On robust pricing-hedging duality in continuous time
aa r X i v : . [ q -f i n . M F ] J u l On robust pricing–hedging duality in continuous time ∗ Zhaoxu Hou † and Jan Ob l´oj ‡ Mathematical Institute, University of OxfordAWB, ROQ, Oxford OX2 6GG, UKJuly 7, 2015
Abstract
We pursue robust approach to pricing and hedging in mathematical finance. We considera continuous time setting in which some underlying assets and options, with continuous paths,are available for dynamic trading and a further set of European options, possibly with varyingmaturities, is available for static trading. Motivated by the notion of prediction set in Mykland[33], we include in our setup modelling beliefs by allowing to specify a set of paths to beconsidered, e.g. super-replication of a contingent claim is required only for paths falling in thegiven set. Our framework thus interpolates between model–independent and model–specificsettings and allows to quantify the impact of making assumptions or gaining information.We obtain a general pricing-hedging duality result: the infimum over superhedging prices isequal to supremum over calibrated martingale measures. In presence of non-trivial beliefs, theequality is between limiting values of perturbed problems. In particular, our results includethe martingale optimal transport duality of Dolinsky and Soner [21] and extend it to multipledimensions and multiple maturities.
Two approaches to pricing and hedging.
The question of pricing and hedging of a contingentclaim lies at the heart of mathematical finance. Following Merton’s seminal contribution [32],we may distinguish two ways of approaching it. First, one may want to make statements “basedon assumption sufficiently weak to gain universal support ,” e.g. market efficiency combined withsome broad mathematical idealisation of the market setting. We will refer to this perspective as themodel-independent approach . While very appealing at first, it has been traditionally criticised forproducing outputs which are too imprecise to be of practical relevance. This is contrasted with thesecond, model-specific approach which focuses on obtaining explicit statements leading to uniqueprices and hedging strategies. “To do so, more structure must be added to the problem throughadditional assumptions at the expense of loosing some agreement1.” Typically this is done byfixing a filtered probability space (Ω , F , ( F t ) t ≥ , P ) with risky assets represented by some adaptedprocess ( S t ). ∗ We are grateful for helpful discussions we have had with Mathias Beiglb¨ock, Bruno Bouchard, Yan Dolinsky,Kostas Kardaras, Marcel Nutz, Mete Soner, Peter Spoida, Nizar Touzi as well as participants in the
Workshop onRobust optimization in Finance in December 2012 and
Workshop on Robust Techniques in Financial Economics in March 2014, both at ETH Zurich,
Labex Luis Bachelier – SIAM – SMAI Conference on Financial Mathematics in June 2014 in Paris, and
SIAM 2014 Conference on Financial Mathematics in Chicago. † Zhaoxu Hou gratefully acknowledges PhD studentship from the Oxford-Man Institute of Quantitative Financeand support from Balliol College in Oxford. E-mail: [email protected] ‡ Jan Ob l´oj gratefully acknowledges funding received from the European Research Council under the EuropeanUnion’s Seventh Framework Programme (FP7/2007-2013) / ERC grant agreement no. 335421 and is also thankfulto the Oxford-Man Institute of Quantitative Finance and St John’s College in Oxford for their financial support.E-mail: [email protected] ; web: Merton [32] classical approach .The original model of Black and Scholes has been extended and generalised, e.g. adding stochasticvolatility and/or stochastic interest rates, trying to account for market complexity observed inpractice. Such generalisations often lead to market incompleteness and lack of unique rationalwarrant prices. Nevertheless, no-arbitrage pricing and hedging was fully characterised in bodyof works on the Fundamental Theorem of Asset Pricing (FTAP) culminating in Delbaen andSchachermayer [18]. The feasible prices for a contingent claim correspond to expectations of the(discounted) payoff under equivalent martingale measures (EMM) and form an interval. Thebounds of the interval are also given by the super- and sub- hedging prices. Put differently, thesupremum of expectations of the payoff under EMMs is equal to the infimum of prices of super-hedging strategies. We refer to this fundamental result as the pricing–hedging duality . Short literature review.
The ability to obtain unique prices and hedging strategies, whichis the strength of the model-specific approach, relies on its primary weakness – the necessity topostulate a fixed probability measure P giving a full probabilistic description of future marketdynamics. Put differently, this approach captures risks within a given model but fails to tellus anything about the model uncertainty, also called the Knightian uncertainty , see Knight [30].Accordingly, researchers extended the classical setup to one where many measures { P α : α ∈ Λ } are simultaneously deemed feasible. This can be seen as weakening assumptions and going backfrom the model-specific towards model-independent. The pioneering works considered uncertainvolatility , see Lyons [31] and Avellaneda et al. [2]. More recently, a systematic approach based onquasi-sure analysis was developed with stochastic integration based on capacity theory in Denisand Martini [19] and on the aggregation method in Soner et al. [42], see also Neufeld and Nutz [35].In discrete time a corresponding generalisation of the FTAP and the pricing-hedging duality wasobtained by Bouchard and Nutz [8] and in continuous time by Biagini et al. [5], see also referencestherein. We also mentions that setups with frictions, e.g. trading constraints, were considered, seeBayraktar and Zhou [3].In parallel, the model-independent approach has also seen a revived interest. This was mainlydriven by the observation that with the increasingly rich market reality this “universally accept-able” setting may actually provide outputs precise enough to be practically relevant. Indeed, incontrast to when Merton [32] was examining this approach, at present typically not only underly-ing is liquidly traded but so are many European options written on it. Accordingly, these shouldbe treated as inputs and hedging instruments, thus reducing the possible universe of no-arbitragescenarios. Breeden and Litzenberger [9] were first to observe that if many (all) European optionsfor a given maturity trade then this is equivalent to fixing the marginal distribution of the stockunder any EMM in classical setting. Hobson [28] in his pioneering work then showed how thiscan be used to compute model-independent prices and hedges of lookback options. Other exoticoptions were analysed in subsequent works, see Brown et al. [10], Cox and Wang [13], Cox andOb l´oj [15]. The resulting no-arbitrage price bounds could still be too wide even for market makingbut the associated hedging strategies were shown to perform remarkably well when compared totraditional delta-vega hedging, see Ob l´oj and Ulmer [37]. Note that the superhedging property hereis understood in a pathwise sense and typically the strategies involve buy-and-hold positions inoptions and simple dynamic trading in the underlying. The universality of the setting and relativeinsensitivity of the outputs to (few) assumptions earned the setup the name of robust approach .In the wake of financial crisis, significant research focus shifted back to the model-independentapproach and many natural questions, such as establishing the pricing-hedging duality and a (ro-bust) version of the FTAP, were pursued. In a one-period setting, the pricing-hedging dualitywas linked to the Karliin-Ishi duality in linear programming by Davis et al. [16]. Beiglb¨ock et al.[4] re-interpreted the problem as a martingale optimal transport problem and established generaldiscrete time pricing-hedging duality as an analogue of the Kantorovich duality in the optimaltransport: here the primal elements are martingale measures, starting in a given point and havingfixed marginal distribution(s) via the Breeden and Litzenberger [9] formula. The dual elements aresub- or super- hedging strategies and the payoff of the contingent claim is the “cost functional.”An analogue result in continuous time, under suitable continuity assumptions, was obtained by2olinsky and Soner [21] who also, more recently, considered the discontinuous setting [22]. Thesetopics remain an active field of research. Acciaio et al. [1] considered pricing-hedging duality andFTAP with an arbitrary market input in discrete time and under significant technical assumptions.These were relaxed offering great insights in a recent work of Burzoni et al. [11]. Galichon et al. [25]applied the methods of stochastic control to deduce the model-independent prices and hedges, seealso Henry-Labord`ere et al. [27]. Several authors considered setups with frictions, e.g. transactionscosts in Dolinsky and Soner [20] or trading constraints in Cox et al. [14] and Fahim and Huang[23]. Main contribution.
The present work contributes to the literature on robust pricing and hedg-ing of contingent claims in two ways. First, inspired by Dolinsky and Soner [21], we study thepricing-hedging duality in continuous time and extend their results to multiple dimensions, dif-ferent market setups and options with uniformly continuous payoffs. Our results are general andobtained in a parsimonious setting. We specify explicitly several important special cases including:the setting when finitely many options are traded, some dynamically and some statically, and thesetting when all European call options for n maturities are traded. The latter gives the martingaleoptimal transport (MOT) duality with n marginal constraints which was also recently studied ina discontinuous setup by Dolinsky and Soner [22] and, in parallel to our work, by Guo et al. [26].Our second main contribution is to propose a robust approach which subsumes the model-independentsetting but allows to include assumptions and move gradually towards the model-specific setting.In this sense, we strive to provide a setup which connects and interpolates between the two endsof spectrum considered by Merton [32]. In contrast, all of the above works on model-independentapproach stay within Merton [32]’s “universally accepted” setting and analyse the implications ofincorporating the ability to trade some options at given market prices for the outputs: prices andhedging strategies of other contingent claims. We amend this setup and allow to express modellingbeliefs. These are articulated in a pathwise manner. More precisely, we allow the modeller todeem certain paths impossible and exclude them from then analysis: the superhedging propertyis only required to hold on the remaining set of paths P . This is reflected in the form of thepricing-hedging duality we obtain.Our framework was inspired by Mykland [33]’s idea of incorporating a prediction set of paths intopricing and hedging problem. On a philosophical level we start with the “universally acceptable”setting and proceed by ruling out more and more scenarios as impossible , see also Cassese [12]. Wemay proceed in this way until we end up with paths supporting a unique martingale measure, e.g.a geometric Brownian motion, giving us essentially a model-specific setting. However, the hedgingarguments are always required to work for all the paths which remain under consideration anda (strong) arbitrage would be given be a strategy which makes positive profit for all remainingpaths, see also the recent work of Burzoni et al. [11]. This should be contrasted with another wayof interpolating between model-independent and model-specific: one which starts from a givenmodel P and proceeds by adding more and more possible scenarios { P α : α ∈ Λ } . This naturallyleads to probabilistic (quasi-sure) hedging and different notions of no-arbitrage, see Bouchard andNutz [8].Our approach to establishing the pricing-hedging duality involves both discretisation, as in Dolinskyand Soner [21], as well as a variational approach as in Galichon et al. [25]. We first prove an“unconstrained” duality result: Theorem 3.2 states that for any derivative with bounded anduniformly continuous payoff function G , the minimal initial set-up cost of a portfolio consisting ofcash and dynamic trading in the risky assets (some of which could be options themselves) whichsuperhedges the payoff G for every non-negative continuous path, is equal to the supremum of theexpected value of G over all non-negative continuous martingale measures . This result is shownthrough an elaborate discretisation procedure building on ideas in [21; 22]. Subsequently, wedevelop a variational formulation which allows us to add statically traded options, or specificationof prediction set P , via Lagrange multipliers. In some cases this leads to “constrained” dualityresult, similar to ones obtained in works cited above, with superhedging portfolios allowed totrade statically the market options and martingale measures required to reprice these options. In Note that here and throughout, we assume that all assets are discounted or, more generally, are expressed interms of some numeraire. asymptotic duality result with the dual and primal problemsdefined through a limiting procedure. The primal value is the limit of superhedging prices on ǫ -neighbourhood of P and the dual value is the limit of supremum of expectation of the payoffover ǫ -(miss)calibrated models, see Definitions 2.2 and 3.16.The paper is organised as follows. Section 2 introduces our robust framework for pricing andhedging and defines the primal (pricing) and dual (hedging) problems. Section 3 contains all themain results. First, in Section 3.1, we present the unconstrained pricing-hedging duality in Theorem3.2 and derive constrained (asymptotic) duality results under suitable compactness assumptions.This allows us in particular to treat the case of finitely many traded options. Then in Sections3.2–3.4 we apply the previous results to the martingale optimal transport case. All the resultexcept Theorem 3.2 are proved in Section 4. Theorem 3.2 is proved in Sections 5 and 6. The proofproceeds via discretisation: of the primal problem in Section 5 and of the dual problem in Section6. Proofs of two auxiliary results are relegated to the Appendix. Notation.
We gather here, principal notation used in this paper. • Ω is the set of all R d + K + valued continuous functions f : [0 , T ] → R d + K + s.t. f = (1 , . . . , • I ⊂ Ω encodes further market information, e.g. the payoff constraints at maturity. • P ⊂ I is the prediction set, i.e. the set of paths the agent wants to consider. • For Banach spaces E and F , C ( E, F ) denotes the set of continuous F -valued function f on E , endowed with the usual sup norm k · k ∞ . • D ([0 , T ] , R d ) is the set of all R d -valued measurable functions f : [0 , T ] → R d + . • D ([0 , T ] , R d ) is the space of all R d -valued right continuous functions f : [0 , T ] → R d with leftlimits. • S = ( S (1) , . . . , S ( d ) ) is the canonical process on Ω and F = ( F i ) ni =1 its natural filtration. • For any m ≥ | · | : R m → R is the norm | x | = sup ≤ i ≤ m | x ( i ) | , where x = ( x ( i ) , . . . , x ( m ) ). • k S k = sup {| S t | : t ∈ [0 , T ] } . • X is the set of market options available for static trading at time t = 0. • P is a linear pricing operator on X specifying the initial prices for X ∈ X . • A is the set of γ such that γ : Ω → D ([0 , T n ] , R d ) is progressively measurable and of boundedvariation, satisfying Z t γ u ( S ) · dS u ≥ − M, ∀ S ∈ I , t ∈ [0 , T n ] , for some M > • For any p ≥ A ( p ) is the set of γ such that γ : Ω → D ([0 , T n ] , R d ) is progressively measurableand of bounded variation, satisfying Z t γ u ( S ) · dS u ≥ − M (1 + sup ≤ u ≤ t | S u | p ) , ∀ S ∈ Ω , t ∈ [0 , T n ] , for some M > • M is defined to be the set of probability measure P on the space (Ω , F T n , F ) such that S isa local martingale under P , see Subsection 2.5. • Let M be the collection of P ∈ M such that P = P W ◦ M − for some continuous martingale M defined on (Ω W , F WT n , F W , P W ), where (Ω W , F WT n , F W , P W ) is a complete probability spacetogether with a finite dimensional Brownian motion { W t } ∞ t =0 and the natural filtration F Wt = σ { W s | s ≤ t } , see Subsection 2.5. • For any notation N which is defined by using M , we write N to denote the one that is definedin the same way as N but by using M instead of M , which is considered as an analogueof N , for example, as defined in Subsection 3.1, P I ( G ) is denoted sup P ∈M E P [ G ( S )], and hence P I ( G ) = sup P ∈M E P [ G ( S )]. 4 d p is the L´evy–Prokhorov’s metric on probability measures on R d + given by d p ( µ, ν ) := sup f ∈ G b ( R d + ) (cid:12)(cid:12)(cid:12) Z f dν − Z f dµ (cid:12)(cid:12)(cid:12) , (1.1)where G b ( R d + ) := (cid:8) f ∈ C ( R d + , R ) : k f k ≤ | f ( x ) − f ( y ) | ≤ | x − y | ∀ x, y (cid:9) (for moredetails, see Bogachev [7], Chapter 8, Theorem 8.3.2.). We consider a financial market with d + 1 assets: a numeraire (e.g. the money market account)and d underlying assets S ( i ) , . . . , S ( d ) , which may be traded at any time t ≤ T n . All prices aredenominated in the units of the numeraire. In particular, the numeraire’s price is thus normalisedand equal to one. We assume that the price path S ( i ) t of each risky asset is continuous. The assetsstart at S = (1 , . . . ,
1) and are assumed to be non-negative. We work on the canonical space C ([0 , T n ] , R d + ), the set of all R d + -valued continuous functions on [0 , T n ].We pursue here a robust approach and do not postulate any probability measure which wouldspecify the dynamics for S . Instead we assume that there is a set X of market traded options withprices known at time zero, P ( X ), X ∈ X . In all generality, an option X ∈ X is just a mapping X : C ([0 , T n ] , R d + ) → R , measurable with respect to the σ -field generated by coordinate process.However most often we will consider European options, i.e. X ( S ) = f ( S T i ) for some f and formaturities 0 < T < . . . < T n = T . The trading is frictionless so prices are linear and options in X may be bought or sold at time zero at their known prices.Further, we allow some of the options to be traded continuously. We do this by augmenting the setof risky assets so that there are d + K assets which may be traded at any time t ≤ T n : d underlyingassets S and K options X ( c )1 ( S ) , . . . , X ( c ) K ( S ). We assume that X ( c )1 , . . . , X ( c ) K are European optionswith maturity T n and have continuous price paths. In addition, they have non-negative payoffsand their prices today P ( X ( c ) i )’s are strictly positive. Hence, by normalisation, we can assumewithout loss of generality the price of each option starts at 1 and never goes below 0. We nowconsider a natural extension of the path spaceΩ = { f ∈ C ([0 , T n ] , R d + K + ) : f = (1 , . . . , } . The coordinate process on Ω is denoted S = ( S t ) ≤ t ≤ T n i.e. S = ( S (1) , . . . , S ( d + K ) ) : [0 , T ] → R d + , and F = ( F t ) ≤ t ≤ T n is its natural filtration. However, not every ω in Ω is a good candidate forprice path of these assets. It can be seen from the fact that the prices of option X ( c ) i and S attime T n should always respect the payoff function X ( c ) i . Therefore, for the purpose of pricing andhedging duality, we only need to consider the set of possible price paths of these d + K assets,denoted I , i.e. I = { ω ∈ Ω : ω ( d + i ) T n = X ( c ) i ( ω (1) T n , . . . , ω ( d ) T n ) / P ( X ( c ) i ) ∀ i ≤ K } . I , called the information space, encodes not only the prices of these underlying assets and optionsat time zero, but also future payoff constraints. A trading strategy consists of two parts. The first part is static hedging X , which is a linearcombination of market traded options. In contrast, the other part, known as the dynamic trading,5eatures a potentially continuous trading in the underlying asset and a few selected Europeanoptions. Heuristically, the capital gain from this trading activity takes the integral form of R γ u ( S ) · dS u . To define this integral properly, we need to impose some regularity condition on γ . Here, wefollow Dolinsky and Soner [21] and consider γ : [0 , T n ] → R d + K of finite variation for which, usingintegration by parts formula, for any continuous S we set Z t γ u ( S ) · dS u = γ t S t − γ S − Z t S u · dγ u , where the last term on the right hand side is a Stieltjes integral.Further, γ is required to be progressively measurable with respect to a filtration which, in ourcontext, is the natural filtration generated by the canonical process. More precisely, we have: Definition 2.1.
We say that a map φ : Ω → D ([0 , T n ] , R d + K ) is progressively measurable , if ∀ υ, ˆ υ ∈ A , υ u = ˆ υ u , ∀ u ∈ [0 , t ] ⇒ φ ( υ ) t = φ (ˆ υ ) t . (2.1)We say γ is admissible if γ : Ω → D ([0 , T n ] , R d + K + ) is progressively measurable and of finitevariation, satisfying Z t γ u ( S ) · dS u ≥ − M, ∀ S ∈ I , t ∈ [0 , T n ] , for some M >
0. (2.2)Let A be the set of such integrands. The set of simple integrands, i.e. γ ∈ A such that γ ( ω ) is asimple function ∀ ω ∈ Ω, is denoted A sp .An admissible (semi-static) trading strategy is a pair ( X, γ ) where X = a + P mi =1 a i X i , for some m , X i ∈ X and a , a i ∈ R , i = 1 , . . . , m and γ ∈ A . The cost of following such a trading strategyis equal to the cost of setting up its static part, i.e. of buying the options at time zero, and is equalto P ( X ) := a + m X i =1 a i P ( X i ) . We denote the class of admissible (semi-static) trading strategies by A X and A sp X for γ ∈ A or A sp respectively. As argued in the Introduction, we allow our agents to express modelling beliefs. These are encodedas restrictions of the pathspace and may come from time series analysis of the past data, oridiosyncratic views about market in the future. Put differently, we are allowed to rule out pathswhich we deem impossible . The paths which remain are referred to as prediction set or beliefs .Note that such beliefs may also encode one agent’s superior information about the market.We will consider pathwise arguments and require that they work provided the price path S fallsinto the predictions set P ⊆ I . Any path falling out of P will be ignored in our considerations.This binary way of specifying beliefs is motivated by the fact that in the end we only see one pathsand hence we are interested in arguments which work pathwise. Nevertheless, the approach is veryparsimonious and as P changes from all paths in I to a support of a given model we essentiallyinterpolate between model-independent and model-specific setups. It also allows to incorporatethe information from time-series of data coherently into the option pricing setup, as no probabilitymeasure is fixed and hence no distinction between real world and risk neutral measures is made.The idea of such a prediction set first appeared in Mykland [33]; also see Nadtochiy and Ob l´oj [34]and [14] for an extended discussion.As the agent rejects more and more paths, i.e. takes P smaller and smaller, the framework’s outputs– the robust price bounds, should get tighter and tighter. This can be seen as a way to quantifythe impact of making assumptions or acquiring additional insights or information.6 .4 Superreplication Our prime interest is in understanding robust pricing and hedging of a derivative with payoff G : Ω → R whose price is not quoted in the market. Our main results will consider boundedpayoffs G and, since the setup is frictionless and there are no trading restrictions, without any lossof generality we may consider only the superhedging price. The subhedging follows by considering − G . Definition 2.2.
1. A portfolio (
X, γ ) ∈ A X is said to super-replicate G on P if X ( S ) + Z T n γ u ( S ) · dS u ≥ G ( S ) , ∀ S ∈ P . (2.3)2. The (minimal) super-replicating cost of G on P is defined as V X , P , P ( G ) := inf n P ( X ) : ∃ ( X, γ ) ∈ A X s.t. ( X, γ ) super-replicates G on P o . (2.4)3. The approximate super-replicating cost of G on P is defined as e V X , P , P ( G ) := inf n P ( X ) : ∃ ( X, γ ) ∈ A X s.t.( X, γ ) super-replicates G on P ǫ for some ǫ > o , (2.5)where P ǫ = { ω ∈ I : inf υ ∈ P k ω − υ k ≤ ǫ } .4. Finally, we let V sp X , P , P ( G ), respectively e V sp X , P , P ( G ), denote the super-replicating cost of G in(2.4), respectively in (2.5), but with ( X, γ ) ∈ A sp X . Our aim is to relate the robust (super)hedging cost, as introduced above, to the classical pricing-by-expectation arguments. To this end we look at all classical models which reprice market tradedoptions.
Definition 2.3.
We denote by M the set of probability measures P on (Ω , F T n , F ) such that S isa P –martingale and let M I be the set of probability measures P ∈ M such that P ( I ) = 1.A probability measure P ∈ M I is called a ( X , P , P )–market model, or simply a calibrated model,if P ( P ) = 1 and E P [ X ] = P ( X ) for all X ∈ X . The set of such measures is denoted M X , P , P .More generally, a probability measure P ∈ M I is called an η − ( X , P , P )–market model if P ( P η ) > − η and | E P [ X ] − P ( X ) | < η for all X ∈ X . The set of such measures is denoted M η X , P , P .Whenever we have P ∈ M X , P , P it provides us with a feasible no-arbitrage price E P [ G ( S )] for aderivative with payoff G . The robust price for G is given as P X , P , P ( G ) := sup P ∈M X , P , P E P [ G ( S )] , where throughout the expectation is defined with the convention that ∞ − ∞ = −∞ . In thecases of particular interest, ( X , P ) will determine uniquely the marginal distributions of S at givenmaturities and P X , P , P ( G ) is then the value of the corresponding martingale optimal transportproblem. We will often use this terminology, even in the case of arbitrary X .In practice, the market prices P are an idealised concept and may be obtained from averagingof bid-ask spread or otherwise. It might not be natural to require a perfect calibration and the7oncept of η –market model allows for a controlled degree of mis-calibration. This leads to theapproximate value given as e P X , P , P ( G ) := lim η ց sup P ∈M η X , P , P E P [ G ( S )] . As we will show below, both the approximate superhedging cost and the approximate robust pricingcost, while being motivated by practical considerations, appear very naturally when consideringabstract pricing-hedging duality.In some instances, for technical reasons, it will be convenient to consider only P arising withina Brownian setup. We denote the collection of P ∈ M such that P = P W ◦ M − for somecontinuous martingale M defined on some probability space satisfying the usual assumptions(Ω W , F WT n , F W , P W ) with a finite dimensional Brownian motion { W t } t ≥ which generates the filtra-tion F W . We write M X , P , P to denote M X , P , P ∩ M , M η X , P , P for M η X , P , P ∩ M and P X , P , P ( G ) :=sup P ∈M X , P , P E P [ G ( S )], e P X , P , P ( G ) := lim η ց sup P ∈M η X , P , P E P [ G ( S )]. Our prime interest, as discussed in the Introduction, is in establishing a general robust pricing–hedging duality. Given a non-traded derivative with payoff G we have two candidate robust pricesfor it. The first one, V X , P , P ( G ), is obtained through pricing-by-hedging arguments. The secondone, P X , P , P ( G ), is obtained by pricing-via-expectation arguments. In a classical setting, theanaloguous two prices are equal. This is trivially true in a complete market and is a fundamentalresult for incomplete markets, see Theorem 5.7 in Delbaen and Schachermayer [18].Within the present pathwise robust approach, the pricing–hedging duality was obtained for specificpayoffs G in literature linking robust approach with the Skorokhod embedding problem, see Hobson[28] or Ob l´oj [36] for discussion. Subsequently, an abstract result was established in Dolinsky andSoner [21], when Ω = I = P , n = d = 1, K = 0 and X is the set of all call and put options with P ( X ) = R ∞ X ( x ) µ ( dx ) for all X ∈ X , where µ is a probability measure on R + with mean equalto 1: V X , P , I ( G ) = P X , P , I ( G ) for a ‘strongly continuous’ class of bounded G .
The result was extended to unbounded claims by broadening the class of admissible strategies andimposing a technical assumption on µ . Below we extend this duality to a much more general settingof abstract X , possibly involving options with multiple maturities, a multidimensional setting andwith an arbitrary prediction set P .Note that, for any Borel G : Ω → R , the inequality V X , P , P ( G ) ≥ P X , P , P ( G ) (3.1)is true as long as there is at least one P ∈ M X , P , P and at least one ( X, γ ) ∈ A X which su-perreplicates G on P . Indeed, since γ is progressively measurable in the sense of (2.1), theintegral R · γ u ( S ) · d S u , defined pathwise via integration by parts, agrees a.s. with the stochas-tic integral under P . Then, by (2.2), the stochastic integral is a P super-martingale and hence E P h R T n γ u ( S ) · d S u i ≤
0. This in turn implies that E P h G ( S ) i ≤ P ( X ) . The result follows since (
X, γ ) and P were arbitrary. We first consider the case without constraints: X = ∅ and P = I . As this context will be ourreference point we introduce notation to denote the super-hedging cost and the robust price. We8et V I ( G ) := inf n x : ∃ γ ∈ A s.t. γ super-replicates G − x on I o , P I ( G ) := sup P ∈M I E P [ G ( S )] . (3.2)We also write P I ( G ) for sup P ∈M I E P [ G ( S )] and V sp I ( G ) for super-replicating cost of G using γ ∈ A sp . Assumption 3.1.
Either K = 0 or X ( c )1 , . . . , X ( c ) K are bounded and uniformly continuous withmarket prices P ( X ( c )1 ) , . . . , P ( X ( c ) K ) satisfying that there exists an ǫ > p k ) ≤ k ≤ K with |P ( X ( c ) k ) − p k | ≤ ǫ for all k ≤ K , M ˜ I = ∅ , where˜ I := { ω ∈ Ω : ω ( d + i ) T n = X ( c ) i ( ω (1) T n , . . . , ω ( d ) T n ) /p i ∀ i ≤ K } . Theorem 3.2.
Under Assumption 3.1, for any bounded and uniformly continuous G : Ω → R wehave V sp I ( G ) = V I ( G ) = P I ( G ) = P I ( G ) . An analogous duality in a quasi-sure setting was obtained in Possama¨ı et al. [38] and earlier papers,as discussed therein. However, while similar in spirit, there is no immediate link between our resultsor proofs and these in [38]. Here, we consider a comparatively smaller set of admissible tradingstrategies and we require a pathwise superhedging property. Consequently, we also need to imposestronger regularity constraints on G . The inequality V I ( G ) ≥ sup P ∈M I E P [ G ( S )]is a special case of (3.1). Sections 5 and 6 are mainly devoted to the proof of the much harderreverse inequality V I ( G ) ≤ sup P ∈M I E P [ G ( S )] , (3.3)which then implies Theorem 3.2. The proof proceeds through discretisation of both the primaland the dual problem.We let Lin( X ) denote the set of finite linear combinations of elements of X andLin N ( X ) = n a + m X i =1 a i X i : m ∈ N , X i ∈ X , m X i =0 | a i | ≤ N o . Then, similarly to e.g. Proposition 5.2 in Henry-Labord`ere et al. [27], a calculus of variationscharacterisation of e V X , P , P is a corollary of Theorem 3.2. From that we are able to deduce pricing-hedging duality between the approximate values. Corollary 3.3.
Under Assumption 3.1, let P be a measurable subset of I and X such that all X ∈ X are uniformly continuous and bounded. Then for any uniformly continuous and bounded G : Ω → R we have: e V sp X , P , P ( G ) = e V X , P , P ( G ) = inf X ∈ Lin N ( X ) , N ≥ n P I ( G − X − N λ P ) + P ( X ) o , (3.4)where λ P ( ω ) := inf υ ∈ P k ω − υ k ∧ Remark 3.4.
As a by-product of the proof of Corollary 3.3, we show that for any bounded G , e V X , P , P ( G ) = inf N ≥ e V X , P , I ( G − N λ P ) and e V sp X , P , P ( G ) = inf N ≥ e V sp X , P , I ( G − N λ P ) . (3.5) Assumption 3.5.
Lin ( X ) is a compact subset of C (Ω , R ) and every X ∈ X is bounded anduniformly continuous. 9 heorem 3.6. Given I , P and X satisfy conditions in Corollary 3.3, if M η X , P , P = ∅ for any η >
0, then for any uniformly continuous and bounded G : Ω → R we have e V X , P , P ( G ) ≥ e P X , P , P ( G ) , (3.6)and if X satisfies Assumption 3.5, then M η X , P , P = ∅ for any η > e V X , P , P ( G ) = e P X , P , P ( G ) = e P X , P , P ( G ) . (3.7) Example 3.7 (Finite X ) . Consider X = { X , . . . , X m } , where X i ’s are bounded and uniformlycontinuous. In this case, Lin ( X ) is a convex and compact subset of C (Ω , R ). Therefore, if M η X , P , P = ∅ for any η >
0, we can apply Theorem 3.6 to conclude e V X , P , P ( G ) = e P X , P , P ( G ).We end this section with consideration if the approximate superhedging and robust prices, e V , e P ,are close to the precise values V, P . First, we focus on the case of finitely many traded put optionsand no beliefs. We consider X = { ( K ( i ) k,j − S ( i ) T j ) + , ≤ i ≤ d, ≤ j ≤ n, ≤ k ≤ m ( i, j ) } , (3.8)where 0 < K ( i ) k,j < K ( i ) k ′ ,j for any k < k ′ and m ( i, j ) ∈ N . To simplify the notation, we write P (( K ( i ) k,j − S ( i ) T j ) + ) = p k,i,j ∀ i, j, k. Assumption 3.8.
Market put prices are such that there exists an ǫ > p k,i,j ) i,j,k with | ˜ p k,i,j − p k,i,j | ≤ ǫ for all i, j, k , there exists a ˜ P ∈ M I such that˜ p k,i,j = E ˜ P [( K ( i ) k,j − S ( i ) T j ) + ] ∀ i, j, k. Remark 3.9.
Assumption 3.8 can be rephrased as saying that the market prices ( X , P ) are in theinterior of the no-arbitrage region. Theorem 3.10.
Let X be given in (3.8), prices P be such that Assumption 3.8 holds and I satisfyAssumption 3.1. Then for any uniformly continuous and bounded G : Ω → R , we have V X , P , I ( G ) = P X , P , I ( G ) . The above result establishes a general robust pricing-hedging duality when finitely many putoptions are traded. It extends in many ways the duality obtained in Davis et al. [16] for thecase of d = n = 1 and K = 0. Note that in general e V X , P , I ( G ) = V X , P , I ( G ) so it follows fromExample 3.7 that in Theorem 3.10 we also have e P X , P , I ( G ) = P X , P , I ( G ). These equalities maystill hold, but may also fail dramatically, when non-trivial beliefs are specified. We present twoexamples to highlight this. Example 3.11.
In this example we consider P corresponding to Black-Scholes model. For sim-plicity, consider the case without any traded options K = 0 , X = ∅ , d = 1 and let P = { ω ∈ Ω : ω admits quadratic variation and d h ω i t = σ ω t dt, ≤ t ≤ T } . Then M P = { P σ } , where S is a geometric Brownian motion with constant volatility σ under P σ .The duality in Theorem 3.6 then gives that for any bounded and uniformly continuous G e V P ( G ) = inf { x : ∃ γ ∈ A s.t. γ super-replicates G − x on P ǫ for some ǫ > } = lim η ց sup P ∈M η P E P [ G ] . However in this case, P has full support on Ω so that P ǫ = Ω and M ǫ P = M for any ǫ >
0. Theabove then boils down to the duality in Theorem 3.2 and we have e V P ( G ) = V I ( G ) = sup P ∈M E P [ G ] ≥ E P σ [ G ] = P P ( G ) , (3.9)where for most G the inequality is strict. See also Step 4 in the proof of Theorem 3.22 in Section 4. xample 3.12. Consider again the case with no traded options, K = 0 , X = ∅ , d = 1 and let P = { ω ∈ Ω : k ω k ≤ b } for some b ≥ . Let G be bounded and uniformly continuous and consider the duality in Theorem 3.6. For each N ∈ N pick P ( N ) ∈ M /N P such that E P ( N ) [ G ] ≥ sup P ∈M /N P E P [ G ] − /N. By Doob’s martingale inequality, P ( N ) ( k S k > M ) ≤ d X i =1 E P ( N ) [ S T ] M ≤ dM . Hence by considering τ M ( S ) = inf { t ≥ k S k > M } ∧ T , we know | E P ( N ) [ G ( S τ M )] − E P ( N ) [ G ( S )] | ≤ d k G k ∞ M and for M > b + 1, P ( N ) ( | S τ M T | > b + 1 /N ) ≤ P ( N ) ( k S τ M k > b + 1 /N ) = P ( N ) ( k S k > b + 1 /N ) ≤ /N, where the last inequality follows from the fact that P ( N ) ∈ M /N P .Write π ( N ) := L P ( N ) ( S τ M T ). π ( N ) ’s are probability measures on a compact subset of R d + , with mean1. It follows that there exists { π ( N k ) } k ≥ , a subsequence of { π ( N ) } N ≥ , converging to some π withmean 1, and by Portemanteau Theorem, for ǫ > π (cid:0) { ~x ∈ R d + : | x i | ≤ b + ǫ ∀ i ≤ d } (cid:1) ≥ lim sup k →∞ π ( N k ) (cid:0) { ~x ∈ R d + : | x i | ≤ b + ǫ ∀ i ≤ d } (cid:1) = 1 . Since ǫ > π (cid:0) { ~x ∈ R d + : | x i | ≤ b ∀ i ≤ d } (cid:1) = 1 by Dominated Convergence Theorem.It follows from Theorem 3.6 that e V P ( G ) = lim N →∞ sup P ∈M /N P E P [ G ] ≤ lim sup k →∞ sup P ∈M π ( Nk ) E P [ G ] + 1 N k + 2 d k G k ∞ M ≤ P π ( G ) + 2 d k G k ∞ M . where the last inequality follows from Lemma 4.4. It is straightforward to see that any P ∈ M π issupported on P , and hence from above we have, for all large M , e V P ( G ) ≤ P P ( G ) + k G k ∞ M and hence e V P ( G ) ≤ P P ( G ) . We conclude that in this example e V P ( G ) = e P P ( G ) = P P ( G ) = V P ( G ) . We focus now on the cases when ( X , P ) determine uniquely certain distributional properties of S under any P ∈ M X , P , P . We start with the case when X is large enough so that the marketprices P pin down the (joint) distribution of ( S (1) T , . . . , S ( d ) T ) under any calibrated model. Later weconsider the case when only marginal distributions of S iT for i ≤ d are fixed. In the former case welimit ourselves to one maturity and P = I which simplifies the exposition. It is possible to extendthese results along the lines of the latter case, when we consider prices at multiple maturities anda non-trivial prediction set, however this would increase the complexity of the proof significantly.11et n = 1 and T = T n . We assume market prices for a rich family of basket options are available.We consider X s.t. Lin( X ) is a dense subset of { f ( S (1) T , . . . , S ( d ) T ) | f : R d + → R bounded, Lip. cont. } . (3.10)In particular, X is large enough to determine uniquely the distribution of S T under any calibratedmodel, i.e. there exists a unique probability distribution π on R d + such that E P [ X ] = P ( X ) = Z R d + X ( s , . . . , s d ) π (d s , . . . , d s d ) , ∀ X ∈ X , P ∈ M X , P , I . (3.11)As an example, we could take X equal to the RHS in (3.10). A martingale measure P ∈ M I is acalibrated model if and only if the distribution of S (1) T , . . . , S ( d ) T under P is π . Accordingly we write M X , P , I = M π, I with M π, I , P π, I etc. defined analogously. Note that in a Brownian setting, wecan always define a continuous martingale M valued in R d + K + with M = 1, ( M (1) T , . . . , M ( d ) T ) ∼ π and M ( d + i ) T = X ( c ) i ( M (1) T , . . . , M ( d ) T ) for every i ≤ K simply by taking conditional expectations ofa suitably chosen random variable distributed according to π and satisfying payoff constraints. Itfollows that the following equivalence holds. Lemma 3.13.
For a probability measure π on R d + , M π, I = ∅ if and only if M π, I = ∅ if and onlyif Z R d + s i π (d s , . . . , d s d ) = 1 , i = 1 , . . . , d. (3.12)Note that if (3.12) fails, then one of the forwards is mispriced leading to arbitrage opportunities .We exclude this situation from our setup. The following is then a multi-dimensional extension ofthe pricing-hedging duality in Dolinsky and Soner [21]. Theorem 3.14.
Consider traded options X and information space I satisfying (3.10) and As-sumption 3.1, with market prices P such that M X , P , I = ∅ . Then for any uniformly continuousand bounded G , we have V X , P , I ( G ) = P π, I ( G ) . It is clear that the above result holds if instead of assuming every X ∈ X is bounded and Lipschitzcontinuous, we allow bounded and uniformly continuous European payoffs, as long as X containsa subset made of bounded and Lipschitz continuous payoffs, which is rich enough to guaranteeuniqueness of π which satisfies (3.11).We now turn to the case when X is much smaller and the market prices determine marginaldistributions of S ( i ) T for i ≤ d . For concreteness, let us consider the case when put options aretraded X = { ( K − S ( i ) T j ) + : i = 1 , . . . , d, j = 1 , . . . , n, K ∈ R + } . (3.13)Arbitrage considerations, see e.g. Cox and Ob l´oj [15] and Cox et al. [14], show that absence of(weak type of) arbitrage is equivalent to M X , P , P = ∅ . Note that the latter is equivalent to marketprices P being encoded by probability measures ( µ ( i ) j ) with p i,j ( K ) = P (( K − S ( i ) T j ) + ) = Z ( K − s ) + µ ( i ) j (d s ) , (3.14)where, for each i = 1 , . . . , d , µ ( i )1 , . . . , µ ( i ) n have finite first moments, mean 1 and increase in convexorder ( µ ( i )1 (cid:22) µ ( i )2 (cid:22) · · · (cid:22) µ ( i ) n ), i.e. R φ ( x ) µ ( i )1 (d x ) ≤ . . . ≤ R φ ( x ) µ ( i ) n (d x ) for any convex function This may be, depending on the sign of mispricing and the admissibility criterion, a strong arbitrage in Cox andOb l´oj [15] or model independent arbitrage in Davis and Hobson [17] and Acciaio et al. [1] or else a weaker type ofapproximate arbitrage, e.g. a weak free lunch of vanishing risk ; see Cox and Ob l´oj [15] and Cox et al. [14]. : R + → R . In fact, as noted already by Breeden and Litzenberger [9], probability measures µ ( i ) j are defined by µ ( i ) j ([0 , K ]) = p ′ i,j ( K +) for K ∈ R + . We may think of ( µ ( i ) j ) and P as the modelling inputs. The set of calibrated market models M X , P , P is simply the set of probability measures P ∈ M such that S ( i ) T j is distributed according to µ ( i ) j ,and P ( P ) = 1. Accordingly, we write M X , P , P = M ~µ, P and P ~µ, P ( G ) = P X , P , P ( G ). Furthermore,since µ ( i ) j ’s all have means equal to 1, under any P ∈ M ~µ, P , S is a (true) martingale. Remark 3.15.
It follows, see Strassen [43], that M ~µ, I = ∅ if and only if µ ( i )1 , . . . , µ ( i ) n have finitefirst moments, mean 1 and increase in convex order , for any i = 1 , . . . , d . However, in general, theadditional constraints associated with a non-trivial P ( I are much harder to understand.In this context we can improve Theorem 3.6 and narrow down the class of approximate marketmodels requiring that they match exactly the marginal distributions at the last maturity. Definition 3.16.
Let M ~µ, P ,η be the set of all measure P ∈ M such that L P ( S ( i ) T j ), the law of S ( i ) T j under P satisfies L P ( S ( i ) T n ) = µ ( i ) n and d p ( L P ( S ( i ) T j ) , µ ( i ) j ) ≤ η, for j = 1 , . . . , n − , i = 1 , . . . , d, and furthermore P ( P η ) ≥ − η . Finally, let e P ~µ, P ( G ) := lim η ց sup P ∈M ~µ, P ,η E P [ G ( S )] . Note that M ~µ, P ,η ⊂ M ǫ ( η ) X , P , P for a suitable choice of ǫ ( η ) which converges to zero as η →
0. Itfollows that e P ~µ, P ( G ) ≤ e P X , P , P ( G ). The following result extends and sharpens the duality obtainedin Theorem 3.6 to the current setting. Theorem 3.17.
Let P be a measurable subset of I , X be given by (3.13) and P be such that,for any η > M ~µ, P ,η = ∅ , where ~µ is defined via (3.14). Then for any uniformly continuous andbounded G the robust pricing-hedging duality holds between the approximate values: e V X , P , P ( G ) = e P X , P , P ( G ) = e P ~µ, P ( G ) . We want to extend Theorem 3.17 to unbounded exotic options, including a lookback option. How-ever, the admissibility condition considered so far, and given by (2.2), is too restrictive and has tobe relaxed. To see this consider d = 1, K = 0, X is given by (3.13) and G ( S ) = sup ≤ t ≤ T n S t . If G could be super-replicated by an admissible trading strategy ( X, γ ) ∈ A X then, following similararguments as for (3.1), we see that P I ( G − X ) ≤ . This is clearly impossible since X is bounded and there exists P ∈ M such that E P [ G ( S )] = ∞ .The argument is similar if instead of puts we took all call options. We conclude that we need toenlarge the set of dynamic trading strategies A .We fix p > γ is admissible if γ : Ω → D [0 , T n ] is progressively measurable and of bounded variation, satisfying Z t γ u ( S ) · dS u ≥ − M (cid:0) ≤ s ≤ t | S s | p (cid:1) , ∀ S ∈ I , t ∈ [0 , T n ] , for some M >
0. (3.15) One can take ǫ ( η ) = √ η + 2 f (1 / √ η ) with f ( K ) = max ≤ i ≤ d (cid:8) p i,n ( K ) − K + 1 (cid:9) .
13o avoid confusion, we denote by A ( p ) the set of all such γ . We also say ( X, γ ) ∈ A ( p ) X if γ ∈ A ( p ) and X = a + P mi =1 a i X i , for some m and X i ∈ X ( p ) given by X ( p ) := { f ( S ( i ) T j ) : | f ( x ) | ≤ K (1 + | x | p ) for some K > , for j = 1 , . . . , n, i = 1 , . . . , d } . As previously with X in (3.13), the above set X ( p ) is large enough to determine uniquely themarginal distributions of S ( i ) T j . That is M X , P , P = ∅ implies that there exist unique probabilitymeasures µ ( i ) j such that L P ( S ( i ) T j ) = µ ( i ) j , i = 1 , . . . , d , j = 1 , . . . , n for any P ∈ M X , P , P = M ~µ, P .We write V ( p ) X , P , P for the superreplication cost V X , P , P ( G ) but with ( X, γ ) ∈ A ( p ) X and e V ( p ) X , P , P forthe approximative value. We need to assume that µ ’s admit p th moment. Assumption 3.18.
Assume ~µ = ( µ ( i ) j : i = 1 , . . . , d, j = 1 , . . . , n ) are probability measures on R + , with mean 1, admitting finite p -th moment for some p > µ ( i )1 (cid:22) µ ( i )2 (cid:22) · · · (cid:22) µ ( i ) n , i = 1 , . . . , d . Theorem 3.19.
Let ~µ satisfy Assumption 3.18, P be a measurable subset of I such that for any η > M ~µ, P ,η = ∅ . Then, under Assumption 3.1, for any uniformly continuous G that satisfies | G ( S ) | ≤ L (1 + sup ≤ t ≤ T n | S t | p ) , the following robust pricing-hedging duality holds e V ( p ) X ( p ) , P , P ( G ) = e P ~µ, P ( G ) , where p is the same as in Assumption 3.18. Theorems 3.14 and 3.19 extend the duality obtained in [21]. In general we obtain an asymptotic duality result with the dual and primal problems defined through a limiting procedure. In thissection, we want to focus on establishing a duality result without any asymptotic approximation.As already seen from Theorem 3.14, in a setting where there is a single marginal and the predictionset is absent, this type of duality result can be obtained without imposing further conditions on thepayoff function G other than uniform continuity. However, to achieve this goal in a more generalsetting, we will impose stricter conditions on the payoff function G and prediction set P . Assumption 3.20.
There exist constants
L > p > G is uniformly continuousw.r.t. sup norm k · k and subject to | G ( S ) | ≤ L (1 + k S k p ) , S ∈ D ([0 , T n ] , R d + )Moreover, let υ, ˆ υ ∈ D ([0 , T n ] , R d + ) be of the form υ t = n X i =1 m i − X j =0 υ i,j [ t i,j ,t i,j +1 ) ( t ) + v n,m n − T n ( t ) , ˆ υ t = n X i =1 m i − X j =1 υ i,j [ˆ t i,j , ˆ t i,j +1 ) ( t ) + v n,m n − T n ( t )where t i, = ˆ t i, = T i ∀ ≤ i ≤ n − t i,m i − = ˆ t i,m i − = T i ∀ ≤ i ≤ n . Then, | G ( υ ) − G (ˆ υ ) | ≤ L k υ k p n X i =1 m i X i =1 | ∆ t i,j − ∆ˆ t i,j | (3.16)where as usual ∆ t i,j := t i,j − t i,j − and ∆ˆ t i,j := ˆ t i,j − ˆ t i,j − .14ote that Assumption 3.20 is close in spirit to Assumption 2.1 in [21]. Despite their proximity,our assumption here is strictly weaker, which can be seen from the fact that it includes Europeanoptions having intermediate maturities, in contrast to Assumption 2.1 in [21]. Definition 3.21.
We say P is time invariant if for any non-decreasing continuous function f : [0 , T n ] → [0 , T n ] such that f (0) = 0 and f ( T i ) = T i for any i = 1 , . . . , n , S ∈ P implies( S f ( t ) ) t ∈ [0 ,T n ] ∈ P . Theorem 3.22.
Let ~µ satisfy Assumption 3.18 and P be closed and time invariant. Then, underAssumption 3.1, for any G that satisfies Assumption 3.20 the following robust pricing-hedgingduality holds e V ( p ) ~µ, P ( G ) = V ( p ) ~µ, P ( G ) = P ~µ, P ( G ) = e P ~µ, P ( G ) , (3.17)where p is the same as in Assumption 3.18. We present in this section proof of all the results except Theorem 3.2 which is shown in Sections5 and 6. We start by describing a discretisation of a continuous path, often referred to as the“Lebesgue discretisation” which will often used. In particular, it will be central to Section 5 butis also employed in the proofs of Lemma 4.3, 4.4, 4.5 and Theorem 3.22 below.
Definition 4.1.
For a positive integer N and any S ∈ Ω, we set τ ( N )0 ( S ) = 0 and m ( N )0 ( S ) = 0,then define τ ( N ) k ( S ) = inf n t ≥ τ ( N ) k − ( S ) : | S t − S τ ( N ) k − ( S ) | = 12 N o ∧ T and let m ( N ) ( S ) = min { k ∈ N : τ ( N ) k ( S ) = T } .Following the observation that m ( N ) ( S ) < ∞ ∀ S ∈ Ω, we say the sequence of stopping times0 = τ ( N )0 < τ ( N )1 < · · · < τ ( N ) m ( N ) = T forms a Lebesgue partition of [0 , T ] on Ω. Similar partitionswere studied previously, see e.g. Vovk [45]. Their main appearances have been as tools to buildpathwise version of the Itˆo’s integral. They can also be interpreted, from a financial point of view,as candidate times for rebalancing portfolio holdings, see Whalley and Wilmott [46]. Remark 4.2.
Note that m ( N − ( S ) ≤ m ( N ) ( ˜ S ) for any S, ˜ S ∈ Ω such that k S − ˜ S k < − N . Tojustify this, notice that for each i < m ( N − ( S ), { ˜ S t : t ∈ ( τ ( N − i − ( S ) , τ ( N − i ( S )] ∩ { k/ N : k ∈ N + } has at least three elements, which implies that for each i < m ( N − ( S ) there exist at leastone j < m ( N ) ( ˜ S ) such that τ ( N ) j ( ˜ S ) ∈ ( τ ( N − i − ( S ) , τ ( N − i ( S )]. In consequence, for any sequence( P ( k ) ) k ≥ converging to P weakly and bounded non-increasing function φ : N → RE P h φ (cid:0) m ( D ) ( S ) (cid:1)i ≤ lim inf k →∞ E P ( k ) h φ (cid:0) m ( D − ( S ) (cid:1)i . (4.1) Note that any (
X, γ ) that super-replicates G − N λ P also super-replicates G − /N on P N . Itfollows that e V sp X , P , P ( G ) = inf n P ( X ) : ∃ ( X, γ ) ∈ A sp X s.t. ( X, γ ) super-replicates G on P ǫ for some ǫ > o ≤ N + inf n P ( X ) : ∃ ( X, γ ) ∈ A sp X s.t. ( X, γ ) super-replicates G − N λ P on I o . N , we have e V sp X , P , P ( G ) ≤ inf N ≥ inf n P ( X ) : ∃ ( X, γ ) ∈ A sp X s.t. and ( X, γ ) super-replicates G − N λ P on I o = inf N ≥ e V sp X , P , I ( G − N λ P ) . Note that by the same argument above we have e V X , P , P ( H ) ≤ inf N ≥ e V X , P , I ( H − N λ P ) and e V sp X , P , P ( H ) ≤ inf N ≥ e V sp X , P , I ( H − N λ P ) (4.2)hold for every bounded measurable H .Notice thatinf n P ( X ) : ∃ ( X, γ ) ∈ A sp X s.t. ( X, γ ) super-replicates G − N λ P on I o = inf X ∈ Lin( X ) (cid:8) P ( X ) + inf n x : ∃ γ ∈ A sp s.t. γ super-replicates G − N λ P − X − x on I (cid:9)o = inf X ∈ Lin( X ) (cid:8) P ( X ) + V sp I ( G − N λ P − X ) (cid:9) = inf X ∈ Lin( X ) n P ( X ) + P I ( G − X − N λ P ) o , where the last equality is justified by Theorem 3.2 as λ P and X are bounded and uniformlycontinuous. Hence, we have e V sp X , P , P ( G ) ≤ inf N ≥ , X ∈ Lin( X ) n P ( X ) + P I ( G − X − N λ P ) o . On the other hand, given any (
X, γ ) ∈ A X and ǫ > X, γ ) super-replicates G on P ǫ ,by the admissibility of ( X, γ ) ∈ A X and boundedness of X and G , if N > X ( S ) + Z T n γ u ( S ) · dS u ≥ G ( S ) − N, and hence ( X, γ ) super-replicates G − N λ P . It follows that e V X , P , P ( G ) = inf n P ( X ) : ∃ ( X, γ ) ∈ A X s.t. ( X, γ ) super-replicates G on P ǫ for some ǫ > o ≥ inf N ≥ inf n P ( X ) : ∃ ( X, γ ) ∈ A X s.t. ( X, γ ) super-replicates G − N λ P on I o = inf N ≥ , X ∈ Lin( X ) (cid:8) P ( X ) + V I ( G − X − N λ P ) (cid:9) = inf N ≥ , X ∈ Lin( X ) n P ( X ) + P I ( G − X − N λ P ) o , where the last equality is again justified by Theorem 3.2. As e V X , P , P ( G ) ≤ e V sp X , P , P ( G ), this estab-lishes the equality in (3.4).Note that by the same argument above we can argue that e V X , P , P ( H ) ≥ inf N ≥ e V X , P , I ( H − N λ P ) and e V sp X , P , P ( H ) ≥ inf N ≥ e V sp X , P , I ( H − N λ P )hold for every bounded measurable H . Therefore, combining this with (4.2), we show (3.5). To establish (3.6), we consider a ( X , γ ) ∈ A X that super-replicates G on P ǫ for some ǫ >
0, i.e. X ( S ) + Z T n γ u d S u ≥ G ( S ) on P ǫ . X is bounded, it follows from the definition of admissibility that there exists M > X ( S ) + Z T n γ u d S u ≥ G ( S ) − M λ P ( S ) . (4.3)Next, for each N ≥
1, we pick P ( N ) ∈ M /N X , P , P such that E P ( N ) [ G ( S )] ≥ sup P ∈M /N X , P , P E P [ G ( S )] − N .
Since γ is progressively measurable in the sense of (2.1), the integral R · γ u ( S ) · d S u , defined pathwisevia integration by parts, agrees a.s. with the stochastic integral under any P ( N ) . Then, by (2.2),the stochastic integral is a P ( N ) –super-martingale and hence E P ( N ) h R T n γ u ( S ) · d S u i ≤
0. Therefore,from (4.3), we can derive that E P ( N ) [ X ( S )] ≥ E P ( N ) (cid:2) G ( S ) − M λ P ( S ) (cid:3) ≥ sup P ∈M /N X , P , P E P [ G ( S )] − N − MN . (4.4)Also note that X takes the form of a + P mi =1 a i X i . Then by definition of M η X , P , P (cid:12)(cid:12) P ( X ) − E P ( N ) [ X ( S )] (cid:12)(cid:12) → N → ∞ . This, together with (4.4), yields P ( X ) ≥ e P X , P , P ( G ) . As (
X, γ ) ∈ A X is arbitrary, we therefore establish (3.6).To show (3.7), we first deduce from Theorem 3.2 and (3.4) that e V X , P , P ( G ) = inf X ∈ Lin( X ) , N ≥ n P I ( G − X − N λ P ) + P ( X ) o = lim N →∞ inf X ∈ Lin N ( X ) n sup P ∈M I E P [ G − X − N λ P ] + P ( X ) o = lim N →∞ sup P ∈M /N X , P , P E P [ G ]= e P X , P , P ( G ) , (4.5)where the crucial third equality follows from (4.6) in Lemma 4.3 below.Last, we show that M η X , P , P = ∅ for any η >
0. By the above and equality between P I = P I inTheorem 3.2 we haveinf X ∈ Lin( X ) , N ≥ n sup P ∈M I E P [ G − X − N λ P ] + P ( X ) o = inf X ∈ Lin( X ) , N ≥ n sup P ∈M I E P [ G − X − N λ P ] + P ( X ) o = e V X , P , P ( G ) = e P X , P , P ( G ) . Then, taking G = 0, as M η X , P , P = ∅ for any η > X ∈ Lin( X ) , N ≥ n sup P ∈M I E P [ − X − N λ P ] + P ( X ) o = e P X , P , P (0) = 0 . Therefore, it follows from the equivalence in Lemma 4.3, with M s = M , that M η X , P , P = ∅ for any η >
0. In addition, by (4.7) in Lemma 4.3 below, e V X , P , P ( G ) = inf X ∈ Lin( X ) , N ≥ n sup P ∈M I E P [ G − X − N λ P ] + P ( X ) o = lim N →∞ sup P ∈M /N X , P , P E P [ G ] = e P X , P , P ( G ) . This completes the proof of Theorem 3.6. It remains to argue the following which is stated in ageneral form and also used in subsequent proofs.17 emma 4.3.
Let P be a measurable subset of I , X satisfy Assumption 3.5 and M s be a non-emptyconvex subset of M I . Then the following two are equivalent:(i) for any η > M η X , P , P T M s = ∅ .(ii) inf X ∈ Lin( X ) , N ≥ n sup P ∈M s E P [ − X − N λ P ] + P ( X ) o = 0.Further, under (i) or (ii), for any uniformly continuous and bounded G : Ω → R we have:inf X ∈ Lin( X ) , N ≥ n sup P ∈M s E P [ G − X − N λ P ] + P ( X ) o = lim N →∞ sup P ∈M /N X , P , P ∩M s E P [ G ] . (4.6)Moreover, for any α, β ≥ D ∈ N .inf X ∈ Lin( X ) , N ≥ n sup P ∈M s E P [ G ( S ) − α ∧ ( β q m ( D ) ( S )) − X ( S ) − N λ P ( S )] + P ( X ) o ≤ lim N →∞ sup P ∈M /N X , P , P ∩M s E P [ G ( S ) − α ∧ ( β q m ( D − ( S ))] , (4.7)where m ( D ) is defined in Definition 4.1. Proof.
Choose κ > ∨ ( k G k ∞ + α ). We first observe thatinf X ∈ Lin( X ) , N ≥ n sup P ∈M s E P [ G − α ∧ ( β p m ( D ) ) − X − N λ P ] + P ( X ) o = lim N →∞ inf X ∈ Lin N ( X ) n sup P ∈M s E P [ G − α ∧ ( β p m ( D ) ) − X − N λ P ] + P ( X ) o . (4.8)Define the function G : Lin N ( X ) × M s → R by G ( X, P ) := lim ǫ ց inf ˜ P ∈M I , d p (˜ P , P ) <ǫ E ˜ P h G ( S ) − α ∧ ( β p m ( D − ) − x ( S ) − N λ P ( S ) i + P ( X )= lim ǫ ց inf ˜ P ∈M I , d p (˜ P , P ) <ǫ E ˜ P h − α ∧ ( β p m ( D − ) i + E P [ G − N λ P − X ] + P ( X ) . Then by (4.1) in Remark 4.2, for any sequence ( P ( k ) ) k ≥ converging to P weakly, E P h − α ∧ ( β q m ( D ) ( S )) i ≤ lim inf k →∞ E P ( k ) h − α ∧ ( β q m ( D − ( S )) i . and hence lim N →∞ inf X ∈ Lin N ( X ) n sup P ∈M s E P [ G − α ∧ ( β p m ( D ) ) − X − N λ P ] + P ( X ) o ≤ lim N →∞ inf X ∈ Lin N ( X ) sup P ∈M s E P [ G ( X, P )] , with equality when α = β = 0.The next step is to interchange the order of the infimum and supremum. Notice that when wefix P , G is affine in the first variable and continuous due to bounded convergence theorem. Inaddition, by definition G is lower-semi continuous in the second variable. Furthermore, G is convexin the second variable. To justify this, we notice that P E P h − α ∧ ( β p m ( D − ( S )) i is a linearfunctional and it follows that for each ǫ > λ ∈ [0 , ˜ P ∈M s , d p (˜ P ,λ P (1) +(1 − λ ) P (2) ) <ǫ E ˜ P h − α ∧ ( β q m ( D − ( S )) i ≤ λ inf ˜ P ∈M s , d p (˜ P , P (1) ) <ǫ E ˜ P h − α ∧ ( β q m ( D − ( S )) i + (1 − λ ) inf ˜ P ∈M s , d p (˜ P , P (2) ) <ǫ E ˜ P h − α ∧ ( β q m ( D − ( S )) i . N ( X ) is convex and compact, it follows that we can now apply Min-Max Theorem (seeCorollary 2 in Terkelsen [44]) to G and derivelim N →∞ inf X ∈ Lin N ( X ) sup P ∈M s E P [ G ( X, P )] = lim N →∞ sup P ∈M s inf X ∈ Lin N ( X ) E P [ G ( X, P )] . Therefore, we havelim N →∞ inf X ∈ Lin N ( X ) n sup P ∈M s E P [ G − α ∧ ( β p m ( D ) ) − X − N λ P ] + P ( X ) o ≤ lim N →∞ sup P ∈M s inf X ∈ Lin N ( X ) n E P [ G − α ∧ ( β p m ( D − ) − X − N λ P ] + P ( X ) o , (4.9)with equality when α = β = 0.Now first consider the case: α = β = 0. In this case, it follows from above thatlim N →∞ inf X ∈ Lin N ( X ) n sup P ∈M s E P [ G − X − N λ P ] + P ( X ) o = lim N →∞ sup P ∈M s n inf X ∈ Lin N ( X ) E P [ G − X − N λ P ] + P ( X ) o . (4.10)Suppose that for any η > M η X , P , P T M s = ∅ . Then we see that for any N , P ∈ M /N X , P , P and X ∈ Lin N ( X ), | E P [ X ] − P ( X ) | ≤ m X i =1 | a i | (cid:12)(cid:12) E P [ X i ] − P ( X i ) (cid:12)(cid:12) ≤ NN = 1 N , where X takes the form of a + P mi =1 a i X i , for some m ∈ N , X i ∈ X and a i ∈ R such that P mi =0 | a i | ≤ N . In addition, λ P ≤ N { S ∈ P /N } + { S / ∈ P /N } leads to E P [ N λ P ] ≤ N P ( S ∈ P /N ) + N P ( S / ∈ P /N ) ≤ N .
Therefore, we can deduce thatlim N →∞ sup P ∈M /N X , P , P ∩M s n inf X ∈ Lin N ( X ) E P [ G − X − N λ P ] + P ( X ) o ≥ lim N →∞ sup P ∈M /N X , P , P ∩M s n E P [ G ] − N o = lim N →∞ sup P ∈M /N X , P , P ∩M s E P [ G ] . (4.11)Consequently, by taking G = 0, using (4.8)–(4.11) and noting that considering a sup over a largerset increases its value, we haveinf X ∈ Lin( X ) , N ≥ n sup P ∈M s E P [ − X − N λ P ] + P ( X ) o ≥ , which leads to inf X ∈ Lin( X ) , N ≥ n sup P ∈M s E P [ − X − N λ P ] + P ( X ) o = 0 . On the other hand, if inf X ∈ Lin( X ) , N ≥ n sup P ∈M s E P [ − X − N λ P ] + P ( X ) o = 0 , (4.12)then we will argue in the following that in the sup term of (4.10) it suffices to consider probabilitymeasures P ∈ M κ/N X , P , P ∩ M s . Suppose P ∈ ( M s \ M κ/N X , P , P ), then either there exist X ∈ X suchthat E P [ X ] − P ( X ) > κ/N or P ( S / ∈ P κ/N ) ≥ κ/N . In the former case, since N X ∈ Lin( X ), E P [ G − N X − N λ P ] + P ( N X ) ≤ E P [ G ] − N ( E P [ X ] − P ( X )) < − κ, E P [ G − N λ P ] < κ − κ = − κ , whilelim N →∞ sup P ∈M s n inf X ∈ Lin N ( X ) E P [ G − X − N λ P ] + P ( X ) o = inf X ∈ Lin( X ) , N ≥ n sup P ∈M s E P [ G − X − N λ P ] + P ( X ) o ≥ inf X ∈ Lin( X ) , N ≥ n sup P ∈M s E P [ − κ − X − N λ P ] + P ( X ) o = − κ, where the last equality follows from (4.10). This argument also implies that M κ/N X , P , P ∩ M s = ∅ for any N ∈ N . Therefore we have the equivalence between ∀ η > M η X , P , P \ M s = ∅ and inf X ∈ Lin( X ) , N ≥ n sup P ∈M s E P ( − X − N λ P ) + P ( X ) o = 0 . Now consider the general case: α, β ≥
0. We begin to verify (4.6) and (4.7). Since M η X , P , P T M s = ∅ ∀ η >
0, sup P ∈M s E P [ X − N λ P ] − P ( X ) ≥ , ∀ X ∈ X , N ∈ R + . Hence for every N and X ∈ Lin N ( X )sup P ∈M s E P [ G − α ∧ ( β p m ( D ) ) − X − N λ P ] + P ( X ) ≥ − k G k ∞ − α + sup P ∈M s E P [ X − N λ P ] − P ( X ) ≥ − κ, and thereforelim N →∞ inf X ∈ Lin N ( X ) n sup P ∈M s E P [ G − α ∧ ( β p m ( D ) ) − X − N λ P ] + P ( X ) o ≥ − κ. Then, by using the same argument as above, we can argue that in the sup term of (4.9) it sufficesto consider probability measures P ∈ M κ/N X , P , P ∩ M s and hence we havelim N →∞ inf X ∈ Lin N ( X ) n sup P ∈M s E P [ G − α ∧ ( β p m ( D ) ) − X − N λ P ] + P ( X ) o ≤ lim N →∞ inf X ∈ Lin N ( X ) n sup P ∈M κ/N X , P , P ∩M s E P [ G − α ∧ ( β p m ( D − ) − X ] + P ( X ) o ≤ lim N →∞ sup P ∈M κ/N X , P , P ∩M s E P [ G − α ∧ ( β p m ( D − )] , where the second inequality follows from the fact that − X ∈ Lin N ( X ) for every X ∈ Lin N ( X ).This completes the verification of (4.7). In the case that α = β = 0, combining the inequalityabove with (4.11), we then conclude thatinf X ∈ Lin( X ) , N ≥ n sup P ∈M s E P [ G − X − N λ P ] + P ( X ) o = lim N →∞ sup P ∈M /N X , P , P ∩M s E P [ G ] . From Theorem 3.6, as worked out in Example 3.7, we know that V X , P , I ( G ) = e P X , P , I ( G ) = lim N →∞ sup P ∈M /N X , P , I E P [ G ] . (4.13)20ow for every positive integer N , we pick P ( N ) ∈ M /N X , P , I such that E P ( N ) [ G ] + 1 /N ≥ sup P ∈M /N X , P , I E P [ G ] . We write p ( N ) k,i,j := E P ( N ) [( K ( i ) k,j − S ( i ) k,j ) + ]for any i = 1 , . . . d , j = 1 , . . . , n , k = 1 , . . . , m ( i, j ), and define ˜ p ( N ) k,i,j ’s by˜ p ( N ) k,i,j = √ N (cid:0) p k,i,j − (1 − / √ N ) p ( N ) k,i,j (cid:1) . Note that | ˜ p ( N ) k,i,j − p k,i,j | = ( √ N − | p k,i,j − p ( N ) k,i,j | ≤ √ NN = 1 √ N ∀ i, j, k. (4.14)Then, it follows from Assumption 3.8 that when N is large enough there exists a ˜ P ( N ) ∈ M I suchthat ˜ p ( N ) k,i,j := E ˜ P ( N ) [( K ( i ) k,j − S ( i ) k,j ) + ] ∀ i, j, k. Now we consider Q := (1 − / √ N ) P ( N ) + ˜ P ( N ) / √ N . It follows that E Q [( K ( i ) k,j − S ( i ) k,j ) + ] =(1 − / √ N ) E P ( N ) [( K ( i ) k,j − S ( i ) k,j ) + ] + 1 √ N E ˜ P ( N ) [( K ( i ) k,j − S ( i ) k,j ) + ]=(1 − / √ N ) p ( N ) k,i,j + ˜ p ( N ) k,i,j / √ N = p k,i,j and hence Q ∈ M X , P , I . In addition, (cid:12)(cid:12) E Q [ G ] − E P ( N ) [ G ] (cid:12)(cid:12) ≤ √ N ( E P ( N ) [ | G | ] + E ˜ P ( N ) [ | G | ]) ≤ k G k ∞ √ N .
Therefore, we have sup P ∈M /N X , P , I E P [ G ] ≤ sup P ∈M X , P , I E P [ G ] − k G k ∞ √ N − N , which leads us to conclude e P X , P , I ( G ) = lim N →∞ sup P ∈M /N X , P , I E P [ G ] ≤ P X , P , I ( G ) . Together with (4.13) and (3.1) this completes the proof.
Let Y = { f ( S (1) T , . . . , S ( d ) T ) : f ∈ C ( R d + , R ) s.t. sup ~x = ~y | f ( ~x ) − f ( ~y ) || ~x − ~y | ≤ , k f k ∞ ≤ } . Then, as Lin( X ) is dense in Lin( Y ), V X , P , I ( G ) = V Y , P , I ( G ) . (4.15)We now consider Y M := { f ( S (1) T ∧ M, . . . , S ( d ) T ∧ M ) : f ∈ Y} . Y M is a subset of Y for each M ∈ N ,and in consequence, V Y , P , I ( G ) ≤ V Y M , P , I ( G ) , ∀ M ∈ N . (4.16)21bserve that, for any X ∈ X , X ( S (1) T ∧ M, . . . , S ( d ) T ∧ M ) ≤ X ( S (1) T , . . . , S ( d ) T ) + 2 M − d X i =1 S ( i ) T and hence (cid:12)(cid:12)(cid:12) Z R d + X ( s ∧ M, . . . , s d ∧ M ) π (d s , . . . , d s d ) − Z R d + X ( s , . . . , s d ) π (d s , . . . , d s d ) (cid:12)(cid:12)(cid:12) ≤ dM . Note that by definition Y M is closed and convex. Also, by Arzel´a-Ascoli theorem, Y M is compact.Hence Lin ( Y M ) satisfies Assumption 3.5. Therefore, applying Theorem 3.6 to Y M , we have V Y M , P , I ( G ) = e P Y M , P , I ( G ) ∀ M > . (4.17)Then, by putting (4.15), (4.16) and (4.17) together V X , P , I ( G ) ≤ lim M →∞ e P Y M , P , I ( G ) = lim M →∞ lim N →∞ sup P ∈M /N Y M , P , I E P [ G ] ≤ lim M →∞ sup P ∈M /M Y M , P , I E P [ G ] . For every N ∈ N , take P ( N ) ∈ M /N Y N , P , I such thatsup P ∈M /N Y N , P , I E P [ G ] ≤ E P ( N ) [ G ] + 1 N .
Let π ( N ) be the law of ( S (1) T , . . . , S ( d ) T ) under P ( N ) . It is a probability measure on R d + with meanequal to 1. It follows that the family { π ( N ) } N ≥ is tight. By Prokhorov theorem, there exists { π ( N k ) } k ≥ , a subsequence of { π ( N ) } N ≥ , converging to some ˜ π . In the following, we are going toargue that ˜ π is in fact π . For any X ∈ Y and N ∈ N , (cid:12)(cid:12)(cid:12) E P ( N ) [ X ( S T )] − Z R d + X ( s , . . . , s d ) π (d s , . . . , d s d ) (cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12) E P ( N ) [ X ( S (1) T ∧ N, . . . , S ( d ) T ∧ N )] − Z R d + X ( s ∧ N, . . . , s d ∧ N ) π (d s , . . . , d s d ) (cid:12)(cid:12)(cid:12) + 2 E P ( N ) (cid:2) n max ≤ d ≤ n { S ( i ) T >N } o (cid:3) + 2 Z R d + { max ≤ d ≤ n { s i >N } } π (d s , . . . , d s d ) ≤ N + 2 dN + 2 dN = 4 d + 1 N .
By weak convergence of π ( N ) , along a subsequence of { π ( N ) } N ≥ , for every X ∈ Y Z R d + X ( s , . . . , s d ) π ( N ) (d s , . . . , d s d ) → Z R d + X ( s , . . . , s d )˜ π (d s , . . . , d s d ) as N → ∞ . Therefore, for every X ∈ Y Z R d + X ( s , . . . , s d ) π (d s , . . . , d s d ) = Z R d + X ( s , . . . , s d )˜ π (d s , . . . , d s d ) , which implies that π = ˜ π as Y is rich enough to guarantee uniqueness of π .It follows that V X , P , I ( G ) ≤ lim N →∞ sup P ∈M /N Y N, P , I E P [ G ] ≤ lim sup k →∞ sup P ∈M π ( Nk ) , I E P [ G ] ≤ P π, I ( G ) , where the last inequality follows from the following lemma.22 emma 4.4. Assume π ( N ) and π are probability measures on R d + such that π ( N ) and π satisfies(3.12) and π ( N ) converges to π weakly. Then, for any bounded and uniformly continuous G , α, β ≥ D ∈ N .lim sup N →∞ P π ( N ) , I (cid:0) G ( S ) − α ∧ ( β q m ( D ) ( S )) (cid:1) ≤ P π, I (cid:0) G ( S ) − α ∧ ( β q m ( D − ( S )) (cid:1) , where m ( D ) is defined in Definition 4.1. Proof.
Choose f e : R + → R + such that | G ( ω ) − G ( υ ) | ≤ f e ( k ω − υ k ) for any ω, υ ∈ Ω, | X ( c ) i ( ~x ) − X ( c ) i ( ~y ) | ≤ f e ( | ~x − ~y | ) for any ~x, ~y ∈ R d + and lim x ց f e ( x ) = 0. Now fix N and P ( N ) ∈ M π ( N ) , I .By definition of M , there exists a complete probability space (Ω W , F WT , F W , P W ) together a finitedimensional Brownian motion ( W t ) t ≥ and the natural filtration F Wt = σ { W s | s ≤ t } , and acontinuous martingale M defined on (Ω W , F WT , F W , P W ) such that P ( N ) = P W ◦ M − .Write ǫ N := d p ( π ( N ) , π ). π ( N ) converges to π weakly is equivalent to saying that ǫ N → N → ∞ . Fix N . If ǫ N = 0, then it is trivially true that P π ( N ) , I ( G ) = P π, I ( G ). Therefore, weonly consider the case that ǫ N >
0. By Strassen’s theorem, Corollary of Theorem 11 on page 438in Strassen [43] or theorem 4 on page 358 in Shiryaev [41], we can find a F WT measurable randomvariable Λ such that Λ ( d + i ) = X ( c ) i (Λ (1) , . . . , Λ (1) ) for every i ≤ K ,(Λ (1) , . . . , Λ ( d ) ) ∼ P W π and P W ( | Λ ( i ) − M ( i ) T | > ǫ N ) < ǫ N ∀ i ≤ d. (4.18)We now construct a continuous martingale from Λ by taking conditional expectation, i.e.Γ t = E W [Λ |F Wt ] , t ∈ [0 , T ] , where E W is the expectation with respect to P W . Note that by uniform continuity of X ( c ) i | Λ ( d + i ) − M ( d + i ) T | ≤ f e (2 ǫ N ) ∀ i ≤ K, whenever | Λ ( j ) − M ( j ) T | ≤ ǫ N ∀ j ≤ d Hence, for every i ≤ K P W (cid:0) | Λ ( i + d ) − M ( i + d ) T | > f e (2 ǫ N ) (cid:1) ≤ dǫ N . (4.19)Observe that E W [Λ ( i ) ] = E W [ M ( i ) T ] = 1 and Λ ( i ) ≥ P W -a.s. ∀ i . Then, using (4.18), E W [ | Λ ( i ) − M ( i ) T | ] = 2 E W [(Λ ( i ) − M ( i ) T ) + ] − E W [Λ ( i ) − M ( i ) T ]= 2 E W [(Λ ( i ) − M ( i ) T ) + ] ≤ ǫ N + 2 E W [Λ ( i ) {| Λ ( i ) − M ( i ) T | > ǫ N } ] ≤ ǫ N + 2 E W [Λ ( i ) {| Λ ( i ) − M ( i ) T | > ǫ N } { Λ > / √ ǫ N } ] + 4 √ ǫ N ≤ ǫ N + 2 Z { x i ≥ √ ǫN }∩ R d + x i π (d x , . . . , d x d ) + 4 √ ǫ N , ∀ i = 1 , . . . , d. Similarly, for every i ≤ K , E W [ | Λ ( d + i ) − M ( d + i ) T | ] =2 E W [(Λ ( d + i ) − M ( d + i ) T ) + ] ≤ f e (2 ǫ N ) + 2 E W [Λ ( i ) {| Λ ( d + i ) − M ( d + i ) T | >f e (2 ǫ N ) } ] ≤ f e (2 ǫ N ) + 4 d k X ( c ) i k ∞ P ( X ( c ) i ) ǫ N . Now define η N by η N = 2 f e (2 ǫ N ) + 4 ǫ N + 4 d K X i =1 k X ( c ) i k ∞ P ( X ( c ) i ) ǫ N + 2 d X i =1 Z { x i ≥ √ ǫN }∩ R d + x i π (d x , . . . , d x d ) + 4 √ ǫ N η N → N → ∞ . Then by Doob’s martingale inequality P W ( k Γ − M k ≥ η / N ) ≤ η − / N d + K X i =1 E W [ | Λ ( i ) − M ( i ) T | ] ≤ ( d + K ) η / N . (4.20)It follows that (cid:12)(cid:12) E W [ G (Γ) − G ( M )] (cid:12)(cid:12) ≤ d + K ) k G k ∞ η / N + E W (cid:2)(cid:12)(cid:12) G (Γ) − G ( M ) (cid:12)(cid:12) {k Γ − M k <η / N } (cid:3) ≤ d + K ) k G k ∞ η / N + f e ( η / N ) . Note that by (4.1) in Remark 4.2 for N sufficiently large, E W [ α ∧ ( β q m ( D ) ( M ))] ≥ E W [ α ∧ ( β q m ( D − (Γ))] − αη / N . As P ( N ) ∈ M π ( N ) , I is arbitrary, P π ( N ) , I (cid:0) G − α ∧ ( β p m ( D ) ) (cid:1) ≤ P π, I (cid:0) G − α ∧ ( β p m ( D − ) (cid:1) − (cid:16) ( d + K ) k G k ∞ η / N + f e ( η / N ) + αη / N (cid:17) . Therefore, we can conclude thatlim sup N →∞ P π ( N ) , I (cid:0) G − α ∧ ( β p m ( D ) ) (cid:1) ≤ P π, I (cid:0) G − α ∧ ( β p m ( D − ) (cid:1) , as required. From Theorem 3.6, we know that e V X , P , P ( G ) ≥ e P X , P , P ( G ) . We also have the observation that e P ~µ, P ( G ) ≤ e P X , P , P ( G ). Then to establish Theorem 3.17, itsuffices to show that e V X , P , P ( G ) ≤ e P ~µ, P ( G ) . (4.21)This follows as a special case ( α = β = 0) of the following crucial lemma which also be used toprove Theorem 3.22 below. Lemma 4.5.
Let P be a measurable subset of I , X be given by (3.13) and P be such that, forany η > M ~µ, P ,η = ∅ , where ~µ is defined via (3.14). Then for any uniformly continuous andbounded G and α, β ≥ e V X , P , P (cid:16) G − α ∧ ( β p m ( D ) ) (cid:17) ≤ e P ~µ, P (cid:16) G − α ∧ ( β p m ( D − ) (cid:17) . where m ( D ) is defined in Definition 4.1. Proof.
Recall that G N ( R d + ) := n f ∈ C ( R d + , R ) : sup ~x = ~y | f ( ~x ) − f ( ~y ) || ~x − ~y | ≤ N, k f k ∞ ≤ N, and f ( x , . . . , x d ) = f ( x ∧ N , . . . , x d ∧ N ) ∀ ( x , . . . , x d ) ∈ R d + o and G ( R d + ) = ∪ N> G N ( R d + ). 24et Z M = { f ( S ( i ) T n ) : f ∈ G M ( R + ) , i = 1 , . . . , d } and Y M = { f ( S ( i ) T j ) : f ∈ G M ( R + ) , i =1 , . . . , d, j = 1 , . . . , n − } . We also write Z = [ M ≥ Z M and Y = [ M ≥ Y M . Notice that given any f ∈ C b ( R + , R ), ǫ > µ on R + which has finite first moment,there is some u : R + → R taking the form a + P ni =1 a i ( s − K i ) + such that u ≥ f and R ( u − f ) dµ < ǫ .It follows that e V X , P , P (cid:16) G − α ∧ ( β p m ( D ) ) (cid:17) = e V Z∪Y , P , P (cid:16) G − α ∧ ( β p m ( D ) ) (cid:17) (4.22)= inf X ∈ Lin(
Z∪Y ) , N ≥ n V I (cid:16) G − X − α ∧ ( β p m ( D ) ) − N λ P (cid:17) + P ( X ) o (4.23) ≤ inf X ∈ Lin(
Z∪Y ) , N ≥ n P I (cid:16) G − X − α ∧ ( β p m ( D − ) − N λ P (cid:17) + P ( X ) o (4.24)= inf Y ∈ Lin( Y ) , N ≥ inf M ≥ inf Z ∈ Lin( Z M ) n P I (cid:16) G − Y − Z − α ∧ ( β p m ( D − ) − N λ P (cid:17) + P ( Y + Z ) o (4.25) ≤ inf Y ∈ Lin( Y ) inf M ≥ , N ≥ lim L →∞ sup P ∈M /L Z M , P , I E P [ G − α ∧ ( β p m ( D − ) − Y − N λ P + P ( Y )] (4.26) ≤ inf Y ∈ Lin( Y ) inf M ≥ , N ≥ sup P ∈M /M Z M , P , I E P [ G − α ∧ ( β p m ( D − ) − Y − N λ P + P ( Y )] . (4.27)where the equality between (4.22) and (4.23) follows from Remark 3.4, the inequality between(4.23) and (4.24) is justified by Theorem 5.2. Finally the inequality between (4.25) and (4.26)is given by Lemma 4.3. To justify this, we first observe that G ( R + ) is a convex and compactsubset of C ( R + , R ). Then, since Lin ( Z M ) = Z M for any M , Lin ( Z M ) satisfies Assumption 3.5.Therefore, by keeping Y and N fixed and applying Lemma 4.3 toinf Z ∈ Lin( Z M ) n P I ( G − α ∧ ( β p m ( D − ) − Y − Z − N λ P ) + P ( Y ) + P ( Z ) o , we establish the inequality.For any P ∈ M /M Z M , P , I , let ǫ P = max { d p ( µ ( i ) n , L P ( S ( i ) T n )) : 1 ≤ i ≤ d } . Since d p , the L´evy–Prokhorov’s metric on probability measures on R d + , is given by d p ( µ, ν ) := sup f ∈ G b ( R d + ) (cid:12)(cid:12)(cid:12) Z f dν − Z f dµ (cid:12)(cid:12)(cid:12) , where G b ( R d + ) := (cid:8) f ∈ C ( R d + , R ) : k f k ≤ | f ( ~x ) − f ( ~y ) | ≤ | ~x − ~y | ∀ ~x = ~y (cid:9) , we can pick g ∈ G b ( R + ) such that (cid:12)(cid:12)(cid:12) Z R + g ( x ) µ ( i ) n (d x ) − E P [ g ( S ( i ) T n )] (cid:12)(cid:12)(cid:12) > ǫ P / i = 1 , . . . , d, and define ˆ g ∈ G bM ( R + ) via ˆ g ( x ) = M g ( x ∧ M ). Then, (cid:12)(cid:12)(cid:12) Z R + ˆ g ( x ) µ ( i ) n (d x ) − E P [ˆ g ( S ( i ) T n )] (cid:12)(cid:12)(cid:12) (4.28) ≥ M (cid:12)(cid:12)(cid:12) Z gdµ ( i ) n − E P [ g ( S ( i ) T n )] (cid:12)(cid:12)(cid:12) − ( M + 1) µ ( i ) n ( {| x | > M } ) − ( M + 1) P ( | S ( i ) T n | > M ) ≥ M ǫ P / − M + 1) M .
25y definition of M /M Z M , P , I , (cid:12)(cid:12)(cid:12) Z R + ˆ g ( x ) µ ( i ) n (d x ) − E P [ˆ g ( S ( i ) T n )] (cid:12)(cid:12)(cid:12) ≤ /M. Hence, ǫ P ≤ /M + 2( M + 1) /M ≤ /M when M is sufficiently large. It follows that P ∈M µ n , I , /M and hence M /M Z M , P , I ⊆ M µ n , I , /M when M is sufficiently large. In consequenceinf M ≥ sup P ∈M /M Z M , P , I E P [ G − α ∧ ( β p m ( D − ) − Y − N λ P + P ( Y )] ≤ inf M ≥ sup P ∈M µn, I , /M E P [ G − α ∧ ( β p m ( D − ) − Y − N λ P + P ( Y )] . For every M ∈ N + , take P ( M ) ∈ M µ n , I , /M such that E P ( M ) h G − α ∧ ( β p m ( D − ) − Y − N λ P + P ( Y ) i ≥ sup P ∈M µn, I , /M E P [ G − α ∧ ( β p m ( D − ) − Y − N λ P + P ( Y )] − M .
Let π ( M ) n be the law of ( S (1) T n , . . . , S ( d ) T n ) under P ( M ) . It is a probability measure on R d + with mean 1.It follows that the family { π ( M ) n } M ≥ is tight. By Prokhorov theorem, there exists a subsequence { π ( M k ) n } converging to some π n . Note that the marginal distributions of π n are µ ( i ) n ’s. By Lemma4.4, it follows that lim M →∞ sup P ∈M µn, I , /M E P [ G − α ∧ ( β p m ( D − ) − Y − N λ P + P ( Y )] ≤ sup P ∈M µn, I E P [ G − α ∧ ( β p m ( D − ) − Y − N λ P + P ( Y )] . With the result above, we continue with (4.27): e V X , P , P ( G ) ≤ inf Y ∈ Lin( Y ) inf M ≥ , N ≥ sup P ∈M /M Z M , P , I E P [ G − α ∧ ( β p m ( D − ) − Y − N λ P + P ( Y )] ≤ inf Y ∈ Lin( Y ) , N ≥ sup P ∈M µn, I E P [ G − α ∧ ( β p m ( D − ) − Y − N λ P + P ( Y )] ≤ inf M ≥ inf Y ∈ Lin( Y M ) ,N ≥ sup P ∈M µn, I E P [ G − α ∧ ( β p m ( D − ) − Y − N λ P + P ( Y )] ≤ inf M ≥ lim N →∞ sup P ∈M µn, I ∩M /N Y M , P , P E P [ G − α ∧ ( β p m ( D − )] ≤ inf M ≥ sup P ∈M µn, I ∩M /M Y M , P , P E P [ G − α ∧ ( β p m ( D − )] . To justify the second last inequality, we first notice that M η Y M , P , P = ∅ for any M ∈ N and η > M ~µ, P ,η = ∅ for any η >
0, and hence it follows from Lemma 4.3, with M s = M µ n , I , thatinf M ≥ inf Y ∈ Lin( Y M ) ,N ≥ sup P ∈M µn, I E P [ G − α ∧ ( β p m ( D − ) − Y − N λ P + P ( Y )] ≤ inf M ≥ lim N →∞ sup P ∈M µn, I ∩M /N Y M, P , P E P [ G − α ∧ ( β p m ( D − )] ≤ inf M ≥ sup P ∈M µn, I ∩M /M Y M, P , P E P [ G − α ∧ ( β p m ( D − )] . M /M Y M , P , P ∩ M µ n , I ⊆ M ~µ, P , /M when M is large enough. Fix P ∈ M /M Y M , P , P ∩ M µ n , I and let ˜ ǫ P = max { d p ( µ ( i ) j , L P ( S ( i ) T j )) : i ≤ d, j ≤ n } . We can pick g ∈ G b ( R + ) such that (cid:12)(cid:12)(cid:12) Z R + ˆ g ( x ) µ ( i ) j (d x ) − E P [ˆ g ( S ( i ) T j )] (cid:12)(cid:12)(cid:12) > ˜ ǫ P / i ≤ d, j ≤ n − . By following the same argument as above, we can show that ˜ ǫ P ≤ /M + 2( M + 1) /M . Hence, M /M Y M , P , P ∩ M µ n , I ⊆ M ~µ, P , /M when M is sufficiently large. Therefore, we have e V X , P , P ( G ) ≤ inf M ≥ sup P ∈M µn, I ∩M /M Y M, P , P E P [ G − α ∧ ( β p m ( D − )] ≤ lim N →∞ sup P ∈M ~µ, P , /N E P [ G − α ∧ ( β p m ( D − )] = e P ~µ, P ( G − α ∧ ( β p m ( D − )) . We start with a key lemma, analogous to the one obtained in Dolinsky and Soner [21].
Lemma 4.6.
Consider α D ( S ) := (cid:0) max ≤ i ≤ d k S ( i ) k p + 1 (cid:1) { max ≤ i ≤ d k S ( i ) k +1 ≥ D } + max ≤ i ≤ d k S ( i ) k p D . (4.29)Then, given that ( µ ( i ) j ) satisfies Assumption 3.18, for any P ∈ M such that L P ( S ( i ) T j ) = µ ( i ) j ∀ i ≤ d, j ≤ n E P [ α D ( S )] ≤ e ( ~µ n , D ) , (4.30)where e ( ~µ n , D ) := (cid:16) pp − (cid:17) p P di =1 (cid:16) R | x |≥ ( p − p )( D − | x | p µ ( i ) n (d x ) + K R | x | p µ ( i ) n (d x ) (cid:17) → D →∞ . Proof.
First define h D : R → R by h D ( x ) = pD p − (cid:16) | x | − (cid:16) p − p (cid:17) D (cid:17) { ( p − p ) D ≤| x | 27e now proceed with the proof the Theorem 3.19. We first show that e P ~µ, P ( G ) ≤ e V ( p ) X ( p ) , P , P ( G ).Given ( X , γ ) ∈ A ( p ) X such that ( X , γ ) super-replicates G on P ǫ for some ǫ > 0, since X is bounded,it follows from the definition of A ( p ) that there exists M > X ( S (1) , . . . , S ( d ) ) + Z T n γ u d S u ≥ G ( S ) − M (1 + sup ≤ t ≤ T n | S t | p ) { S / ∈ P ǫ } on I . (4.31)Next, for each N ≥ 1, we pick P ( N ) ∈ M ~µ, P , /N such that E P ( N ) [ G ( S )] ≥ sup P ∈M ~µ, P , /N E P [ G ( S )] − N .We first notice that X ( S (1) , . . . , S ( d ) ) is of the form P di =1 P nj =1 f i,j ( S ( i ) j ) for some f i,j such that ∀ i ≤ d, j ≤ n f i,j is continuous and bounded by M (1 + | S ( i ) T j | ) for some M . Since by Jensen’sinequality, for any P ∈ M such that L P ( S ( i ) T n ) = µ ( i ) n ∀ i ≤ d E P [ | S ( i ) T j | p ] ≤ E P [ | S ( i ) T n | p ] ≤ Z [0 , ∞ ) x p µ ( i ) n ( dx ) < ∞ ∀ i ≤ d, j ≤ n, it follows from weak convergence of measures, definition of M ~µ, P ,ǫ and Lemma 4.6 that (cid:12)(cid:12) P ( X ) − E P ( N ) [ X ( S (1) , . . . , S ( d ) )] (cid:12)(cid:12) → N → ∞ . (4.32)Since γ is progressively measurable in the sense of (2.1), the integral R · γ u ( S ) · d S u , defined pathwisevia integration by parts, agrees a.s. with the stochastic integral under any P ( N ) . Then, by (2.2),the stochastic integral is a P ( N ) super-martingale and hence E P ( N ) h R T n γ u ( S ) · d S u i ≤ 0. Therefore,by Lemma 4.6 E P ( N ) [ X ( S )] ≥ E P ( N ) (cid:2) G ( S ) − M (1 + | S | p ) { S / ∈ P ǫ } (cid:3) ≥ sup P ∈M ~µ, P , /N E P [ G ( S )] − N − M E P ( N ) (cid:2) (1 + | S | p ) { S / ∈ P ǫ } { k S k≤ N / p } (cid:3) − M E P ( N ) (cid:2) (1 + | S | p ) { S / ∈ P ǫ } { k S k >N / p } (cid:3) ≥ sup P ∈M ~µ, P , /N E P [ G ( S )] − N − M (1 + √ N ) N − M N − e ( ~µ n , N / p ) . This, together with (4.32), yields P ( X ) ≥ e P ~µ, P ( G ) . As ( X, γ ) ∈ A X is arbitrary, we therefore establish e P ~µ, P ( G ) ≤ e V ( p ) X ( p ) , P , P ( G ).Let D > G D by G D = G ∧ D ∨ ( − D ). Then it is clear that G D is bounded anduniformly continuous. Therefore, by Theorem 3.17 e V ( p ) X ( p ) , P , P ( G L + D ) = lim η ց sup P ∈M ~µ, P ,η E P [ G L + D ( S )] ≤ lim η ց sup P ∈M ~µ, P ,η E P [ G ( S )] + 2 L lim η ց sup P ∈M ~µ, P ,η E P h (1 + k S k p ) { k S k≥ ( DL ) /p } i , where the second inequality follows from G L + D ( S ) ≤ G ( S ) + 2 L (cid:0) k S k p { k S k≥ ( DL ) /p } (cid:1) .We know from Assumption 3.1 that any S ∈ I satisfies k S ( i ) k ≤ κ ∀ i > d , where κ is the smallestnumber such that X ( c ) i / P ( X ( c ) i )’s are bounded by κ . It follows from Lemma 4.6 that for any D ≥ Lκ p and P ∈ M ~µ, P ,η E P h (1 + k S k p ) { k S k≥ ( DL ) /p } i = E P h max ≤ i ≤ d k S ( i ) k p { max ≤ i ≤ d k S ( i ) k p ≥ ( DL ) /p } i ≤ e ( ~µ n , D/L ) → , as D → ∞ , D →∞ e V ( p ) X ( p ) , P , P ( G D + L ) ≤ e P ~µ, P ( G ) . On the other hand, by the linearity of the market, e V ( p ) X ( p ) , P , P ( G ) ≤ e V ( p ) X ( p ) , P , P ( G D + L ) + e V ( p ) X ( p ) , P , P (2 L (1 + k S k p ) { k S k≥ ( DL ) /p } ) . Since M ~µ, P ,η = ∅ for any η > e V ( p ) X ( p ) , P , P ( α D ) ≥ e P ~µ, P ( α D ) ≥ 0. Then it follows from lemma 4.1in Dolinsky and Soner [21] and the obvious fact that e V ( p ) X ( p ) , P , Ω ( α D ) ≥ e V ( p ) X ( p ) , P , P ( α D ) thatlim sup D →∞ e V ( p ) X ( p ) , P , P ( α D ) = 0 . Hence we conclude that e P ~µ, P ( G ) ≤ e V ( p ) X ( p ) , P , P ( G ) ≤ lim sup D →∞ e V ( p ) X ( p ) , P , P ( G D + L ) ≤ e P ~µ, P ( G )and therefore we have equalities throughout. We first make two simple observations. Remark 4.7. If P is a non-empty closed subset of Ω with respect to sup norm, then P = \ ǫ> P ǫ = \ ǫ> P ǫ , where P ǫ is the closure of P ǫ . Lemma 4.8. If P is time invariant, then for every ǫ > P ǫ is also time invariant. Proof. Fix ǫ > S ∈ P ǫ and a non-decreasing continuous function f : [0 , T n ] → [0 , T n ] such that f (0) = 0 and f ( T i ) = T i for any i = 1 , . . . , n . By definition, there exist S ( N ) ∈ P such that k S ( N ) − S k ≤ ǫ + 1 N Now write ˜ S t = S f ( t ) and ˜ S ( N ) t = S ( N ) f ( t ) . Note that ˜ S ( N ) ∈ P as P is time invariant. Then it isclear that k ˜ S ( N ) − ˜ S k = k S ( N ) − S k ≤ ǫ + 1 N , which implies that ˜ S ∈ P ǫ . Since S ∈ P ǫ and f are arbitrary, we can therefore conclude that P ǫ is time invariant.We now proceed with the proof the Theorem 3.22. As argued in the proof of Theorem 3.19 above, byLemma 4.6, it suffices to argue (3.17) for bounded G . Further, note that the inequalities e V ( p ) ~µ, P ( G ) ≥ V ( p ) ~µ, P ( G ) ≥ P ~µ, P ( G ) hold in general. In addition, according to Theorem 3.19, e V ( p ) ~µ, P ( G ) = e P ~µ, P ( G ).Therefore, we only need to show e P ~µ, P ( G ) = P ~µ, P ( G ). Our proof of this equality is divided intosix steps. First, using Lemma 4.5, we argue that it suffices to consider measures with “goodcontrol” on the expectation of m ( D ) ( S ). Next, we perform three time changes within each tradingperiod [ T i , T i +1 ]. The resulting time change of S , denoted ¨ S , allows for a “good control” overits quadratic variation process. At the same time, we keep G ( S ) and G (¨ S ) “close” and given ameasure P ∈ M ~µ, P ,η with good control on E P [ m ( D ) ( S )], since P η is time invariant, the law of thetime-changed price process ¨ S remains an element of M ~µ, P ,η . Then, in Step 5, given a sequence29f models with improved calibration precisions, we show tightness of quadratic variation processof the time-changed price process ¨ S under these measures. This then leads to tightness of imagesmeasures via ¨ S . In Step 6, we deduce the duality e P ~µ, P ( G ) = P ~µ, P ( G ) from tightness and conclude. Step 1: Reducing to measures P with good control on E P [ m ( D ) ( S )] . Let G be bounded and satisfy Assumption 3.20. Choose κ ∈ R + such that k G k ≤ κ and let f e : R d + K + → R + be the modulus of continuity of G , i.e. | G ( ω ) − G ( υ ) | ≤ f e ( | ω − υ | ) for any ω, υ ∈ Ωwith lim x → f e ( x ) = 0. Fix D ∈ N . Consider a random variable X D ( S ) = vuuut m ( D ) ( S ) X j =1 d + K X i =1 | S ( i ) τ ( D ) j ( S ) − S ( i ) τ ( D ) j − ( S ) | ≥ − D q m ( D ) ( S ) − ≥ − D ( q m ( D ) ( S ) − , where τ ( D ) i ’s and m ( D ) are defined in Definition 4.1.Then by Lemma 5.4 in Dolinsky and Soner [22]0 ≤ V ( p ) ~µ, P ( X D ( S )) ≤ V ( p ) ~µ, I ( X D ( S )) ≤ dV ( p ) ~µ, I ( k S k p ) < ∞ . It follows that from the linearity of the market and the estimate above e V ( p ) ~µ, P ( G ( S )) ≤ e V ( p ) ~µ, P (cid:16) G ( S ) − κ D ∧ X D ( S )2 D (cid:17) + e V ( p ) ~µ, P ( X D ( S ) / D ) ≤ e V ( p ) ~µ, P (cid:16) G ( S ) − κ D ∧ p m ( D ) ( S )2 D (cid:17) + c / D ≤ e V ~µ, P (cid:16) G ( S ) − κ D ∧ p m ( D ) ( S )2 D (cid:17) + c / D ≤ e P ~µ, P (cid:16) G ( S ) − κ D ∧ p m ( D − ( S )2 D (cid:17) + c / D = lim N →∞ sup P ∈M ~µ, P , /N E P h G ( S ) − κ D ∧ p m ( D − ( S )2 D i + c / D . where c is a constant and the last inequality follows from Lemma 4.5.Next we denote f M I the set of P ∈ M I such that E P h κ D ∧ p m ( D − ( S )2 D i ≤ κ + 2 . (4.33)We notice that if P ∈ M ~µ, P , /N such that P / ∈ f M I , then E P h G ( S ) − κ D ∧ p m ( D − ( S )2 D i < κ − κ − − κ − . While for N sufficiently large,sup P ∈M ~µ, P , /N E P h G ( S ) − κ D ∧ p m ( D − ( S )2 D i ≥ e P ~µ, P (cid:16) G ( S ) − κ D ∧ p m ( D − ( S )2 D (cid:17) ≥ V ( p ) ~µ, P ( G ( S )) − c / D ≥ − κ − D . It follows thatlim N →∞ sup P ∈M ~µ, P , /N E P h G ( S ) − κ D ∧ p m ( D − ( S )2 D i = lim N →∞ sup P ∈ f M I ∩M ~µ, P , /N E P h G ( S ) − κ D ∧ p m ( D − ( S )2 D i . 30n particular, f M I ∩ M ~µ, P , /N = ∅ for N large enough. Step 2: First time change: “squeezing paths and adding constant paths”. Now for every N ∈ N , take P ( N ) ∈ f M I ∩ M ~µ, P , /N such that E P ( N ) [ G ( S )] ≥ sup P ∈ ˜ M I ∩M ~µ, P , /N E P [ G ( S )] − /N. Define an increasing function f : [0 , T n ] [0 , T n ] by f ( t ) = n X i =1 (cid:16) T i ∧ (cid:16) T i − + ( T i − T i − )( t − T i − ) T i − T i − − /D (cid:17)(cid:17) { T i − For ease of notation it is helpful to rename the elements of the set { τ ( N ) j : j ≤ m ( N ) j } ∪ { T i : i = 1 , . . . , n } as follows. We define a sequence of stopping times τ ( N ) i,j : Ω → [ T i − , T i ] and m ( N ) i : Ω → N + in arecursive manner: set m ( N )0 ( S ) = 0 and τ ( N )0 , − ( S ) = 0, and ∀ i = 1 , . . . , n , set τ ( N ) i, ( S ) = T i − and let τ ( N ) i, ( S ) = inf n t ≥ T i − : | S t − S τ ( N ) i − ,m ( N ) i − S ) − ( S ) | = N } ∧ T i ,τ ( N ) i,k ( S ) = inf n t ≥ τ i,k − ( S ) : | S t − S τ ( N ) i,k − ( S ) | = N } ∧ T i ,m ( N ) i ( S ) = m ( N ) i − + min { k ∈ N : τ ( N ) i,k ( S ) = T i } . It follows that for any S ∈ I m ( D − ( S ) ≤ m ( D − n ( S ) ≤ m ( D − ( S ) + n − . (4.35)Set Θ = 2 ⌈ κ D ⌉ + n and δ = 1 / (4 D Θ ). We now define a sequence of stopping times σ i,j : Ω → [0 , T n ]. Fix any S ∈ Ω as follows. Firstly, set σ i, ( S ) = T i − and σ i, Θ+1 ( S ) = T i . Then, for j ≤ Θ, σ i,j ( S ) = (cid:16) τ ( D − i,j ( S )+ δj (cid:17) ∧ (cid:0) T i − / (2 D ) (cid:1) if j < m ( D − i ( S ), and σ i,j ( S ) = T i − / (2 D ) otherwise.Then it follows from the definition that T i − = σ i, ( S ) ≤ σ i, ( S ) ≤ . . . ≤ σ i, Θ ( S ) < σ i, Θ+1 ( S ) = T i .We also note that since ˜ S is always constant on [ T i − /D, T i ], τ ( D − i,j (˜ S ) ≤ T i − /D and hencefor j ≤ Θ ∧ ( m ( D − i (˜ S ) − σ i,j (˜ S ) ≤ τ ( D − i,m ( D − i − (˜ S ) + δ (Θ − ≤ T i − D + 14 D Θ < T i − D . Therefore, ∀ j = 1 , . . . , (cid:16) Θ ∧ ( m ( D − i (˜ S ) − (cid:17) σ i,j (˜ S ) − σ i,j − (˜ S ) = δ + (cid:0) τ ( D − i,j (˜ S ) − τ ( D − i,j − (˜ S ) (cid:1) ≥ δ. (4.36)31efine a process ˇ S byˇ S t = n − X i =0 Θ − X j =0 n ˜ S τ ( D − i,j (˜ S )+( t − σ i,j (˜ S ) − δ ) + [ σ i,j (˜ S ) ,σ i,j +1 (˜ S )) ( t )+ ˜ S (cid:0) τ ( D − i − , Θ (˜ S )+ Ti − t − Ti − σi, Θ(˜ S ) (cid:1) ∧ T i [ σ i, Θ (˜ S ) ,T i ] ( t ) o . Equivalently, ˇ S can be obtained by time changing ˜ S via an increasing and continuous process g : [0 , T n ] × I → [0 , T n ], defined by g t ( S ) = n − X i =0 Θ − X j =0 n(cid:16) τ ( D − i,j ( S ) + ( t − σ i,j ( S ) − δ ) + (cid:17) [ σ i,j ( S ) ,σ i,j +1 ( S )) ( t )+ T i ∧ (cid:16) τ ( D − i, Θ − ( S ) + ( σ i, Θ ( S ) − σ i, Θ − ( S ) − δ ) + + 1 T i − t − T i − σ i, Θ (˜ S ) (cid:17) [ σ i, Θ ( S ) ,T i ] ( t ) o . In particular, it follows from (4.36) that g t (˜ S ) = n − X i =0 Θ − X j =0 n(cid:16) τ ( D − i,j (˜ S ) + ( t − σ i,j (˜ S ) − δ ) + (cid:17) [ σ i,j (˜ S ) ,σ i,j +1 (˜ S )) ( t )+ T i ∧ (cid:16) σ i, Θ (˜ S ) + 1 T i − t − T i − σ i, Θ (˜ S ) (cid:17) [ σ i, Θ ( S ) ,T i ] ( t ) o . Furthermore, g is adapted to F and hence predictable with respect to F P ( N ) – the usual augmen-tation of F (since g is continuous). Therefore, it is clear that ˇ S is a local martingale with respectto F P ( N ) under P ( N ) . Moreover, ˜ S T i = ˇ S T i = S T i for any i ≤ n . This implies that ˇ S is a martingalewith respect to F P ( N ) under P ( N ) and further P ( N ) ◦ (ˇ S t ) − ∈ M ~µ, P , /N .Observe that for any S ∈ Ω such that m ( D − n ( S ) ≤ Θ it follows from (3.16) – the time continuityproperty of G | G (˜ S ( S )) − G (ˇ S ( S )) | ≤ | G (˜ S ( S )) − G ( F ( D − (˜ S ( S ))) | + | G (ˇ S ( S )) − G ( F ( D − (ˇ S ( S ))) | + | G ( F ( D − ( S )( S )) − G ( F ( D − (ˇ S ( S ))) |≤ f e (2 − D +9 ) + 2 nL k ˜ S k Θ δ ≤ f e (2 − D +9 ) + 2 nL k S ( S ) k /D, (4.37)when D is sufficiently large, where F ( D − is defined in (5.1). From (4.33), Markov inequality gives P ( N ) ( { S ∈ I : m ( D − ( S ) ≥ Θ − n + 2 } ) ≤ κ + 2 κD . (4.38)and hence by (4.35) P ( N ) ( { S ∈ I : m ( D − n ( S ) ≥ Θ + 1 } ) ≤ κ + 2 κD . (4.39)Furthermore, by (4.37) and (4.39) (cid:12)(cid:12) E P ( N ) [ G (˜ S )] − E P ( N ) [ G (ˇ S )] (cid:12)(cid:12) ≤ κ P ( N ) ( m ( D − (˜ S ) > Θ) + 2 f e (2 − D +9 ) + 2 nL E P ( N ) [ k S k ] /D ≤ κ + 42 D + 2 f e (2 − D +9 ) + 2 LnV ( p ) µ n , I ( k S k ) /D. (4.40) Step 4: Third time change: controlling increments of quadratic variation. We say ω ∈ C ([0 , T n ] , R ) admits quadratic variation if m ( N ) ( ω ) − X k =0 (cid:16) ω τ ( N ) k ( ω ) − ω τ ( N ) k +1 ( ω ) (cid:17) converges to a limit as N → ∞ for any i ≤ d + K .32e let h ω i be that limit if ω admits quadratic variation and zero otherwise. In addition, for S ∈ Ω,we say S admits quadratic variation if S ( i ) admits quadratic variation for any i ≤ d + K .It follows from Theorem 4.30.1 in Rogers and Williams [39] that for any P ∈ M , h S i := (cid:0) h S (1) i , . . . , h S ( d + K ) i (cid:1) agrees with the classical definition of quadratic variation of S under P P -a.s.Now Doob’s inequality gives ∀ i ≤ d E P ( N ) [ k ˇ S ( i ) k p ] ≤ (cid:16) pp − (cid:17) p Z [0 , ∞ ) x p µ ( i ) n (d x ) . (4.41)And, by BDG-inequalities, we know there exist constants c p , C p ∈ (0 , ∞ ) such that c p E P ( N ) (cid:2) h ˇ S ( i ) i p/ T n (cid:3) ≤ E P ( N ) [ k ˇ S ( i ) k p ] ≤ C p E P ( N ) (cid:2) h ˇ S ( i ) i p/ T n (cid:3) . (4.42)It follows that E P ( N ) h d + K X i =1 h ˇ S ( i ) i p/ T n i ≤ K , (4.43)where K := (cid:16) c p (cid:17)(cid:16)(cid:16) pp − (cid:17) p P di =1 R [0 , ∞ ) x p µ ( i ) n (d x ) + Kκ p (cid:17) .In the following we want to modify ˇ S on˜ I := { S ∈ I : ˇ S ( S ) admits quadratic variation } = { S ∈ I : S admits quadratic variation } using time change technique to obtain another process ¨ S , the law of which is in M ~µ, P , /N . In fact,¨ S is a time change of ˇ S on each interval [ σ i,j (˜ S ) , σ i,j +1 (˜ S )). Then by continuity of G , it follows that | G (ˇ S ( S )) − G (¨ S ( S )) | ≤ f e (2 − D +9 ) ∀ S ∈ ˜ I ∩ { ˜ S ∈ I : m ( D − n (˜ S ( ˜ S )) ≤ Θ } . This, together with (4.39) and the fact that P (˜ I ) = 1 for any P ∈ M I , yields (cid:12)(cid:12) E P ( N ) [ G (ˇ S ) − G (¨ S )] (cid:12)(cid:12) ≤ f e (2 − D +9 ) + 2 κ P ( N ) ( { S ∈ I : m ( D − n (˜ S ( S )) ≥ Θ + 1 } ) ≤ f e (2 − D +9 ) + 4 κ + 4 D . Hence, by (4.34) and (4.40), (cid:12)(cid:12) E P ( N ) [ G ( S ) − G (¨ S )] (cid:12)(cid:12) ≤ f e (2 − D +9 ) + 2 Ln k S k D + 8 κ + 82 D + 2 LnV ( p ) µ n , I ( k S k ) D . (4.44)First, for every i, j, k , define ρ ( i,j,k ) : Ω → [0 , T n ] by ρ ( i,j,k ) ( S ) = σ i,j (ˇ S ( S )) + δ (1 − − k +1 ) ∀ S ∈ Ω.Then, ∀ i = 1 , . . . , n , j = 0 , , . . . , and k = 1 , , . . . , consider change of time θ ( i,j,k ) : I × [0 , T n ] → [0 , T n ] defined as follows: if S ∈ ˜ I , θ ( i,j,k ) t ( S ) = t ∀ t ≤ ρ ( i,j,k ) ( S ) and for t > ρ ( i,j,k ) ( S ) θ ( i,j,k ) t ( S ) = inf { u ≥ ρ ( i,j,k ) ( S ) : d + K X l =1 (cid:0) h ˇ S ( l ) ( S ) i u − h ˇ S ( l ) ( S ) i ρ ( i,j,k ) (cid:1) > k ( t − ρ ( i,j,k ) ) /δ } ∧ ρ ( i,j,k +1) ∧ σ i,j +1 (ˇ S ( S )) , otherwise θ ( i,j,k ) t ( S ) = t on [0 , T n ].Now, by considering ¨ S – a time change of ˇ S via θ ( i,j,k ) ’s, defined by ¨ S t := ˇ S ( θ ( i,j,k ) t ( S )) − on[ ρ ( i,j,k ) ( S ) , ρ ( i,j,k +1) ( S )) ∀ i, j, k , we see that for any i, j, k and any S ∈ ˜ I the quadratic variation of¨ S ( S ) t grows linearly at rate 2 k on [ ρ ( i,j,k ) ( S ) , ρ ( i,j,k +1) ( S )) with ρ ( i,j,k +1) ( S ) − ρ ( i,j,k ) ( S ) = 2 − k if σ i,j +1 (˜ S ( S )) − σ i,j (˜ S ( S )) > S is a continuous process on ˜ I andfurthermore d + K X l =1 (cid:0) h ¨ S ( l ) ( S ) i t − h ¨ S ( l ) ( S ) i s (cid:1) ≤ k | t − s | /δ ∀ s, t s.t. σ i,j (˜ S ( S )) ≤ s ≤ t ≤ σ i,j +1 (˜ S ( S )) , S ∈ ˜ I is such that P d + Ki =1 h ˇ S ( i ) ( S ) i T n ≤ k . Therefore, on { S ∈ ˜ I , : P d + Ki =1 h ˇ S ( i ) ( S ) i T n ≤ k } , d + K X l =1 (cid:0) h ¨ S ( l ) i t − h ¨ S ( l ) i s (cid:1) ≤ k +1 | t − s | /δ ∀ s, t ∈ [0 , T n ] with | t − s | ≤ δ. (4.45)Hence by Markov inequality P ( N ) (cid:16) d + K X i =1 h ¨ S ( i ) i T n > k (cid:17) = P ( N ) (cid:16) d + K X i =1 h ˇ S ( i ) i T n > k (cid:17) ≤ d + K X i =1 P ( N ) (cid:16) h ˇ S ( i ) i T n > k/ ( d + K ) (cid:17) ≤ E P ( N ) (cid:2) P d + Ki =1 h ˇ S ( i ) i p/ T n (cid:3) ( d + K ) p/ k p/ ≤ ( d + K ) p/ K k − p/ . Step 5: Tightness of measures through tightness of quadratic variation processes. Together with (4.45), by Arzel´a-Ascoli theorem, this implies that { P ( N ) ◦ ( h ¨ S i t ) − } N ∈ N is tight (in C ([0 , T n ] , R d ). Then by Lemma 6.4.13 in Jacod and Shiryaev [29], { P ( N ) ◦ (¨ S t ) − } N ∈ N is tight (in D ([0 , T n ] , R d )), which by Theorem 6.3.21 in Jacod and Shiryaev [29] implies that ∀ ǫ > , η > N ∈ N and θ > N ≥ N ⇒ P ( N ) ( w ′ T n ( S , θ ) ≥ η ) ≤ ǫ, where w ′ T n is defined by w ′ T n ( S, θ ) = inf n max i ≤ r sup t i − ≤ s ≤ t 0, there exist t , . . . , t r with 0 = t < . . . < t r = T n andinf i Then there exists a converging subsequence { P ( N k ) ◦ (¨ S t ) − } such that P ( N k ) ◦ (¨ S t ) − → P weaklyfor some probability measure P on Ω. Consequently,lim k →∞ E P ( Nk ) [ G (¨ S )] = E P [ G ( S )] . In addition, if P is an element of M ~µ, P , then V ( p ) ~µ, P ( G ) ≤ e V ( p ) ~µ, P ( G ) ≤ lim N →∞ sup P ∈M ~µ, I , /N E P h G ( S ) − κ D ∧ p m ( D − ( S )2 D i + c / D ≤ lim inf N →∞ E P ( N ) [ G ( S )] + c / D ≤ lim inf N →∞ E P ( N ) [ G (¨ S )] + e ( D ) ≤ lim k →∞ E P ( Nk ) [ G (¨ S )] + e ( D ) ≤ E P [ G ( S )] + e ( D ) ≤ P ~µ, P ( G ) + e ( D ) . (4.46)34here e ( x ) := 5 f e (2 − x +9 ) + Ln k S k x + c +8 κ +82 x + LnV ( p ) µn, I ( k S k ) x and the third inequality follows from(4.44).It remains to argue that P is an element of M ~µ, P . First, it is straightforward to see that S is a P –martingale and L P ( S T i ) = µ i for any i ≤ n . To show that P ( { S ∈ P } ) = 1, notice that byportemanteau theorem, for every ǫ > P ( { S ∈ P ǫ } ) ≥ lim sup k →∞ P ( N k ) ( { S ∈ P ǫ } ) ≥ lim sup k →∞ P ( N k ) ( { S ∈ P /N k } ) = 1 . Therefore, it follows from Remark 4.7 and monotone convergence theorem that P ( { S ∈ P } ) = lim ǫ> P ( { S ∈ P ǫ } ) = 1 , and hence P ∈ M ~µ, P .To conclude, as D is arbitrary, (4.46) yields that V ( p ) ~µ, P ( G ) ≤ P ~µ, P ( G ) , which then implies that e V ( p ) ~µ, P ( G ) = V ( p ) ~µ, P ( G ) = P ~µ, P ( G ) = e P ~µ, P ( G ) . This and the subsequent section, are devoted to the proof of (3.3) which in turn implies Theorem3.2. The strategy of the proof is inspired by Dolinsky and Soner [21] and proceeds via discretisation,of the dual side in this section and of the primal side in Section 6. The duality between discretecounterparts is obtained using classical probabilistic results of F¨ollmer and Kramkov [24]. The proof of (3.3) is based on a discretisation method involving a discretisation of the path spaceinto a countable set of piece-wise constant functions. These are obtained as a “shift” of the“Lebesgue discretisation” of a path. Recall from Definition 4.1 that for a positive integer N andany S ∈ Ω, τ ( N )0 ( S ) = 0, m ( N )0 ( S ) = 0, τ ( N ) k ( S ) = inf n t ≥ τ ( N ) k − ( S ) : | S t − S τ ( N ) k − ( S ) | = 12 N o ∧ T and m ( N ) ( S ) = min { k ∈ N : τ ( N ) k ( S ) = T } .Now denote by A N the set of γ ∈ A for which we only allow trading in the risky assets to takeplace at the moments 0 = τ ( N )0 ( S ) < τ ( N )1 ( S ) < · · · < τ ( N ) m ( N ) ( S ) ( S ) = T and | γ | ≤ N . Set V ( N ) I ( G ) := inf n x : ∃ γ ∈ A N s.t. γ super-replicates G − x o . Then it is obvious from the definition of V ( N ) I that V ( N ) I ( G ) ≥ V ( N ) I ( G ) ≥ V I ( G ) for any N ≥ N , and in fact, the following result states that V ( N ) I ( G ) converges to V I ( G ) asymptotically. Theorem 5.1. Under the assumptions of Theorem 3.2,lim N →∞ V ( N ) I ( G ) = V I ( G ) . Theorem 5.2. For any α, β ≥ D ∈ N V I ( G − α ∧ ( β p m ( D ) )) ≤ P I ( G − α ∧ ( β p m ( D − )) , where m ( D ) is defined in Definition 4.1. 35 .2 A countable class of piecewise constant functions In this section, we construct a countable set of piecewise constant functions which can give ap-proximations to any continuous function S to a certain degree. It will be achieved in three steps.The first step is to use the Lebesgue partition defined in the last section to discretise a continuousfunction into a piecewise constant function whose jump times are the stopping times. Due to thearbitrary nature of jump times and jump sizes, F ( N ) ( S ), the piecewise constant function generatedthrough this procedure, will take values in an uncountable set. To overcome this, in the subsequenttwo steps, we restrict the jump times and the jump sizes to a countable set and hence define aclass of approximating schemes. Step 1. Let τ ( N ) k ( S ) and m ( N ) ( S ) be defined as in Subsection 5.1. To simplify notations, in thissection we often suppress their dependences on S and N and write m = m ( N ) ( S ) , τ k = τ ( N ) k ( S ) for any k, N. Our first naive approximation F ( N ) : Ω → D ([0 , T ] , R d + K ) is as follows: F ( N ) t ( S ) = m − X k =0 S τ k [ τ k ,τ k +1 ) ( t ) + S T { T } ( t ) for t ∈ [0 , T ], S ∈ Ω. (5.1)Note that F ( N ) ( S ) is piecewise constant and k F ( N ) ( S ) − S k ≤ / N . Step 2. Define a map π ( N ) : R d + → A ( N ) := { − N k : k = ( k , . . . , k d + K ) ∈ N d + K } as π ( N ) ( x ) i := 2 − N ⌈ N x i ⌉ , i = 1 , . . . , d + K. We then define our second approximation ˇ F ( N ) : Ω → D ([0 , T ] , R d + K ) byˇ F ( N ) t ( S ) =( S − π ( N +1) ( S τ )) + m − X k =0 π ( N + k +1) ( S τ k +1 ) [ τ k ,τ k +1 ) ( t )+ π ( N + m ) ( S τ m ) [ τ m − ,T ] ( t ) t ∈ [0 , T ] . Step 3. We now construct the shifted jump times ˆ τ ( N ) k : Ω → Q + ∪ { T } . Firstly, set ˆ τ ( N )0 = 0. Then, forany S ∈ Ω and k = 1 , · · · , m ( N ) ( S ) let∆ˆ τ ( N ) k = ( p k q k with ( p k , q k ) = argmin { p + q : ( p, q ) ∈ N , τ ( N ) k − − ˆ τ ( N ) k − < pq ≤ ∆ τ ( N ) k } if k < m ( N ) ( S ) T − ˆ τ ( N ) m ( N ) − otherwise,where ∆ τ ( N ) k := τ ( N ) k − τ ( N ) k − . Lastly, define ˆ τ ( N ) k := P ki =1 ∆ˆ τ ( N ) i . Here we also suppress thedependences of these shifted jump times on S and N and writeˆ τ k = ˆ τ ( N ) k ( S ) for any k, N. Clearly 0 = ˆ τ < ˆ τ < ˆ τ · · · < ˆ τ m = T , τ k − < ˆ τ k ≤ τ k ∀ k < m and ˆ τ m = τ m = T . These ˆ τ ’s arethe shifted versions of τ ’s, and are uniquely defined for any S . We are going to use ˆ τ ’s to define aclass of approximating schemes.We can define an approximation ˆ F ( N ) : Ω → D ([0 , T ] , R d + K ) byˆ F ( N ) t ( S ) =( S − π ( N +1) ( S τ )) + m − X k =0 π ( N + k +1) ( S τ k +1 ) [ˆ τ k , ˆ τ k +1 ) ( t )+ π ( N + m ) ( S τ m ) [ˆ τ m − ,T ] ( t ) t ∈ [0 , T ] . F ( N ) ( S ) is piecewise constant and k ˆ F ( N ) ( S ) − S k ≤k ˆ F ( N ) ( S ) − ˇ F ( N ) ( S ) k + k ˇ F ( N ) ( S ) − F ( N ) ( S ) k + k F ( N ) ( S ) − S k≤ N − + 22 N + 12 N < N − . (5.2) Definition 5.3. Let ˆ D ( N ) ⊂ D ([0 , T ] , R d + K ) be the set of functions f = ( f ( i ) ) d + Ki =1 which satisfythe following,1. for any i = 1 , . . . , d + K , f ( i ) (0) = 1,2. f is piecewise constant with jumps at times t , · · · , t l − ∈ Q + for some l < ∞ ,where t = t l = 0 < t < t < · · · < t l − < T ,3. for any k = 1 , . . . , l − i = 1 , . . . , d + K , f ( i ) ( t k ) − f ( i ) ( t k − ) = j/ N + k , for j ∈ Z with | j | ≤ k ,4. inf t ∈ [0 ,T ] , ≤ i ≤ d + K f ( i ) ( t ) ≥ − − N +3 ,5. k f ( i ) k ≤ κ + 1 for i = d + 1 , . . . , d + K , where κ = max ≤ j ≤ K k X ( c ) j k ∞ P ( X ( c ) j ) ,6. if f ( i ) ( t k ) = − − N +3 for some i ≤ d + K and k ≤ l − 1, then f ( t j ) = f ( t k ) ∀ k < j < l ,7. if f ( i ) ( t k ) = κ + 1 for some i > d and k ≤ l − 1, then f ( t j ) = f ( t k ) ∀ k < j < l .It is clear that ˆ D ( N ) is countable. Let ˆΩ := D ([0 , T ] , R d + K ) be the space of all right continuous functions f : [0 , T ] → R d + K withleft-hand limits. Denote by ˆ S = (ˆ S t ) ≤ t ≤ T the canonical process on the space ˆΩ.The set ˆ D ( N ) is a countable subset of ˆΩ. There exists a local martingale measure ˆ P ( N ) on ˆΩwhich satisfies ˆ P ( N ) ( ˆ D ( N ) ) = 1 and ˆ P ( N ) ( { f } ) > f ∈ ˆ D ( N ) . In fact, such a local martingalemeasure ˆ P ( N ) on ˆ D ( N ) can be constructed ‘by hand’. Indeed, we can construct a continuous Markovchain that undergoes transitions in the finite number of allowed values in the way that the meanis preserved, with jump times decided via an exponential clock. Let ˆ F ( N ) := { ˆ F ( N ) t } ≤ t ≤ T be thefiltration generated by the process ˆ S and satisfying the usual assumptions (right continuous andcontains ˆ P ( N ) -null sets).In the last section, we saw definitions of ˆ τ ( N ) k on Ω. Here we extend their definitions to S N ∈ N ˆ D ( N ) .Define the jump times by setting ˆ τ (ˆ S ) = 0 and for k > τ k (ˆ S ) = inf (cid:8) t > ˆ τ k − (ˆ S ) : ˆ S t = ˆ S t − (cid:9) ∧ T. (5.3)Next we introduce the random time before Tm (ˆ S ) := min { k : ˆ τ k (ˆ S ) = T } . Observe that for S ∈ Ω, ˆ F ( N ) ( S ) ∈ ˆ D ( N ) , ˆ τ k ( ˆ F ( N ) ( S )) = ˆ τ k ( S ) for all k and m ( ˆ F ( N ) ( S )) = m ( N ) ( S ). It follows that the definitions are consistent.In this context, a trading strategy (ˆ γ t ) Tt =0 on the filtered probability space ( ˆΩ , ˆ F ( N ) , ˆ P ( N ) ) is a pre-dictable stochastic process. Thus, ˆ γ is a map from D ([0 , T ] , R d + K ) to D ([0 , T ] , R d + K ). Choose a ∈D ([0 , T ] , R d + K ) such that a ˆ γ ( ˆ D ( N ) ) and then define a map φ : D ([0 , T ] , R d + K ) → D ([0 , T ] , R d + K )by φ ( ˆ S ) = ˆ γ ( ˆ S ) if ˆ S ∈ ˆ D ( N ) , and equal to a otherwise. Since ˆ P ( N ) has full support on ˆ D ( N ) , ˆ γ = φ (ˆ S )ˆ P ( N ) -a.s.. In particular, for any A ∈ B ( R ), the symmetric difference of { ˆ γ t ∈ A } and { φ (ˆ S ) t ∈ A } 37s a null set for ˆ P ( N ) . Thus φ is a predictable map. Furthermore, since ˆ P ( N ) charges all elementsin ˆ D ( N ) , for any υ, ˜ υ ∈ D ([0 , T ] , R d + K ) and t ∈ [0 , T ]. υ u = ˜ υ u ∀ u ∈ [0 , t ) ⇒ φ ( υ ) t = φ (˜ υ ) t . Indeed, suppose these exist t ∈ [0 , T ] and υ, ˜ υ ∈ ˆ D ( N ) such that υ u = ˜ υ u for all u ∈ [0 , t ) and φ ( υ ) t = φ (˜ υ ) t . Since ˆ γ is predictable, we haveˆ F ( N ) t − ∋ { ˆ γ t = φ ( v ) t } ∩ { S u = v u , u < t } = { ˆ γ t = φ ( v ) t } ∩ { S u = v u , u < t } ∩ { S t = ˜ v t } , which is a contradiction since { ˆ γ t = φ ( v ) t } ∩ { S u = v u , u < t } is not a null set and hence not inˆ F ( N ) t − . We conclude that any predictable process ˆ γ has a version φ that is progressively measurablein the sense of (2.1). In what follows we always take this version.In this section, we formally define the probabilistic super-replicating problem and later build aconnection between the probabilistic super-replication problem on the discretised space and thepath-wise discretised robust hedging problem. For the rest of the section, we write R t t to mean R ( t ,t ] .As G is defined only on Ω, to consider paths in ˆΩ, we need to extend the domain of G to ˆΩ.For most of the financial contracts, the extension is natural. However, here we pursue a generalapproach. We first define a projection function ` : ˆΩ → C ([0 , T ] , R d + K ) by ` ( ˆ S ) = ˆ S if ˆ S is continuous P m ( ˆ S ) − k =0 (cid:16) ˆ S ˆ τk +1 − ˆ S ˆ τk ˆ τ k +1 − ˆ τ k ( t − ˆ τ k ) + ˆ S ˆ τ k (cid:17) [ˆ τ k , ˆ τ k +1 ) ( t ) if ˆ S ∈ S N ∈ N ˆ D ( N ) ω otherwise,where ω is the constant path equal to 1. In fact, when ˆ S ∈ S N ∈ N ˆ D ( N ) , ` ( ˆ S ) is the minimum of0 and the linear interpolation function of (cid:0) (ˆ τ ( ˆ S ) , ˆ S ˆ τ ( ˆ S ) ) , . . . , (ˆ τ m ( ˆ S ) ( ˆ S ) , ˆ S ˆ τ m ( ˆ S ) ( ˆ S ) ) (cid:1) . We then can define ˆ G : ˆΩ → Ω via this explicit projection ` by ˆ G ( ˆ S ) = G ( ` ( ˆ S ) ∨ S ∨ (cid:0) ( ˆ S (1) t ∨ , . . . , ˆ S ( d + K ) t ∨ (cid:1) ≤ t ≤ T for any S ∈ ˆΩ.Note that G and ˆ G are equal on Ω. In addition, for every N ∈ N and ˆ S ∈ ˆ D ( N ) , we have k ` ( ˆ S ) − ˆ S k ≤ − N +1 . (5.4)Therefore, we can deduce that k ` ( ˆ F ( S )) ∨ − S k ≤k ` ( ˆ F ( S )) ∨ − ˆ F ( S ) ∨ k + k ˆ F ( S ) ∨ − S k≤ − N +1 + 2 − N +3 ∀ S ∈ Ω . (5.5)where the last inequality follows from (5.2) and (5.4).Similarly, for each D ∈ N , we define ˆ m ( D ) : ˆΩ → Ω by ˆ m ( D ) ( ˆ S ) = m ( D ) ( ` ( ˆ S ) ∨ N is sufficiently large,ˆ m ( D − ( ˆ F ( N ) ( S ))) ≤ m ( D ) ( S ) ∀ S ∈ Ω . (5.6) Definition 5.4. 1. ˆ γ : ˆΩ → D ([0 , T ] , R d + K ) is ˆ P ( N ) -admissible if ˆ γ is predictable and bounded by N , and thestochastic integral ( R t ˆ γ u (ˆ S ) · d ˆ S u ) ≤ t ≤ T is well defined under ˆ P ( N ) , satisfying that ∃ M > Z t ˆ γ u (ˆ S ) · d ˆ S u ≥ − M ˆ P ( N ) − a.s., t ∈ [0 , T ) . (5.7)38. An admissible strategy ˆ γ is said to ˆ P ( N ) - super-replicate ˆ G if Z T ˆ γ u (ˆ S ) · d ˆ S u ≥ ˆ G (ˆ S ) , ˆ P ( N ) − a.s.. (5.8)3. The super-replicating cost of ˆ G is defined asˆ V ( N ) := inf { x : ∃ ˆ γ s.t. ˆ γ is ˆ P ( N ) -admissible and ˆ P ( N ) -super-replicates ˆ G − x } For the rest of the section we will establish connections between probabilistic super-hedging prob-lems and discretised robust hedging problems. Our reasoning is close to the one in Dolinsky andSoner [21]. Definition 5.5. Given a predictable stochastic process (ˆ γ t ) Tt =0 on ( ˆΩ , ˆ F ( N ) , ˆ P ( N ) ), we define γ ( N ) :Ω → D ([0 , T ] , R d + K ) by γ ( N ) t ( S ) := m − X k =0 ˆ γ ˆ τ k ( ˆ F ( N ) ( S )) ( τ k ,τ k +1 ] ( t ) , (5.9)where τ k = τ ( N ) k ( S ), m = m ( N ) ( S ) are given in Definition 4.1 and ˆ τ k = ˆ τ k ( ˆ F ( N ) ( S )) are given in(5.3). Lemma 5.6. For any admissible process ˆ γ in the sense of Definition 5.4, γ ( N ) defined in (5.9) isprogressively measurable in the sense of (2.1). Proof. To see γ ( N ) is progressively measurable, we need to show γ ( N ) t ( ω ) = γ ( N ) t ( υ ) . for any ω, υ ∈ Ω such that ω u = υ u ∀ u ≤ t for some t ∈ (0 , T ], the case t = 0 being true bydefinition. Let t ∈ (0 , T ] and set k t ( ω ) = k ( N ) t ( ω ) := min { i ≥ τ ( N ) i ( ω ) ≥ t } − . (5.10)It is clear that k t ( ω ) = k t ( υ ), τ k t ( ω ) ( ω ) = τ k t ( υ ) ( υ ) and ω u = υ u for all u ≤ τ k t ( ω ) ( ω ).Write θ := τ k t ( ω ) ( ω ). It follows from the definition of ˆ F ( N ) and ˆ τ ’s thatˆ τ k t ( ˆ F ( N ) ( ω )) ( ˆ F ( N ) ( ω )) = ˆ τ k t ( ˆ F ( N ) ( υ )) ( ˆ F ( N ) ( υ )) , ˆ F ( N ) u ( ω ) = ˆ F ( N ) u ( υ ) ∀ u ∈ [0 , θ ) . From (5.9), γ ( N ) t ( ω ) = ˆ γ θ (cid:0) ˆ F ( N ) ( υ ) (cid:1) , γ ( N ) t ( υ ) = ˆ γ θ (cid:0) ˆ F ( N ) ( υ ) (cid:1) . (5.11)Therefore, by the progressive measurability of ˆ γ as argued above, we conclude that γ ( N ) t ( ω ) = γ ( N ) t ( υ ).The following theorem is crucial. It states that the probabilistic super-replicating value is asymp-totically larger than the value of the discretised robust hedging problem. Recall that λ I ( ω ) :=inf υ ∈I k ω − υ k ∧ Theorem 5.7. For uniformly continuous and bounded G , α, β ≥ D ∈ N , we havelim inf N →∞ V ( N ) I ( G ( S ) − α ∧ ( β q m ( D ) ( S ))) ≤ lim inf N →∞ ˆ V ( N ) (cid:16) ˆ G (ˆ S ) − α ∧ ( β q ˆ m ( D − (ˆ S )) − N λ I (ˆ S ) (cid:17) . (5.12) Proof. See Appendix 7.1. 39 .4 Duality for the discretised problems Definition 5.8. 1. Let ˆΠ ( N ) be the set of all probability measures ˆ Q which are equivalent to ˆ P ( N ) .2. For any κ ≥ 0, denote ˆ M ( N ) I ( κ ) by the set of all probability measures ˆ Q ∈ ˆΠ ( N ) such thatˆ Q (cid:0) { ω ∈ ˆΩ : inf υ ∈I k ˆ S ( ω ) − υ k ≥ /N } (cid:1) ≤ κN and E ˆ Q (cid:20) m (ˆ S ) X k =1 d + K X i =1 | E ˆ Q [ˆ S ( i )ˆ τ k | ˆ F ˆ τ k − ] − ˆ S ( i )ˆ τ k − | (cid:21) ≤ κN , where ˆ τ k = ˆ τ k (ˆ S ) and m = m (ˆ S ) are as defined in (5.3). Lemma 5.9. Suppose ˆ G is bounded by κ − M I = ∅ . Then, there are at most finitely many N ∈ N such that ˆ M ( N ) I (2 κ ) = ∅ andlim inf N →∞ ˆ V ( N ) (cid:16) ˆ G (ˆ S ) − N λ I (ˆ S ) (cid:17) ≤ lim inf N →∞ sup ˆ Q ∈ ˆ M ( N ) I (2 κ ) E ˆ Q [ ˆ G (ˆ S )] . (5.13) Proof. Since for any ˆ Q ∈ ˆΠ ( N ) the support of ˆ Q is ˆ D ( N ) , of which elements are piece-wise constant,the canonical process ˆ S is therefore a semi-martingale under ˆ Q . Moreover, it has the followingdecomposition, ˆ S = ˆ M ˆ Q + ˆ A ˆ Q whereˆ A ˆ Q t = m (ˆ S ) X k =1 h E ˆ Q [ˆ S ˆ τ k | ˆ F ˆ τ k − ] − ˆ S ˆ τ k − i [ˆ τ k , ˆ τ k +1 ) ( t ) , t < T, ˆ A ˆ Q T := lim t ↑ T ˆ A ˆ Q t is a predictable process of bounded variation and ˆ M ˆ Q is a martingale under ˆ Q . Then, similar toDolinsky and Soner [22], it follows from Example 2.3 and Proposition 4.1 in F¨ollmer and Kramkov[24] thatˆ V ( N ) (cid:16) ˆ G (ˆ S ) − N λ I (ˆ S ) (cid:17) = sup ˆ Q ∈ ˆΠ ( N ) E ˆ Q (cid:20) ˆ G (ˆ S ) − N λ I (ˆ S ) − N m (ˆ S ) X k =1 d + K X i =1 | E ˆ Q [ˆ S ( i )ˆ τ k | ˆ F ˆ τ k − ] − ˆ S ( i )ˆ τ k − | (cid:21) . (5.14)By Theorem 5.7,lim inf N →∞ ˆ V ( N ) (cid:16) ˆ G (ˆ S ) − N λ I (ˆ S ) (cid:17) ≥ lim inf N →∞ V ( N ) I ( G ) ≥ P I ( G ) > − κ. Then, in (5.14), it suffices to consider the supremum over ˆ M ( N ) I (2 κ ). In particular, ˆ M ( N ) I (2 κ ) = ∅ for N large enough. Next, we show that we can lift any discrete martingale measure in ˆ M ( N ) I ( c ) to a continuous mar-tingale measure in M I such that the difference of expected value of G under this continuous40artingale measure and the expected value of ˆ G under the original discrete martingale measureis within a bounded error, which goes to zero as N → ∞ . Through this, we connect the primalproblems on the discretised space to the approximation of the primal problems on the space ofcontinuous functions asymptotically. Proposition 6.1. Under the assumptions of Theorem 3.2, if G and X ( c ) i / P ( X ( c ) i )’s are boundedby κ − κ ≥ 1, then for any α, β ≥ D ∈ N lim sup N →∞ sup ˆ Q ∈ ˆ M ( N ) I (2 κ + α ) E ˆ Q [ ˆ G (ˆ S ) − α ∧ ( β q ˆ m ( D ) (ˆ S ))] ≤ sup P ∈M I E P [ G ( S ) − α ∧ ( β q m ( D − ( S ))] . (6.1) Proof. Let f e : R d + K + → R + be the modulus of continuity of G , i.e. | G ( ω ) − G ( υ ) | ≤ f e ( | ω − υ | ) for any ω, υ ∈ Ωand lim x ց f e ( x ) = 0. Recall from Lemma 5.9 that ˆ M ( N ) I (2 κ + 2 α ) = ∅ for N large enough. Hence,to show (6.1), it suffices to prove that for any ˆ Q ∈ ˆ M ( N ) I (2 κ + 2 α ) E ˆ Q [ ˆ G (ˆ S ) − α ∧ ( β q ˆ m ( D ) (ˆ S ))] ≤ sup P ∈M I E P [ G ( S ) − α ∧ ( β q m ( D − ( S ))] + g (1 /N ) , (6.2)for some g : R + → R + such that lim x ց g ( x ) = 0. We now fix N and ˆ Q ∈ ˆ M ( N ) I (2 κ + 2 α ) andprove (6.2) in four steps. Step 1. We will first construct a semi-martingale ˆ Z = ˆ M + ˆ A on a Wiener space (Ω W , F W , P W )such that (cid:12)(cid:12) E ˆ Q [ ˆ G (ˆ S )] − E W [ ˆ G ( ˆ Z )] (cid:12)(cid:12) ≤ κ − N +1 (6.3)and P W (cid:0) { ω ∈ Ω W : inf υ ∈I k ˆ M ( ω ) + ˆ A ( ω ) − υ k ≥ /N } (cid:1) ≤ κ + 2 αN + 2 − N , (6.4)where ˆ M is constructed from a martingale and both have piece-wise constant paths.Since the measure ˆ Q is supported on ˆ D ( N ) , the canonical process ˆ S is a pure jump process underˆ Q , with a finite number of jumps ˆ Q -a.s. Consequently there exists a deterministic positive integer m (depending on N ) such that ˆ Q ( ˆ m (ˆ S ) > m ) < − N . (6.5)It follows that | E ˆ Q [ ˆ G (ˆ S )] − E ˆ Q [ ˆ G (ˆ S ˆ τ m )] | ≤ κ − N +1 . (6.6)Notice that by definition of ˆ D ( N ) , the law of ˆ S ˆ τ m under ˆ Q is also supported on ˆ D ( N ) .Let (Ω W , F W , P W ) be a complete probability space together with a standard m + 2-dimensionalBrownian motion n W t = ( W (1) t , · · · , W ( m +2) t ) o ∞ t =0 and the natural filtration F Wt = σ { W s | s ≤ t } .With a small modification to Lemma 5.1 in Dolinsky and Soner [21], we can construct a sequenceof stopping times (with respect to Brownian filtration) σ ≤ σ ≤ · · · ≤ σ m together with F Wσ i -measurable random variable Y i ’s such that L P W (cid:0) ( σ , . . . , σ m , Y , . . . , Y m ) (cid:1) = L ˆ Q (cid:0) (ˆ τ , . . . , ˆ τ m , ˆ S ˆ τ − ˆ S ˆ τ , . . . , ˆ S ˆ τ m − ˆ S ˆ τ m − ) (cid:1) . (6.7)(Detailed construction is provided in the Appendix 7.2.)Define X i as X i = E W [ Y i |F Wσ i − ∨ σ ( σ i )] , i = 1 , . . . , m . | X i | ≤ − N . Also by construction of σ i ’s and Y i ’s, we have E W [ Y i |F Wσ i − ∨ σ ( σ i )] = E W [ Y i | ~σ i , ~Y i − ] , where ~σ i := ( σ , . . . , σ i ), ~Y i := ( Y , . . . , Y i ) and E W is the expectation with respect to P W .From these, we can construct a jump process ( ˆ A t ) Tt =0 byˆ A t = m X j =1 X j [ σ j ,T ] . In particular, for k ≤ m ˆ A σ k = k X j =1 X j . Set a martingale ( M t ) Tt =0 as M t = 1 + E W h m X j =1 ( Y j − X j ) |F Wt i , t ∈ [0 , T ] . (6.8)Since all Brownian martingales are continuous, so is M . Moreover, Brownian motion incrementsare independent and therefore, M σ k = 1 + k X j =1 ( Y j − X j ) , P W − a.s., k ≤ m. We now introduce a stochastic process ( ˆ M t ) Tt =0 , on the Brownian probability space, by settingˆ M t = M σ k for t ∈ [ σ k , σ k +1 ), k < m and ˆ M t = ˆ M σ m for t ∈ [ σ m , T ]. Note that as | Y i − X i | ≤ − N +1 , for any k ≤ m and t ≤ T (cid:12)(cid:12) ˆ M t ∧ σ k +1 ∨ σ k − M t ∧ σ k +1 ∨ σ k (cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) m X j = k +1 E W [( Y j − X j ) |F Wt ∧ σ k +1 ∨ σ k ] (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) m X j = k +2 E W (cid:2) E W [( Y j − X j ) |F Wσ j − ∨ σ ( σ j )] F Wt ∧ σ k +1 ∨ σ k (cid:3) + E W [ Y k +1 − X k +1 |F Wt ∧ σ k +1 ∨ σ k ] (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) E W [ Y k +1 − X k +1 |F Wt ∧ σ k +1 ∨ σ k ] (cid:12)(cid:12)(cid:12) ≤ E W [ | Y k +1 − X k +1 ||F Wt ∧ σ k +1 ∨ σ k ] ≤ − N +1 and hence k ˆ M − M k < − N +2 . (6.9)We also notice that ˆ Z = ˆ M + ˆ A satisfies ˆ Z = ˆ S and L P W (cid:0) ( σ , . . . , σ m , Y , . . . , Y m ) (cid:1) = L ˆ Q (cid:0) (ˆ τ , . . . , ˆ τ m , ˆ S ˆ τ − ˆ S ˆ τ , . . . , ˆ S ˆ τ m − ˆ S ˆ τ m − ) (cid:1) . It follows that E W [ ˆ G ( ˆ Z )] = E ˆ Q [ ˆ G (ˆ S τ m )] . (6.10)In particular, by (6.6) we see that (6.3) holds and also by (6.5) and definition of ˆ M and ˆ A (6.4)holds. Step 2. We will shortly construct a continuous martingale M θ from M such that M θ isbounded below by − − N +2 − N − and | E W [ ˆ G ( M θ )] − E ˆ Q (cid:2) ˆ G (ˆ S ) (cid:3) | ≤ c N − + 2 f e ( N − + 2 − N +2 ) + 2 − N . (6.11)42s the law of ˆ Z is the same as ˆ S m under ˆ Q , it follows from the fact that ˆ Q is supported on ˆ D ( N ) and any f ∈ ˆ D ( N ) is above − − N +3 thatˆ Z ≥ − N +3 , P W -a.s. . (6.12)Then, by combining this with (5.4) and (6.9), we can deduce that k ` ( ˆ Z ) − M k ≤k ` ( ˆ Z ) − ˆ Z ∨ k + k ˆ Z ∨ − ˆ Z k + k ˆ Z − M k≤ − N +1 + 2 N +3 + k ˆ M − M k + k ˆ A k≤ − N +4 + N − , whenever max ≤ i ≤ d + K m X k =1 | X ( i ) k | ≤ N − . It follows that | ˆ G ( M ) − ˆ G ( ˆ Z ) | = | G ( M ∨ − G ( ` ( ˆ Z ) ∨ |≤ f e (2 − N +4 + N − ) , whenever max ≤ i ≤ d + K m X k =1 | X ( i ) k | ≤ N − , where we use the fact that k ` ( ˆ Z ) ∨ − M ∨ k ≤ k ` ( ˆ Z ) − M k . Hence, since ˆ G is bounded by κ (cid:12)(cid:12) E W [ ˆ G ( M )] − E W [ ˆ G ( ˆ Z )] (cid:12)(cid:12) ≤ f e (2 − N +4 + N − ) + 2 κP W (cid:16) max ≤ i ≤ d + K m X k =1 | X ( i ) k | > N − (cid:17) . Note that X k = E W (cid:2) Y k | ~σ k , ~Y k − (cid:3) ( d ) = E ˆ Q (cid:2) ˆ S ˆ τ k − ˆ S ˆ τ k − | ~ ˆ τ k , ~ ∆ˆ S ˆ τ k − (cid:3) = E ˆ Q (cid:2) ˆ S ˆ τ k − ˆ S ˆ τ k − | ˆ F ˆ τ k − (cid:3) where ∆ˆ S k = ˆ S ˜ τ k − ˆ S ˜ τ k − for k ≤ m and hence E W h d + K X i =1 m X k =1 | X ( i ) k | i = E ˆ Q (cid:20) m X k =1 d + K X i =1 | E ˆ Q [ˆ S ( i )ˆ τ k | ˆ F ˆ τ k − ] − ˆ S ( i )ˆ τ k − | (cid:21) . By Markov inequality and definition of ˆ M ( N ) I (2 κ ), we have P W (cid:16) d + K X i =1 m X k =1 | X ( i ) k | > N − (cid:17) ≤√ N E W h d + K X i =1 m X k =1 | X ( i ) k | i ≤√ N E ˆ Q (cid:20) m X k =1 d + K X i =1 | E ˆ Q [ˆ S ( i )ˆ τ k | ˆ F ˆ τ k − ] − ˆ S ( i )ˆ τ k − | (cid:21) ≤ κN − . (6.13)Therefore, we have | E W [ ˆ G ( M )] − E W [ ˆ G ( ˆ Z )] | ≤ f e (2 − N +4 + N − ) + 4 κ N − . (6.14)By (6.9), (6.12) and (6.13) P W (cid:0) inf ≤ t ≤ T min ≤ i ≤ d + K M ( i ) t > − − N +4 − N − and max d ≤ i ≤ d + K k M ( i ) k < κ + 1 + 2 − N +2 + N − (cid:1) ≥ − κN − . (6.15)Hence a stopped process M θ , with θ := inf (cid:8) t ≥ ≤ i ≤ d + K M ( i ) t ≤ − − N +4 − N − or max d ≤ i ≤ d + K k M ( i ) k ≥ κ + 1 + 2 − N +2 + N − (cid:9) , | E W [ ˆ G ( M )] − E W [ ˆ G ( M θ )] | ≤ κ N − . (6.16)By (6.6), (6.14) and (6.16), it follows that | E W [ ˆ G ( M θ )] − E ˆ Q (cid:2) ˆ G (ˆ S ) (cid:3) |≤| E W [ ˆ G ( M θ )] − E W [ ˆ G ( M )] | + | E W [ ˆ G ( M )] − E W [ ˆ G ( ˆ Z )] | + | E ˆ Q [ ˆ G (ˆ S ˆ τ m )] − E ˆ Q [ ˆ G (ˆ S )] |≤ κ N − + 4 κ N − + f e (2 − N +4 + N − ) + κ − N +1 . (6.17)In addition, by (6.9) and (6.13) we can deduce from (6.4) that P W (cid:0) { ω ∈ Ω W : inf υ ∈I k M θ ( ω ) − υ k ≥ /N + N − + 2 − N +2 } (cid:1) ≤ κ + 2 αN + 2 − N + 2 κN − which for simplicity we notice that it implies for N large enough P W (cid:0) { ω ∈ Ω W : inf υ ∈I k M θ ( ω ) − υ k ≥ κN − } (cid:1) ≤ κN − . (6.18)Similarly, by (6.9) and (6.13), we have P W ( k ˆ Z − M θ k ≥ − N +2 + N − ) ≤ κN − . (6.19) Step 3. The next step is to modify the martingale M θ in such way that Γ, the new continuousmartingale, is non-negative.Write ǫ N = 2 − N +4 + N − and define a non-negative F WT -measurable random variable Λ byΛ = ( M T ∧ θ + ǫ N ) / (1 + ǫ N ). Then | Λ − M T n ∧ θ | = (cid:12)(cid:12)(cid:12) ǫ N − M T ∧ θ ǫ N (cid:12)(cid:12)(cid:12) ≤ ǫ N (1 + | M T ∧ θ | ) . Note that for any i > d k Λ ( i ) k ≤ κ + 1 + 2 − N +2 + N − + ǫ N ≤ κ + 2 for N large enough. We nowconstruct a continuous martingale from the Λ by taking conditional expectations:Γ t = E W [Λ |F Wt ] , t ∈ [0 , T ]and Λ ≥ ( i )0 = 1 ∀ i ≤ d + K . Hence P ( N ) := P W ◦ (Γ t ) − ∈ M .We first notice that E W [ | M ( i ) T ∧ θ | ] = E W [ M ( i ) T ∧ θ − M ( i ) T ∧ θ ) − ] ≤ E W [ M ( i ) T ∧ θ + 2] = 3 ∀ i = 1 , . . . , d + K. Then by Doob’s martingale inequality P W ( k Γ − M θ k ≥ ǫ / N ) ≤ ǫ − / N d + K X i =1 E W [ | Λ ( i ) − M ( i ) T ∧ θ | ] ≤ ǫ − / N d + K ) ǫ N = 4( d + K ) ǫ / N . (6.20)This together with (6.17) yields (cid:12)(cid:12) E W [ G (Γ)] − E ˆ Q [ G (ˆ S )] (cid:12)(cid:12) ≤ E W (cid:2) | G (Γ) − ˆ G ( M θ ) | (cid:3) + (cid:12)(cid:12) E W [ ˆ G ( M θ )] − E ˆ Q [ ˆ G (ˆ S )] (cid:12)(cid:12) ≤ E W (cid:2) | G (Γ) − G ( M θ ∨ | n k Γ − M θ k <ǫ / N o (cid:3) + 8 κ ( d + K ) ǫ / N + 8 κ N − + f e (2 − N +4 + N − ) + κ − N +1 ≤ f e ( ǫ / N ) + 8 κ ( d + K ) ǫ / N + 9 κ ǫ / N + f e ( ǫ / N ) ≤ f e ( ǫ / N ) + 17 κ ( d + K ) ǫ / N . P W (cid:0) { ω ∈ Ω W : inf υ ∈I k Γ( ω ) − υ k ≥ κN − + ǫ / N } (cid:1) ≤ κN − + 4 ǫ / N . (6.21)and P W ( k ˆ Z − Γ k ≥ κN − + ǫ / N ) ≤ κN − + 4 ǫ / N . (6.22) Step 4. The last step is to construct a new process ˜Γ from Γ such that the law of ˜Γ under P W is an element of M I .We write η N = 4 κN − + 4 ǫ / N and p ( N ) i := E P ( N ) [ X ( c ) i ( S (1) T , . . . , S ( d ) T )]for any i = 1 , . . . K , and define ˜ p ( N ) i ’s by˜ p ( N ) i = P ( X ( c ) i ) − (1 − √ η N ) p ( N ) i √ η N . Note that as E P ( N ) [ X ( c ) i ( S (1) T , . . . , S ( d ) T )] = P ( X ( c ) i ), we can deduce that (cid:12)(cid:12) P ( X ( c ) i ) − p ( N ) i (cid:12)(cid:12) ≤ E W [ | X ( c ) i (Γ (1) T , . . . , Γ ( d ) T ) − P ( X ( c ) i )Γ ( d + i ) T | ] ≤P ( X ( c ) i ) η N + E W h | X ( c ) i (Γ (1) T , . . . , Γ ( d ) T ) − P ( X ( c ) i )Γ ( d + i ) T | n | X ( c ) i (Γ (1) T ,..., Γ ( d ) T ) / P ( X ( c ) i ) − Γ ( d + i ) T | >η N o i ≤P ( X ( c ) i ) η N + 2( κ + 1) P ( X ( c ) i ) η N , ∀ i = 1 , . . . , K. It follows immediately that (cid:12)(cid:12) ˜ p ( N ) i − P ( X ( c ) i ) (cid:12)(cid:12) = (cid:16) √ η N − (cid:17)(cid:12)(cid:12)(cid:12) P ( X ( c ) i ) − p ( N ) i (cid:12)(cid:12)(cid:12) ≤ κ + 1) P ( X ( c ) i ) η N √ η N = 2( κ + 1) P ( X ( c ) i ) √ η N ∀ i ≤ K. (6.23)Then, it follows from Assumption 3.1 that when N is large enough there exists a ˜ P ( N ) ∈ M ˜ I suchthat ˜ p ( N ) i := E ˜ P ( N ) [ X i ( S (1) T , . . . , S ( d ) T )] ∀ i ≤ K. Enlarge Wiener space (Ω W , F W , P W ) if necessary, then there are continuous martingales Γ and˜ M which have laws equal to P ( N ) and ˜ P ( N ) respectively, and an F WT -measurable random variable ξ ∈ { , } that is independent of Γ and ˜ M , with P W ( ξ = 1) = 1 − √ η N and P W ( ξ = 0) = √ η N . Define F WT -measurable random variables ˜Λ ( i ) by˜Λ ( i ) = Γ ( i ) T { ξ =1 } + ˜ M ( i ) T { ξ =0 } ∀ i = 1 , . . . , d, ˜Λ ( i ) = X i − d (˜Λ (1) , . . . , ˜Λ ( d ) ) / P ( X ( c ) i − ) ∀ i > d. We now construct a continuous martingale from ˜Λ by taking conditional expectations:˜Γ t = E W [˜Λ |F Wt ] , t ∈ [0 , T ] . 45t follows from the fact that ξ is independent of M and ˜ M ˜Γ ( i )0 = E W [˜Γ ( i ) T |F W ]=(1 − √ η N ) E W [ X i (Γ (1) T , . . . , Γ ( d ) T ) / P ( X ( c ) i )] + √ η N E W [ X i ( ˜ M (1) T , . . . , ˜ M ( d ) T ) / P ( X ( c ) i )]= (1 − √ η N ) p Ni + √ η N ˜ p Ni P ( X ( c ) i ) = 1 ∀ i > d and ˜Γ ( i )0 = E W [˜Γ ( i ) T |F W ] = E W [˜Λ ( i ) T |F W ] = (1 − η N ) E W [Γ ( i ) T ] + η N E W [ ˜ M ( i ) T ] = 1 ∀ i ≤ d. Hence ˜ P := P W ◦ (˜Γ t ) − ∈ M I . Also by independence between ξ and ( M, ˜ M ), we have E W [ | ˜Λ ( i ) − Γ ( i ) T | ] = √ η N E W [ | ˜ M ( i ) T − Γ ( i ) T | ] ≤ √ η N ∀ i ≤ d and by (6.21) P W ( | Γ T − ˜Λ ( i ) | > η N ) ≤ η N + √ η ≤ √ η N ∀ i > d, which implies that E W [ | ˜Λ ( i ) − Γ ( i ) T | ] = 2 E W [(˜Λ ( i ) − Γ ( i ) T ) + ] − E W [˜Λ ( i ) − Γ ( i ) T ]= 2 E W [(˜Λ ( i ) − Γ ( i ) T ) + ] ≤ η N + 2 E W h Λ ( i ) n | ˜Λ ( i ) − Γ ( i ) T | >η N o i ≤ η N + 4( κ + 2) √ η N ≤ κ √ η N , ∀ i = d + 1 , . . . , K. Then by Doob’s martingale inequality P W ( k ˜Γ − Γ k ≥ κη / N ) ≤ κη / N d + K X i =1 E W [ | ˜Λ ( i ) − Γ ( i ) T | ] ≤ d + K ) η / N and hence (cid:12)(cid:12) E ˜ P [ G ( S )] − E P ( N ) [ G ( S )] (cid:12)(cid:12) = (cid:12)(cid:12) E W [ G (˜Γ) − G (Γ)] (cid:12)(cid:12) ≤ f e ( κη / N ) + E W h | G (Γ) − G (Γ) | n k ˜Γ − Γ k≥ κη / N o i ≤ f e ( κη / N ) + 28 κ ( d + K ) η / N . In addition, we can decuce from (6.22) that P W ( k ˆ Z − ˜Γ k ≥ κη / N + 4 κN − + ǫ / N ) ≤ κN − + 4 ǫ / N + 14( d + K ) κη / N . (6.24)Notice that when N is sufficiently large such that κη / N + 4 κN − + ǫ / N < − D − , on the event (cid:8) ω ∈ Ω W : k ˆ Z ( ω ) − ˜Γ( ω ) k < κη / N + 4 κN − + ǫ / N and ˆ Z ( ω ) ∈ ˆ D ( N ) (cid:9) , we can deduce from (5.4)and (6.12) that | ` ( ˆ Z ) ∨ − ˜Γ | ≤ | ` ( ˆ Z ) ∨ − ` ( ˆ Z ) | + | ` ( ˆ Z ) − Z | + | Z − ˜Γ | < − N +3 + 2 − N +1 + 2 − D +1 ≤ − D . and hence by Remark 4.2 the inequality ˆ m ( D ) ( ˆ Z ) ≥ m ( D − (˜Γ) holds on (cid:8) ω ∈ Ω W : k ˆ Z ( ω ) − ˜Γ( ω ) k < κη / N + 4 κN − + ǫ / N and ˆ Z ( ω ) ∈ ˆ D ( N ) (cid:9) .It follows that E ˆ Q [ α ∧ ( β q ˆ m ( D ) (ˆ S ))] ≥ E ˆ Q [ α ∧ ( β q ˆ m ( D ) (ˆ S m ))]= E W [ α ∧ ( β q ˆ m ( D ) ( ˆ Z ))] ≥ E W [ α ∧ ( β q m ( D − (˜Γ))] − α (cid:16) κN − + 4 ǫ / N + 14( d + K ) κη / N (cid:17) . Appendix Proof. Fix N ≥ 6. Choose f e : R + → R + such that | G | ≤ κ , | G ( ω ) − G ( υ ) | ≤ f e ( | ω − υ | ) for any ω, υ ∈ Ω and lim x → f e ( x ) = 0. Define G ( N ) : Ω → R as G ( N ) ( S ) := ˆ G ( S ) − f e (2 − N +4 ) − d + K ) N N . and V ( N ) I ( G − α ∧ ( β p m ( D ) )) = V ( N ) I ( G ( N ) − α ∧ ( β p m ( D ) )) + f e (2 − N +4 ) + 14( d + K ) N N . Hence, to show (5.12), it suffices to show V ( N ) I ( G ( N ) − α ∧ ( β p m ( D ) )) ≤ ˆ V ( N ) (cid:16) ˆ G − α ∧ ( β p ˆ m ( D − ) − N λ I (cid:17) . (7.1)The rest of proof is structured to establish (7.1). Given a probabilistic semi-static portfolio ˆ γ which super-replicates ˆ G − α ∧ ( β √ ˆ m ( D − ) − N λ I − x , we will argue that the lifted progressivelymeasurable trading strategy γ ( N ) super-replicates G ( N ) − α ∧ ( β √ m ( D ) ) − x on I . To simplifynotations, throughout the rest of the proof, we fix S ∈ I and write ˆ F := ˆ F ( N ) ( S ). Super-replication: We first notice that for any j < m − | ( S τ j +1 − S τ j ) − ( ˆ F ˆ τ j − ˆ F ˆ τ j − ) |≤| S τ j +1 − ˆ F ˆ τ j | + | S τ j − ˆ F ˆ τ j − | ≤ N + j +1 + 12 N + j = 32 N + j +1 . It follows that for any k < m , (cid:12)(cid:12)(cid:12) Z τ k γ ( N ) u ( S ) · dS u − Z ˆ τ k ˆ γ u ( ˆ F ) · d ˆ F u |≤ (cid:12)(cid:12)(cid:12) k − X j =0 ˆ γ ˆ τ j ( ˆ F ) · ( S τ j +1 − S τ j ) − k − X j =0 ˆ γ ˆ τ j +1 ( ˆ F ) · ( ˆ F ˆ τ j +1 − ˆ F ˆ τ j ) (cid:12)(cid:12)(cid:12) ≤ k − X j =0 (cid:12)(cid:12)(cid:12) ˆ γ ˆ τ j +1 ( ˆ F ) · (cid:0) ( S τ j +2 − S τ j +1 ) − ( ˆ F ˆ τ j +1 − ˆ F ˆ τ j ) (cid:1)(cid:12)(cid:12)(cid:12) + 2( d + K ) N N − ≤ ∞ X j =0 N ( d + K )2 N + j +2 + 2( d + K ) N N − ≤ d + K ) N N . (7.2)In addition, (cid:12)(cid:12)(cid:12) Z Tτ m − γ ( N ) u ( S ) · dS u − Z T ˆ τ m − ˆ γ u ( ˆ F ) · d ˆ F u (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) ˆ γ ˆ τ m − ( ˆ F ) · ( S T − S τ m − ) − ˆ γ ˆ τ m ( ˆ F ) · ( ˆ F ˆ τ m − ˆ F ˆ τ m − ) (cid:12)(cid:12)(cid:12) ≤ N ( d + K )2 N . (7.3)Hence, x + Z T γ ( N ) u ( S ) · dS u ≥ x + Z T ˆ γ u ( ˆ F ) · d ˆ F u − d + K ) N N − ( d + K ) N N (7.4) ≥ ˆ G ( ˆ F ) − α ∧ ( β q ˆ m ( D − ( ˆ F )) − N λ I ( ˆ F ) − d + K ) N N (7.5) ≥ ˆ G ( ˆ F ) − α ∧ ( β q ˆ m ( D − ( ˆ F )) − N/ N − − d + K ) N N (7.6) ≥ G ( S ) − α ∧ ( β q m ( D ) ( S )) − f e (2 − N +4 ) − d + K ) N N = G ( N ) ( S )47here the inequality between (7.4) and (7.5) follows from the super-replicating property of ˆ γ andthe fact that ˆ P ( N ) ( f ) > ∀ f ∈ ˆ D ( N ) , the inequality between (7.5) and (7.6) is justified by (5.2)andthe last inequality is given by (5.5) and 5.6. Admissibility :Now, for a given t < T , let k < m be the largest integer so that τ k ( S ) ≤ t . It follows from (7.2)and (7.3) that Z t γ ( N ) u ( S ) · dS u = Z τ k γ ( N ) u ( S ) · dS u + Z tτ k γ ( N ) u ( S ) · dS u ≥ Z ˆ τ k ˆ γ u ( ˆ F ) · d ˆ F u − d + K ) N N − N ( d + K ) max i | S ( i ) t − S ( i ) τ k | (7.7) ≥ − M − d + K ) N N . (7.8)where the inequality between (7.7) and (7.8) follows from the admissibility of ˆ γ and the fact thatˆ P ( N ) ( f ) > , ∀ f ∈ ˆ D ( N ) . Hence, π ( N ) is admissible. σ ’s and Y in Theorem 6.1 For an integer m and given x , · · · , x m , introduce the notation ~x m := ( x , · · · , x m ) . Set T := Q + ∪ { a ≥ a = T − b for some b ∈ Q + } = { t l } ∞ l , S k := { N + k ( a , . . . , a d + K ) : a j ∈ Z , | a j | ≤ k , j = 1 , . . . , d + K } , where ( t l ) l ≥ is a decreasing sequence of strictly positive numbers t k ց t = T . For k = 1 , · · · , m , define the functions Ψ k , Φ k : T k × S × . . . × S k − → [0 , 1] byΨ k ( ~α k ; ~β k − ) := ˆ Q (ˆ τ k − ˆ τ k − ≥ α k | B ) , (7.9)where B := (cid:8) ˆ τ i − ˆ τ i − = α i , ˆ S ˆ τ i − ˆ S ˆ τ i − = β i , i ≤ k − (cid:9) , and Φ k ( ~α k ; ~β k − ; β ) := ˆ Q (ˆ S ˆ τ k − ˆ S ˆ τ k − = β | C ) , β ∈ S k , (7.10)where C := (cid:8) ˆ τ k ≤ T, ˆ τ j − ˆ τ j − = α j , ˆ S ˆ τ i − ˆ S ˆ τ i − = β i , j ≤ k, i ≤ k − (cid:9) . As usual we set ˆ Q ( ·|∅ ) ≡ 0. Next, for k ≤ m , we define the maps Υ k : T k × S × . . . × S k − → [ −∞ , ∞ ] and Θ k : T k × S × . . . × S k − → [ −∞ , ∞ ], as the unique solutions of the followingequations, P W (cid:16) W (1) α k < Υ k ( ~α k ; ~β k − ; s k,l ) (cid:17) = l X j =1 Φ k ( ~α k ; ~β k − ; s k,j ) (7.11)where { s k, , s k, , . . . , s k,l , . . . } is an enumeration of S k , and P W (cid:16) W (1) t l − W (1) t l +1 < Θ k ( ~α k ; ~β k − ) (cid:17) = Ψ k ( ~α k − , t l ; ~β k − )Ψ k ( ~α k − , t l +1 ; ~β k − ) , (7.12)where l ∈ N is given by α k = t l ∈ T . From the definitions it follows that Ψ k ( ~α k − , t l ; ~β k − ) ≤ Ψ k ( ~α k − , t l +1 ; ~β k − ). Thus if Ψ k ( ~α k − , t l +1 ; ~β k − ) = 0 for some l , then Ψ k ( ~α k − , t l ; ~β k − ) = 0. We48et 0 / ≡ σ ≡ σ , . . . , σ m , Y , . . . , Y m by the following recursiverelations σ = ∞ X k =1 t k n W (1) tk − W (1) tk +1 > Θ ( t k ) o ∞ Y j = k +1 n W (1) tj − W (1) tj +1 < Θ ( t j ) o ,Y = ∞ X j =1 s ,j n Υ ( σ ; s ,j − ) ≤ W (2) σ < Υ ( σ ; s ,j ) o , and for i ≥ σ i = σ i − + ∆ i (7.13) Y i = { σ i Applied Math Finance (2), 73–88.[3] Bayraktar, E. and Zhou, Z. [2014], On arbitrage and duality under model uncertainty andportfolio constraints. arXiv:1402.2596v3.[4] Beiglb¨ock, M., Henry-Labord`ere, P. and Penkner, F. [2013], ‘Model-independent bounds foroption prices: a mass transport approach’, Finance and Stochastics (3), 477–501.[5] Biagini, S., Bouchard, B., Kardaras, C. and Nutz, M. [2014], ‘Robust fundamental theoremfor continuous processes’, arXiv:1410.4962 .[6] Black, F. and Scholes, M. [1973], ‘The pricing of options and corporate liabilities’, Journal ofpolitical economy (3), 637–654.[7] Bogachev, V. I. [2007], Measure theory. Vol. I, II , Springer-Verlag, Berlin.[8] Bouchard, B. and Nutz, M. [2015], ‘Arbitrage and duality in nondominated discrete-timemodels’, Ann. Appl. Prob (2), 823–859.[9] Breeden, D. T. and Litzenberger, R. H. [1978], ‘Prices of state-contingent claims implicit inoption prices’, Journal of Business pp. 621–651.[10] Brown, H., Hobson, D. and Rogers, L. C. G. [2001], ‘Robust hedging of barrier options’, Math.Finance (3), 285–314.[11] Burzoni, M., Frittelli, M. and Maggis, M. [2015], Universal arbitrage aggregator in discretetime markets under uncertainty. arXiv:1407.0948v2.[12] Cassese, G. [2008], ‘Asset pricing with no exogenous probability measure’, Math. Finance (1), 23–54.[13] Cox, A. M. G. and Wang, J. [2013], ‘Root’s barrier: Construction, optimality and applicationsto variance options’, Annals of Applied Probability (3), 859–894.[14] Cox, A. M., Hou, Z. and Ob l´oj, J. [2014], ‘Robust pricing and hedging under trading restric-tions and the emergence of local martingale models’, arXiv:1406.0551 .[15] Cox, A. M. and Ob l´oj, J. [2011], ‘Robust pricing and hedging of double no-touch options’, Finance and Stochastics (3), 573–605.[16] Davis, M. H. A., Ob l´oj, J. and Raval, V. [2014], ‘Arbitrage bounds for prices of weightedvariance swaps’, Mathematical Finance , 821–854.[17] Davis, M. H. and Hobson, D. G. [2007], ‘The range of traded option prices’, MathematicalFinance (1), 1–14.[18] Delbaen, F. and Schachermayer, W. [1994], ‘A general version of the fundamental theorem ofasset pricing’, Mathematische Annalen (1), 463–520.[19] Denis, L. and Martini, C. [2006], ‘A theoretical framework for the pricing of contingent claimsin the presence of model uncertainty’, Ann. Appl. Probab. (2), 827–852.[20] Dolinsky, Y. and Soner, H. [2014 a ], ‘Robust hedging with proportional transaction costs’, Finance and Stochastics (2), 327–347.[21] Dolinsky, Y. and Soner, H. M. [2013], ‘Martingale optimal transport and robust hedging incontinuous time’, Probability Theory and Related Fields (1-2), 391–427.5022] Dolinsky, Y. and Soner, H. M. [2014 b ], ‘Martingale optimal transport in the Skorokhod space’, arXiv:1404.1516 .[23] Fahim, A. and Huang, Y. [2014], Model-independent superhedging under portfolio constraints.arXiv:1402.2599v2.[24] F¨ollmer, H. and Kramkov, D. [1997], ‘Optional decompositions under constraints’, ProbabilityTheory and Related Fields (1), 1–25.[25] Galichon, A., Henry-Labord`ere, P. and Touzi, N. [2014], ‘A stochastic control approach tono-arbitrage bounds given marginals, with an application to lookback options’, Ann. Appl.Prob (1), 312–336.[26] Guo, G., Tan, X. and Touzi, N. [2015], Tightness and duality of martingale transport on theSkorokhod space. private communication.[27] Henry-Labord`ere, P., Ob l´oj, J., Spoida, P. and Touzi, N. [2015], ‘The maximum maximum ofa martingale with given n marginals’, Ann. Appl. Prob . to appear, arXiv:1203.6877v3.[28] Hobson, D. G. [1998], ‘Robust hedging of the lookback option’, Finance and Stochastics (4), 329–347.[29] Jacod, J. and Shiryaev, A. N. [2002], Limit theorems for stochastic processes , Vol. 288, SpringerBerlin.[30] Knight, F. [1921], Risk, Uncertainty and Profit , Boston: Houghton Mifflin.[31] Lyons, T. J. [1995], ‘Uncertain volatility and the risk-free synthesis of derivatives’, AppliedMath Finance (2), 117–133.[32] Merton, R. C. [1973], ‘Theory of rational option pricing’, Bell Journal of Economics (1), 141–183.[33] Mykland, P. A. [2003], ‘Financial options and statistical prediction intervals’, Ann. Statist. (5), 1413–1438.[34] Nadtochiy, S. and Ob l´oj, J. [2015], Robust pricing and hedging of barrier options with beliefson implied volatility. in preparation.[35] Neufeld, A. and Nutz, M. [2013], ‘Superreplication under volatility uncertainty for measurableclaims’, Electron. J. Probab. (48), 1–14.[36] Ob l´oj, J. [2010], Skorokhod Embedding, in R. Cont, ed., ‘Encyclopedia of Quantitative Fi-nance’, Wiley, pp. 1653–1657.[37] Ob l´oj, J. and Ulmer, F. [2012], ‘Performance of robust hedges for digital double barrier op-tions’, International J.Theoretical and Applied Finance (1), 1–34.[38] Possama¨ı, D., Royer, G. and Touzi, N. [2013], ‘On the robust superhedging of measurableclaims’, Electron. Commun. Probab. , 1–13.[39] Rogers, L. C. G. and Williams, D. [2000], Diffusions, Markov processes and martingales:Volume 2, Itˆo calculus , Vol. 2, Cambridge university press.[40] Samuelson, P. [1965], ‘Rational theory of warrant pricing’, Industrial Management Rev. , 13–31.[41] Shiryaev, A. N. [1984], Probability , Springer-Verlag, New York.[42] Soner, M., Touzi, N. and Zhang, J. [2011], ‘Quasi-sure stochastic analysis through aggrega-tion’, Electron. J. Probab. (2), 1844–1879.[43] Strassen, V. [1965], ‘The existence of probability measures with given marginals’, The Annalsof Mathematical Statistics (2), pp. 423–439.5144] Terkelsen, F. [1972], ‘Some minimax theorems.’, Mathematica Scandinavica , 405–413.[45] Vovk, V. [2012], ‘Continuous-time trading and the emergence of probability’, Finance andStochastics (4), 561–609.[46] Whalley, A. E. and Wilmott, P. [1997], ‘An asymptotic analysis of an optimal hedging modelfor option pricing with transaction costs’, Mathematical Finance7