1+ ∞ otherwise. . In particuar D ( Q ) = 0 for every Q ∈ Mart(Ω). Moreover we observe that S U ( ϕ ) = U ( ϕ ) for every ϕ ∈ E . Applying Corollary 2.8 (whose assumptions are clearly satisfied here), from Equation (50)we get thatinf Q ∈ Mart(Ω) E Q [ c ] = sup ( T X t =0 m t | m , . . . , m T ∈ R s.t. ∃ ∆ ∈ H with T X t =0 m t + I ∆ ≤ c ) . We recognize in the RHS above the RHS of (64). Equation (65) can be obtained in a similar wayusing Corollary 2.8 Equation (51).
In this section we change our perspective. Instead of starting from a given U , we will give aparticular form of the penalization term D and proceed in identifying the corresponding U in thespirit of Remark 2.2. For each t = 0 , . . . , T we suppose that finite sequences ( c t,n ) ≤ n ≤ N t ⊆ R and ( f t,n ) ≤ n ≤ N t ⊆ C b ( K t ) are given. The functions ( f t,n ) ≤ n ≤ N t ⊆ C b ( K t ) represent payoffs ofoptions whose prices ( c t,n ) ≤ n ≤ N t ⊆ R are known from the market. Furthermore, we considerpenalization functions Ψ n,t : R → ( −∞ , + ∞ ] which are convex, null in 0, symmetric in 0, properand lower semicontinuous. For such functions we define the conjugates Ψ ∗ t,n : R → ( −∞ , + ∞ ] asΨ ∗ t,n ( y ) = sup x ∈ R ( xy − Ψ t,n ( x )).Define Mart t ( K t ) = { γ t ∈ Prob( K t ) | ∃ Q ∈ Mart(Ω) with γ t ≡ Q t } ⊆ ca( K t )and for γ t ∈ ca( K t ) D Ψ t ( γ t ) := P N t n =1 Ψ t,n (cid:16)(cid:12)(cid:12)(cid:12)R K t f t,n d γ t − c t,n (cid:12)(cid:12)(cid:12)(cid:17) for γ t ∈ Mart t ( K t )+ ∞ otherwise Proposition . Suppose that the martingale measure b Q ∈ Mart(Ω) in Standing Assumption 4.1also satisfies (cid:12)(cid:12)(cid:12)R K t f t,n d b Q t − c t,n (cid:12)(cid:12)(cid:12) ∈ dom(Ψ t,n ) for every t = 0 , . . . , T , n = 0 , . . . , N t . Then settingfor n = 1 , . . . , N t , t = 0 , . . . , T g t,n := f t,n − c t,n ∈ C b ( K t ) we have Subhedging Duality : inf Q ∈ Mart(Ω) E Q [ c ] + T X t =0 D Ψ t ( Q t ) ! = sup ( T X t =0 U Ψ t ( ϕ t ) | ϕ ∈ S sub ( c ) ) , (66) where U Ψ t ( ϕ t ) := sup y t ∈ R Nt Π sub ϕ t + N t X n =1 y t,n g t,n ! − N t X n =1 Ψ ∗ t,n ( | y t,n | ) ! is a stock additive functional and Π sub is given in (64) . uperhedging Duality : sup Q ∈ Mart(Ω) E Q [ d ] − T X t =0 D Ψ t ( Q t ) ! = inf ( T X t =0 V Ψ t ( ϕ t ) | ϕ ∈ S sup ( d ) ) , (67) where V Ψ t ( ϕ t ) = − U Ψ t ( − ϕ t ) = inf y t ∈ R Nt Π sup ϕ t − N t X n =1 y t,n g t,n ! + N t X n =1 Ψ ∗ t,n ( | y t,n | ) ! is a stock additive functional and Π sup is given in (65) . Before providing a proof, we state an auxiliary result.
Lemma . Suppose K , . . . , K t ⊆ R are compact. Then Mart t ( K t ) is σ (ca( K t ) , C b ( K t )) -compact.Proof. We see that Mart(Ω) is a σ (ca(Ω) , C b (Ω))-closed subset of the σ (ca(Ω) , C b (Ω))-compact setProb(Ω) (which is compact since Ω is a compact Polish Space, see [2] Theorem 15.11), hence it iscompact himself. Mart t ( K t ) is then the image of a compact set via the marginal map γ γ t whichis σ (ca(Ω) , C b (Ω)) − σ (ca( K t ) , C b ( K t )) continuous, hence it is σ (ca( K t ) , C b ( K t )) compact. Proof of Proposition 4.9.
We focus on (66) first.
STEP 1 : for any t ∈ { , . . . , T } we prove the following: the functional D Ψ t is σ (ca( K t ) , C b ( K t ))-lower semicontinuous and for every ϕ t ∈ C b ( K t ) its Fenchel-Moreau (convex) conjugate satisfies V Ψ t ( ϕ t ) := sup γ t ∈ ca( K t ) (cid:18)Z K t ϕ t d γ t − D Ψ t ( γ t ) (cid:19) = inf y t ∈ R Nt Π sup ϕ t − N t X n =1 y t,n g t,n ! + N t X n =1 Ψ ∗ t,n ( | y t,n | ) ! , and thus U Ψ t ( ϕ t ) := − V Ψ t ( − ϕ t ) = sup y t ∈ R Nt Π sub ϕ t + N t X n =1 y t,n g t,n ! − N t X n =1 Ψ ∗ t,n ( | y t,n | ) ! . (68)We observe that D Ψ t is σ (ca( K t ) , C b ( K t ))-lower semicontinuous (it is a sum of functions, each beingcomposition of a lower semicontinuous function and a continuous function on Mart t ( K t ) which is σ (ca( K t ) , C b ( K t ))-compact by Lemma 4.10). We now need to compute V Ψ t ( ϕ t ) = sup γ t ∈ ca( K t ) (cid:18)Z K t ϕ t d γ t − D Ψ t ( γ t ) (cid:19) = sup Q t ∈ Mart t ( K t ) (cid:18)Z K t ϕ t d Q t − D Ψ t ( Q t ) (cid:19) . Recall now that from Fenchel-Moreau Theorem and symmetry Ψ t,n ( | x | ) = Ψ t,n ( x ) = sup y ∈ R ( xy − Ψ ∗ t,n ( y )) = sup y ∈ R ( xy − Ψ ∗ t,n ( | y | )). Hence, setting g t,n = f t,n − c t,n , V Ψ t ( ϕ t ) = sup Q t ∈ Mart t ( K t ) Z K t ϕ t d Q t − N t X n =1 sup y t,n ∈ R (cid:18) y t,n Z K t g t,n d Q t − Ψ ∗ t,n ( | y t,n | ) (cid:19)! = sup Q t ∈ Mart t ( K t ) Z K t ϕ t d Q t − N t X n =1 sup y t,n ∈ dom(Ψ ∗ t,n ) (cid:18) y t,n Z K t g t,n d Q t − Ψ ∗ t,n ( | y t,n | ) (cid:19)! = sup Q t ∈ Mart t ( K t ) inf y t ∈ dom Z K t ϕ t − N t X n =1 y t,n g t,n ! d Q t + N t X n =1 Ψ ∗ t,n ( | y t,n | ) ! =: sup Q t ∈ Mart( K t ) inf y t ∈ dom T ( y t , Q t ) , ∗ t, ) × · · · × dom(Ψ ∗ t,N t ) ⊆ R N t . We now see that T is real valued on dom × Mart t ( K t ), is convex in the first variable and concave in the second. Moreover, {T ( y t , · ) ≥ C } is σ (Mart t ( K t ) , C b ( K t ))-closed in Mart t (Ω) for every y t ∈ dom, and Mart t ( K t ) is σ (Mart t ( K t ) , C b ( K t ))-compact (by Lemma 4.10). As a consequence T ( y t , · ) is σ (Mart t ( K t ) , C b ( K t ))-lower semicontinuouson Mart t ( K t ). We can apply [47] Theorem 3.1 with A = dom and B = Mart t ( K t ) endowed withthe topology σ (Mart t ( K t ) , C b ( K t )), and interchange inf and sup. From our previous computationswe then get V Ψ t ( ϕ t ) = sup Q t ∈ Mart t ( K t ) inf y t ∈ dom T ( y t , Q t ) = inf y t ∈ dom sup Q t ∈ Mart t ( K t ) T ( y t , Q t )= inf y t ∈ dom sup Q t ∈ Mart t ( K t ) Z K t ϕ t − N t X n =1 y t,n g t,n ! d Q t + N t X n =1 Ψ ∗ t,n ( | y t,n | ) ! = inf y t ∈ dom sup Q ∈ Mart(Ω) Z Ω ϕ t − N t X n =1 y t,n g t,n ! d Q + N t X n =1 Ψ ∗ t,n ( | y t,n | ) ! ( ) = inf y t ∈ dom Π sup ϕ t − N t X n =1 y t,n g t,n ! + N t X n =1 Ψ ∗ t,n ( | y t,n | ) ! = inf y t ∈ R Nt Π sup ϕ t − N t X n =1 y t,n g t,n ! + N t X n =1 Ψ ∗ t,n ( | y t,n | ) ! . Equation (68) can be obtained with minor manipulations.
STEP 2 : conclusion. We are clearly in the setup of Theorem 2.4 with D given as in Setup 3.4from D Ψ0 , . . . , D Ψ T , and by definition dom( D Ψ t ) ⊆ Prob( K t ) for each t = 0 , . . . , T . Using Lemma 3.5Item 2, together with the compuatations in STEP 1 and the fact that clearly S U Ψ t ≡ U Ψ t by cashadditivity of U Ψ t , we get the desired equality from Corollary 2.8 Equation (50): observe that ourassumption on the existence of the measure b Q ∈ Mart(Ω) guarantees, together with the fact that D is clearly lower bounded on Meas(Ω), that the hypotheses of Theorem 2.4 are satisfied (henceso is Standing Assumption 2.1). Equality (67) can now be obtained similarly to (66). Remark . Our assumption of existence of a particular b Q ∈ Mart(Ω) in Proposition 4.9 expressesthe fact that we are assuming our market prices ( c t,n ) t,n are close enough to those given byexpectations under some martingale measure. Remark . Proposition 4.9 covers a wide range of penalizations. For example, we might usepower-like penalizations, i.e. Ψ t,n ( x ) = | x | pt,n p t,n for p t,n ∈ (1 , + ∞ ). In such a case Ψ ∗ t,n ( x ) = | x | qt,n q t,n for p t,n + q t,n = 1. Alternatively, we might impose a threshold for the fitting, that is take intoaccount only those martingale measure Q such that (cid:12)(cid:12)R Ω f t,n d Q t − c t,n (cid:12)(cid:12) ≤ ε t,n for some ε t,n > x, y ∈ R Ψ t,n ( x ) = | x | ≤ ε t,n + ∞ otherwise = ⇒ Ψ ∗ t,n ( y ) = | y | ε t,n We now explore the versatility of Corollary 2.8, which can be used beyond the semistatic subhedgingand superhedging problems in Section 4.1. Note that in Section 4.1 we chose for static hedging28ortfolios the sets E t , t = 0 , . . . , T consisting of deterministic amounts, units of underlying stock attime t and call options with different strike prices and same maturity t . This affected the primalproblem in the fact that the penalty D turned out to depend solely on the (one dimensional)marginals of b Q . Nonetheless, Theorem 2.4 allows to choose for each t = 0 , . . . , T a subspace E t ⊆ C b ( K × · · · × K t ), potentially allowing to consider also Asian and path dependent options inthe sets E t . We expect that this would translate in the penalty D depending no more only on theone dimesnional marginals of b Q . The study of these less restrictive, yet technically more complexcases is left for future research.In the following we will treat a slightly different problem, which however helps understanding howalso the extreme case E t = C b ( K × · · · × K t ) , t = 0 , . . . , T is of interest. Theorem 2.4 yields the following dual robust representation of the generalized Optimized CertaintyEquivalent associated to the indirect utility function. We stress here the fact that, again, b Q ∈ Mart(Ω) is a fixed martingale measure, but we will not focus anymore on its marginals only, aswill become clear in the following.
Theorem . Take u : R → R such that u = . . . , u T := u satisfy Assumption 3.6 and let v ∗ bedefined in (55) with u in place of u t . Let U H b Q : C b (Ω) → R be the associated indirect utility U H b Q ( ϕ ) := sup ∆ ∈H Z Ω u ( ϕ + I ∆ ) d b Q . and S U H b Q be the associated Optimized Certainty Equivalent defined according to (24) , namely S U H b Q ( ϕ ) := sup ξ ∈ R (cid:16) U H b Q ( ϕ + ξ ) − ξ (cid:17) ϕ ∈ C b (Ω) . Then for every c ∈ C b (Ω) S U H b Q ( c ) = inf Q ∈ Mart(Ω) (cid:18)Z Ω c d Q + D b Q ( Q ) (cid:19) where for µ ∈ Meas(Ω) D b Q ( µ ) := R Ω v ∗ (cid:16) d µ d b Q (cid:17) d b Q if µ ≪ b Q + ∞ otherwise . Proof.
Take E t = C b ( K × · · · × K t ) for t = 0 , . . . , T . Define for ψ ∈ E = E × ... × E T U ( ψ ) := U H b Q (cid:16)P Tt =0 ψ t (cid:17) . Clearly U ( ψ ) > −∞ for any ψ ∈ E , and since b Q ∈ Mart(Ω) and u ( x ) ≤ x for all x ∈ R we also have U ( ψ ) ≤ P Tt =0 k ϕ t k ∞ < + ∞ . Moreover it is easy to verify that defining D asin (26) for any Q ∈ Mart(Ω) we have D ( Q ) := sup ψ ∈E U ( ψ ) − Z Ω T X t =0 ψ t ! d Q ! = sup ϕ ∈C b (Ω) (cid:18)Z Ω u ( ϕ ) d b Q − Z Ω ϕ d Q (cid:19) and arguing as in Proposition 3.8 we get D ( Q ) = D b Q ( Q ). From the fact that u ( x ) ≤ x for every x ∈ R we have v ∗ (1) < + ∞ , hence from Assumption 4.1 b Q ∈ dom( D ). This and c ∈ L ( b Q ) in29urns yields b Q ∈ N (see (36)). Moreover dom( U ) = E and by definition of D for any µ ∈ Meas(Ω)we have D ( µ ) ≥ U (0) − D is lower bounded on the whole Meas(Ω). We conclude that U and D satisfy the assumptions of Theorem 2.4. We then getinf Q ∈ Mart(Ω) (cid:16) E Q [ c ( X )] + D b Q ( Q ) (cid:17) = inf Q ∈ Mart(Ω) ( E Q [ c ( X )] + D ( Q )) = sup ∆ ∈H sup ψ ∈ Φ ∆ ( c ) S U ( ψ ) . Observe now that S U satisfies S U ( ψ ) := sup λ ∈ R T +1 U ( ψ + λ ) − T X t =0 λ t ! = sup λ ∈ R T +1 U H b Q T X t =0 ψ t + T X t =0 λ t ! − T X t =0 λ t ! = sup ξ ∈ R U H b Q T X t =0 ψ t + ξ ! − ξ ! =: S U H b Q T X t =0 ψ t ! .S U H b Q : C b (Ω) → R is (IA) and is nondecreasing, thussup ∆ ∈H sup ψ ∈ Φ ∆ ( c ) S U H b Q T X t =0 ψ t ! = sup ∆ ∈H sup ψ ∈ Φ ∆ ( c ) S U H b Q T X t =0 ψ t + I ∆ ! = S U H b Q ( c )by definition of Φ ∆ ( c ) and since c ∈ C b (Ω). 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 Borelsigma algebra B (Ω) generated by its open sets. A set function µ : B (Ω) → R is a finite signedmeasure if µ ( ∅ ) = 0 and µ is σ -additive. A finite measure µ is a finite signed measure such that µ ( B ) ≥ B ∈ B (Ω). A finite measure µ such that µ (Ω) = 1 will be called a probabilitymeasure . Recall from Section 2 the notations for ca(Ω) , Meas(Ω), Prob(Ω). The following resultis well known, see e.g. [11] 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 > there exists a compact K ε ⊆ B such that µ ( B \ K ε ) ≤ ε . A measure µ ∈ Meas(Ω) has full support if µ ( A ) > A ⊆ Ω. Wealso introduce for M ∈ N , M ≥ C b (Ω) := C b (Ω , R ) = { ϕ : Ω → R | ϕ is bounded and continuous on Ω } , C b (Ω , R M ) := ( C b (Ω)) M = { ϕ : Ω → R M | ϕ is bounded and continuous on Ω } , LSC b (Ω) := LSC b (Ω , R ) = { ϕ : Ω → R | ϕ is bounded and lower semicontinuous on Ω } . Given a vector subspace
E ⊆ C b (Ω , R M +1 ) we will consider the dual pair (ca(Ω) , C b (Ω , R M +1 ))with pairing given by the bilinear functional ( γ, ϕ ) R Ω (cid:16)P Mm =0 ϕ m (cid:17) d γ . We will induce onca(Ω) the topology σ (ca(Ω) , E ), which is the coarsest topology on ca(Ω) making the functional γ R Ω (cid:16)P Mm =0 ϕ m (cid:17) d γ continuous for each ϕ ∈ E . Similarly, we will induce on E the topology σ ( E , ca(Ω)) which is the coarsest topology on E making the functional γ R Ω (cid:16)P Mm =0 ϕ m (cid:17) d γ continuous for each γ ∈ ca(Ω). 30 .1.2 Weak and Narrow Topology Definition
A.2 . The
Weak Topology on Meas(Ω) is the coarsest (Hausdorff ) topology for which allmaps µ R Ω ϕ d µ are continuous, for all ϕ ∈ C b (Ω) . The Narrow Topology is the coarsest (Haus-dorff ) topology for which all maps µ R Ω ϕ d µ are lower semicontinuous, for all ϕ ∈ LSC b (Ω) .Remark A.3 . The weak topology on Meas(Ω) is the topology σ (Meas(Ω) , C b (Ω , R ), which is therelative topology σ (ca(Ω) , C b (Ω , R ) | Meas(Ω) induced by σ (ca(Ω) , C b (Ω , R ) on Meas(Ω) ⊆ ca(Ω) (see[2] Lemma 2.53). Proposition
A.4 . When Ω is a Polish Space, the weak and narrow topologies coincide.Proof. See [46] page 371.
Remark
A.5 . Even though the two topologies coincide in our setting, because of their differentdefinitions we will find more convenient to exploit the one or the other topology in our proofs.We now turn our attention to compactness issues in Meas(Ω) under the narrow topology. We recallfirst that a family Γ ⊆ Meas(Ω) is bounded if sup µ ∈ Γ µ (Ω) < + ∞ and tight if for every ε > K ε ⊆ Ω such that sup µ ∈ Γ µ (Ω \ K ε ) ≤ ε . The following generalization ofProkhorov’s Theorem holds: Theorem
A.6 . If a subset Γ ⊆ Meas(Ω) is bounded and tight, it is relatively compact in the narrowtopology.Proof.
See [46] Theorem 3 pg. 379.
A.2 Auxiliary Results and Proofs
Lemma
A.7 . Take compact K , . . . , K T ⊆ R , and suppose that K = { x } and card ( K t +1 ) ≥ card ( K t ) for every t = 0 , . . . , T − . Take E = E × · · · × 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 (8) , such that P Tt =0 ϕ t = P Tt =0 ψ t + I ∆ . Then there exist constants k , . . . , k T , h , . . . , 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 t =0 S t ( ϕ t ) = T X t =0 S t ( ψ t ) . and for V := P Tt =0 E t + I (see (9) ) the map v = T X t =0 ϕ t + I ∆ S ( v ) := T X t =0 S t ( ϕ t ) is well defined on V , (CA) and (IA).Proof. STEP 1 : we prove that if P Tt =0 ϕ t = P Tt =0 ψ t + I ∆ then ∆ = [∆ , . . . , ∆ T − ] ∈ H is a deterministicvector ∆ ∈ R T . If card( K T ) = 1 this is trivial. We can then suppose card( K T ) ≥ ϕ T ( x T ) − ψ T ( x T ) = T − X t =0 ( ψ ( x t ) − ϕ t ( x t )) + T − X t =0 ∆ t ( x , . . . , x t )( x t +1 − x t )+31∆ T − ( x , . . . , x T − )( x T − x T − ) = f ( x , . . . , x T − ) + ∆ T − ( x , . . . , x T − ) x T for some function f . If ∆ T − 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 uniquesolution, contradicting the fact that all the equalities need to hold on the whole K , . . . , K T andin particular for two different values of x T . We proceed one step backward. If card( K T − ) = 1,the claim trivially follows, given our previous step. If card( K T − ) ≥
2, similarly to the previouscomputation ϕ T − ( x T − ) − ψ T − ( x T − ) = X s = T − ( ψ s ( x s ) − ϕ s ( x s )) + T − X t =0 ∆ t ( x , . . . , x t )( x t +1 − x t )++∆ T − ( x , . . . , x T − )( x T − − x T − ) + ∆ T − ( x T − x T − )= f ( x s , s = T −
1) + (∆ T − ( x , . . . , x T − ) − ∆ T − ) x T − . An argument similar to the one we used in the previous time step shows that ∆ T − ( x , . . . , x T − ) − ∆ T − is constant, hence so is ∆ T − . Our argument can be clearly be iterated up to ∆ . STEP 2 : we prove existence of the vectors k, h ∈ R T +1 , as stated in the Lemma. From Step 1 itis clear that there exist constants k , . . . , k T such that I ∆ ( x ) = P Tt =0 k t x t . Hence P Tt =0 ϕ t ( x t ) = P Tt =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 , . . . , k T , h , . . . , h T ∈ R satisfy ourrequirements. The last claim P Tt =0 S Ut ( ϕ t ) = P Tt =0 S Ut ( ψ t ) is then an easy consequence of stockadditivity. STEP 3 : well posedness and properties of S . Observe that whenever ϕ, ψ ∈ E , ∆ , H ∈ H aregiven with P Tt =0 ϕ t + I ∆ = P Tt =0 ψ t + I H we have by Steps 1-2 that P Tt =0 S Ut ( ϕ t ) = P Tt =0 S Ut ( ψ t ) . As a consequence, S is well defined. Cash Additivity is inherited from S , . . . , S T while IntegralAdditivity is trivial from the definition. Proof of Lemma 3.2.
We will only focus on (52), since the remaining claims are easily checked.We have that D ( γ ) = sup ϕ ∈E T X t =0 U t ( ϕ t ) − T X t =0 Z K t ϕ t d γ ! = T X t =0 sup ϕ t ∈E t (cid:18) U t ( ϕ t ) − Z K t ϕ t d γ (cid:19) = T X t =0 D t ( γ t ) = T X t =0 D t ( γ ) . As to the second claim in (52), we observe thatsup ξ ∈ R T +1 U ( ϕ + ξ ) − T X t =0 ξ t ! = T X t =0 sup ξ ∈ R ( U t ( ϕ t + ξ ) − ξ ) = T X t =0 S U t ( ϕ t ) . roof of Lemma 3.5. Item 1 . For each t = 0 , . . . , T D t ( γ ) = D t ◦ π t ( γ ) , where D t is σ (ca( K t ) , E t )-lower semicontinuousand π t , the projection to the t -th marginal, is σ (ca(Ω) , E ) − σ (ca( K t ) , E t ) continuous. Hence, foreach t = 0 , . . . , T γ
7→ D t ( γ ) is σ (ca(Ω) , E )-lower semicontinuous. Lower semicontinuity of D is thena consequence of the fact that the sum of lower semicontinuous functions is lower semicontinuous. Item 2, equation (53) . We have that for ψ = − ϕ − U ( ϕ ) = V ( ψ ) = sup µ ∈ ca(Ω) Z Ω T X t =0 ψ t ! d µ − D ( µ ) ! = sup µ ∈ ca(Ω) T X t =0 (cid:18)Z K t ψ t d µ − D t ( µ t ) (cid:19) ( i ) = sup ( T X t =0 (cid:18)Z K t ψ t d γ t − D t ( γ t ) (cid:19) | γ ∈ ca(Ω) with γ t ∈ Prob( K t ) ∀ t = 0 , . . . , T ) ( ii ) = sup ( T X t =0 (cid:18)Z K t ψ t d Q t − D t ( Q t ) (cid:19) | [ Q , . . . , Q T ] ∈ Prob( K ) × · · · × Prob( K T ) ) = T X t =0 sup Q t ∈ Prob( K t ) (cid:18)Z K t ψ t d Q t − D t ( Q t ) (cid:19) ( iii ) = T X t =0 sup γ t ∈ ca( K t ) (cid:18)Z K t ψ t d γ t − D t ( γ t ) (cid:19) = T X t =0 V t ( ψ t ) = T X t =0 − U t ( ϕ t )Note that ( i ) follows from dom( D ) ⊆ Z := { γ ∈ ca(Ω) | γ t ∈ Prob( K t ) ∀ t = 0 , . . . , T } . In ( ii )we used the facts that any vector of probability measures ( Q , . . . , Q T ) with Q t ∈ Prob( K t ), t = 0 , . . . , T , identifies γ := Q ⊗ · · · ⊗ Q T ∈ Z with D ( γ ) = P Tt =0 D t ( Q t ) (note that this does nothold for a general vector of signed measures, which is why we need the additional assumption onthe domains of the penalization functionals for Item 2) and that for every γ ∈ Z , setting Q t := γ t ∈ Prob( K t ), we have D ( γ ) = P Tt =0 D t ( Q t ). Equality ( iii ) follows from dom( D t ) ⊆ Prob( K t )for each t = 0 , . . . , T . Item2, equation (54) . The argument is identical to the one in the proof of Lemma 3.2, usingthe additive structure of U we obtained in the previous step of the proof. Proof of Proposition 3.8.
We will use [42] Theorem 2.7 and [42] Remark 2.8. To do so, let usrename F := v ∗ t ((see (55) for the definition of v ∗ ), which implies that F ◦ ( y ) := − F ∗ ( − y ) of[42] Equation (2.45) satisfies F ◦ ( y ) := − F ∗ ( − y ) = − v ∗∗ t ( − y ) = − v t ( − y ) = u t ( y ), by Fenchel-Moreau Theorem. All the assumptions of [42] Section 2.3 on F are satisfied, since for every y ≥ F ( y ) ≥ u t (0) − y = 0 and F (1) = sup x ∈ R ( u t ( x ) − x ) ≤ u t ( x ) ≤ x, ∀ x ∈ R ). Also,since dom( u t ) = R , lim y → + ∞ F ( y ) y = F ′∞ = + ∞ . We can then apply [42] Theorem 2.7 and [42]Remark 2.8, obtaining (57). We stress the fact that since u t is finite valued on the whole R , itis continuous there and for every ϕ t ∈ C b ( K t ), F ◦ ( ϕ t ) = u t ( ϕ t ) ∈ C b ( K t ), hence the additionalconstraint F ◦ ( ϕ t ) ∈ C b ( K t ) (below [42] (2.49)) would be redundant in our setup. Proof of Proposition 3.9.
We will exploit again [42] Theorem 2.7 and [42] Remark 2.8 (with u t in place of F ◦ ) , as we explain now. Since u t is nondecreasing, either its domain is in the form[ M, + ∞ ) or ( M, + ∞ ), with M ≤
0. Given a ϕ t ∈ C b ( K t ) and a µ ∈ Meas( K t )33 Either inf( ϕ t ( R )) > M , in which case u t ( ϕ t ) ∈ C b ( K t ) since u t is continuous on the interiorof its domain. • Or inf( ϕ t ( R )) < M , in which case { ϕ t < M } is open nonempty and hence has positive b Q t measure, as b Q t has full support. Thus R K t u t ( ϕ t ) d b Q t = −∞ . • Or inf( ϕ t ( R )) = M in which case u t ( ϕ t ) = lim ε ↓ u t (max( ϕ t , M + ε )) (since u t is nonde-creasing and upper semicontinuous) u t (max( ϕ t , M + ε )) ∈ C b ( K t ) (see first bullet) and byMonotone Convergence Theorem Z K t u t ( ϕ t ) d b Q t − Z K t ϕ t d µ = lim ε ↓ (cid:18)Z K t u t (max( ϕ t , M + ε )) d b Q t − Z K t max( ϕ t , M + ε ) d µ (cid:19) . Then we infer thatsup ϕ t ∈C b ( K t ) (cid:18)Z K t ϕ t d µ − Z K t v t ( ϕ t ) d b Q t (cid:19) = sup ϕ t ∈C b ( K t ) (cid:18)Z K t u t ( ϕ t ) d b Q t − Z K t ϕ t d µ (cid:19) = sup (cid:26)Z K t u t ( ϕ t ) d b Q t − Z K t ϕ t d µ | ϕ t , u t ( ϕ t ) ∈ C b ( K t ) (cid:27) , (69)and from [42] Theorem 2.7, [42] Remark 2.8 and from (69) we conclude the thesis. A.3 On Minimax Duality Theorem
The following theorem is stated, without the proof, in [42], Th. 2.4. For the sake of completenessand without claiming any originality, we here provide the short proof.
Theorem
A.8 (Minimax Duality Theorem) . Let
A, B be nonempty convex subsets of some vectorspaces and suppose A is endowed with a Hausdorff topology. Let L : A × B → R be a function suchthat1. a L ( a, b ) is convex and lower semicontinuous in A for every b ∈ B .2. b L ( a, b ) is concave in B for every a ∈ A When α := sup b ∈ B inf a ∈ A L ( a, b ) < + ∞ , suppose that there exist C > α and b ⋆ ∈ B such that { a ∈ A | L ( a, b ⋆ ) ≤ C } is compact in A . Then inf a ∈ A sup b ∈ B L ( a, b ) = sup b ∈ B inf a ∈ A L ( a, b ) . (70) Proof.
We start observing that in general inf a ∈ A sup b ∈ B L ( a, b ) ≥ sup b ∈ B inf a ∈ A L ( a, b ), hence if α = + ∞ then (70) trivially holds. We then assume α < + ∞ and modify the proof of [47] Theorem3.1. Let b , . . . , b N ∈ B be given and set b = b ⋆ . By [47] Lemma 2.1.(a), using f i ( · ) := L ( · , b i ) weget constants λ , . . . , λ N ≥ P Ni =0 λ i = 1 such thatinf a ∈ A (cid:18) max i =0 ,...,N L ( a, b i ) (cid:19) = inf a ∈ A N X i =0 λ i L ( a, b i ) ! ≤ inf a ∈ A L a, N X i =0 λ i b i ! ≤ sup b ∈ B inf a ∈ A L ( a, b ) = α , B to obtain the first inequality. We now observe that for all ε > a ∈ A such that a ∈ (cid:26) max i =0 ,...,N L ( a, b i ) ≤ α + ε (cid:27) = N \ i =0 { L ( a, b i ) ≤ α + ε } = N \ i =1 { L ( a, b i ) ≤ α + ε }∩{ L ( a, b ⋆ ) ≤ α + ε } . Hence for A ⋆ = { L ( a, b ⋆ ) ≤ α + ε } the family A εb := { a ∈ A ⋆ | L ( a, b ) ≤ α + ε } is a collection ofclosed subsets of A ⋆ having the finite intersection property. Now take ε > α + ε < C .Then A ⋆ is Hasudorff and compact, being a closed subset of the compact set { a ∈ A | L ( a, b ⋆ ) ≤ C } .As a consequence T b ∈ B A εb = ∅ . This yields the existence of an a ⋆ such that a ⋆ ∈ A ⋆ and L ( a ⋆ , b ) ≤ α + ε ∀ b ∈ B . Henceinf a ∈ A sup b ∈ B L ( a, b ) ≤ sup b ∈ B L ( a ⋆ , b ) ≤ ε + α and letting ε ↓ a ∈ A sup b ∈ B L ( a, b ) ≤ sup b ∈ B inf a ∈ A L ( a, b ) ≤ inf a ∈ A sup b ∈ B L ( a, b ) . References [1] B. Acciaio, M. Beiglb¨ock, F. Penkner, and W. Schachermayer. A model-free version of thefundamental theorem of asset pricing and the super-replication theorem.
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