A unified Framework for Robust Modelling of Financial Markets in discrete time
AA UNIFIED FRAMEWORK FOR ROBUST MODELLING OFFINANCIAL MARKETS IN DISCRETE TIME
JAN OB(cid:32)L ´OJ AND JOHANNES WIESEL
Abstract.
We unify and establish equivalence between the pathwise and thequasi-sure approaches to robust modelling of financial markets in discrete time.In particular, we prove a Fundamental Theorem of Asset Pricing and a Super-hedging Theorem, which encompass the formulations of (Bouchard and Nutz,2015) and (Burzoni et al., 2019a). In bringing the two streams of literaturetogether, we also examine and relate their many different notions of arbitrage.We also clarify the relation between robust and classical P -specific results.Furthermore, we prove when a superhedging property w.r.t. the set of martin-gale measures supported on a set of paths Ω may be extended to a pathwisesuperhedging on Ω without changing the superhedging price. Introduction
Mathematical models of financial markets are of great significance in economicsand finance and have played a key role in the theory of pricing and hedging ofderivatives and of risk management. Classical models, going back to (Samuelson,1965) and (Black and Scholes, 1973) in continuous time, specify a fixed probabilitymeasure P to describe the asset price dynamics. They led to a powerful theoryof complete, and later incomplete, financial markets. The original models haveundergone a myriad of variations including, amongst others, local and stochasticvolatility models and have been widely applied. However, they also faced importantcriticism for ignoring the issue of model uncertainty, particularly so in the wake ofthe 2007/08 financial crisis. Consequently, inspired by the theoretical developmentsgoing back to (Knight, 1921), new modelling approaches emerged which aim toaddress this fundamental issue. These can be broadly divided into two streamsbased on the so-called quasi-sure and pathwise approaches respectively.The quasi-sure approach introduces a set of priors P representing possible marketscenarios. These priors can be very different and P typically contains measureswhich are mutually singular. This presents significant mathematical challenges andled to the theory of quasi-sure stochastic analysis (see, e.g., (Peng, 2004; Denisand Martini, 2006)). In discrete time, this framework was abstracted in (Bouchardand Nutz, 2015), which we call the quasi-sure formulation in the rest of this pa-per. By varying the set of probability measures P between the “extreme” casesof one fixed probability measure, P = { P } , and that of considering all probabilitymeasures, P = P ( X ), this formulation allows for widely different specifications ofmarket dynamics. The quasi-sure approach has been employed to consider modeluncertainty along market frictions and other related problems, see e.g. (Bayraktarand Zhou, 2017; Bayraktar and Zhang, 2016). The pathwise approach addressesKnightian uncertainty in market modelling by describing the set of market scenar-ios in absence of a probability measure or any similar relative weighting of such Date : December 4, 2019.We gratefully acknowledge funding received from the European Research Council under theEuropean Union’s Seventh Framework Programme (FP7/2007-2013) / ERC grant agreement no.335421) and from St. John’s College in Oxford. We also thank Matteo Burzoni for his helpfulremarks and comments. JW further acknowledges support from the German Academic ScholarshipFoundation. a r X i v : . [ q -f i n . M F ] D ec JAN OB(cid:32)L ´OJ AND JOHANNES WIESEL scenarios. It is also referred to as the pointwise, or ω by ω , approach and it bearssimilarity to the way central banks carry out stress tests using scenario generators.In discrete time a suitable theory was obtained in (Burzoni et al., 2019a), basedon earlier developments in (Burzoni et al., 2017, 2016). The methodology buildson the notion of prediction sets introduced in (Mykland et al., 2003) and used incontinuous time in (Hou and Ob(cid:32)l´oj, 2018). The particular case of including all sce-narios is often referred to as the model-independent framework and was pioneeredin (Davis and Hobson, 2007) and (Acciaio et al., 2016). From here, a further modelspecification is carried out by including additional assumptions, which representthe different agents’ beliefs. In this manner paths deemed impossible by all agentsare eliminated. The remaining set of paths is then called the prediction set, or themodel.Both approaches, the quasi-sure and the pathwise, allow thus to interpolate betweenthe two ends of the modelling spectrum, as identified by (Merton, 1973): the model-independent and the model-specific settings (see Figure 1). In doing so, they allowto capture how their outputs change in function of adding or removing modellingassumptions, thus allowing to quantify the impact and risk that a given set ofassumptions bear on the problem at hand, see (Cont, 2006). Both approaches were Quasisure ap-proach:
Addprobability measuresto arrive at P Model-specific approach: Fixed P Universally acceptable settingThis paper Pathwise robustapproach:
Start withgeneral Polish space X , rule out impossiblepaths to get ⌦ Figure 1.
Different approaches to modelling financial marketssuccessful in developing suitable notions of arbitrage and extending the core resultsfrom the classical P -a.s. setting to their more general context. In particular, in bothapproaches, it is possible to establish a Fundamental Theorem of Asset pricing ofthe form No Arbitrage ⇔ Existence of martingale measures Q and a Superhedging Theorem of the formsup Q E Q [ g ] = inf { x | x is the initial capital of a superhedging strategy of g } . Our main contribution is to unify these two approaches to model uncertainty. Weshow that, under mild technical assumptions, the pathwise and quasi-sure Fun-damental Theorems of Asset Pricing and Superhedging Dualities can be inferredfrom one another and are thus equivalent. Our statements follow a meta-structureoutlined below:
Metatheorem.
Suppose we are in the quasi-sure setting with a given set of priors P . Then, there exists a suitable selection of scenarios Ω P such that the pathwiseresult for Ω P implies the quasi-sure result for P .Conversely, suppose we are given a selection of scenarios Ω . Then, there is a set ofpriors P Ω such that the quasi-sure result for P Ω implies the pathwise result for Ω . OBUST MODELLING OF FINANCIAL MARKETS 3
Establishing such equivalence allows us to gain significant additional insights intothe core objects in both approaches, as well as clarify links to the classical model-specific setting. In particular, when transposing the results from the pointwise tothe quasi-sure setup, the key technical analytic product structure assumption inBouchard and Nutz (2015), see Definition 2.1 below, is deduced naturally from theanalyticity of the set of scenarios in (Burzoni et al., 2019a). When establishingthe Superhedging Theorem, we not only show that the pathwise superhedging priceof g is equal to the quasi-sure one, but we also show that both are equal to themodel-specific P -superhedging price, where P depends on the setting, i.e., on P orequivalently on Ω, but also on the payoff g . Finally, the key implication in the proofof the robust Fundamental Theorem of Asset Pricing, i.e., (5) ⇒ (1) in Theorem2.7 below, is obtained by carefully constructing a suitable P ∈ P which does notadmit an arbitrage in the classical sense and hence admits an equivalent martingalemeasure.Furthermore, we survey and relate the concepts of arbitrage used in both ap-proaches. We provide an extensive list of arbitrage notions introduced and usedacross the literature on robust finance and establish clear relations between them.We also investigate in detail the notion of pathwise superhedging. As noted in(Burzoni et al., 2017), the pathwise superhedging duality does not hold for generalclaims g when superhedging on a general set Ω is required. Instead, one has to con-sider hedging on a smaller “efficient” set Ω ∗ (defined as the largest set supportedby martingale measures and contained in Ω) to retain the pricing-hedging duality.We clarify when this is necessary and when one can extend the superhedging du-ality from Ω ∗ to Ω. Intuitively, since there are arbitrage opportunities on Ω \ Ω ∗ ,one could try to superhedge the claim g on Ω \ Ω ∗ without any additional cost byimplementing an arbitrage strategy. We provide a number of counterexamples toshow this idea is not feasible in general and link this to measurability constraintson arbitrage strategies, which were also encountered in (Burzoni et al., 2016). Wethen show that the above-mentioned intuition is only true for essentially uniformlycontinuous g under certain regularity conditions on Ω.The rest of the paper is organised as follows. Section 2 contains the main results.First, in Section 2.1, we introduce the general setup in which we work. We discussdifferent notions of (robust) arbitrage in Section 2.2. Then, in Section 2.3, we estab-lish our version of the robust Fundamental Theorem of Asset Pricing which unifiesthe quasi-sure and pathwise perspectives. And in Section 2.4, we state a robustSuperhedging Theorem. Section 3 presents complementary results on extendingthe superhedging duality from Ω ∗ to Ω without additional cost and on relationsbetween two strong notions of pathwise arbitrage. Finally, Section 4 contains tech-nical results and most of the proofs. In particular, we give the proofs of Theorems2.6 and 2.7 in Section 4.1 and of Theorem 2.9 in Section 4.2.2. Unified Framework for Robust Modelling of Financial Markets
Trading strategies and pricing measures.
We use notation similar to(Bouchard and Nutz, 2015) and work in their setting, so we only recall the mainobjects of interest here and refer to (Bouchard and Nutz, 2015) and (Bertsekas andShreve, 1978, Chapter 7) for technical details. Let T ∈ N and X be a Polish space.We define for t ∈ { , . . . T } the Cartesian product X t := X t and define X := X T ,with the convention that X is a singleton. We denote by B ( X ) the Borel setson X , by P ( X ) the set of probability measures on B ( X ) and define the functionproj t : X → X which projects ω ∈ X to the t -th coordinate, i.e., proj t ( ω ) = ω t .Next we specify the financial market. Let d ∈ N , F an arbitrary filtration and let S t = ( S t , . . . , S dt ) : X t → R d be Borel-measurable, 0 ≤ t ≤ T , and adapted. Allprices are given in units of a numeraire, S , which itself is thus normalised, S t ≡ JAN OB(cid:32)L ´OJ AND JOHANNES WIESEL ≤ t ≤ T . Trading strategies H ( F ) are defined as the set of F -predictable R d -valuedprocesses. All trading is frictionless and self-financing. Given H ∈ H ( F ), we denote H ◦ S t = t (cid:88) u =1 H u ∆ S u with H ◦ S t representing the cashflow at time t from trading using H . Above,and throughout, H is a row vector, S is a column vector and 1 denotes eithera scalar or a column vector (1 , . . . , T . We let Φ denote the vector of payoffsof the statically traded assets Φ = ( φ λ : λ ∈ Λ), where Λ is some index set.For notational convenience, we often identify Φ with the set of its elements. Weassume that each φ ∈ Φ is Borel-measurable. When there are no statically tradedassets we write Φ = 0. These assets, which we think of as options, can only bebought or sold at time zero (without loss of generality at zero cost) and are helduntil maturity T . A trading position h can only hold finitely many of these assets, h ∈ c (Λ) the space of sequences of reals indexed by Λ with only finitely many non-zero elements, and generates the payoff h · Φ = (cid:80) λ ∈ Λ h λ φ λ at time T. We call a pair( h, H ) ∈ c (Λ) × H ( F ) a semistatic trading strategy. The class of such strategies isdenoted A Φ ( F ) := c (Λ) × H ( F ). For technical reasons we also introduce the levelsets of S , which are denoted byΣ ωt = { ˜ ω ∈ X | S t ( ω ) = S t (˜ ω ) } for t ∈ { , . . . , T } and ω ∈ X t , where S t := ( S , . . . S t ). Finally, we denote by F = ( F t ) t =0 ,...,T the natural filtration generated by S and let F U t be the universalcompletion of F t , t = 0 , . . . , T . Furthermore we write ( X, F U ) for ( X T , F U T ) andoften consider ( X t , F U t ) as a subspace of ( X, F U ).Within this setup, the literature on robust pricing and hedging adopts two ap-proaches to model an agent’s beliefs. One stream is scenario-based and proceeds byspecifying a prediction set Ω ⊆ X , which describes the possible price trajectories.The other stream proceeds by specifying a set of probability measures P ⊆ P ( X ),which determines the set of negligible outcomes. We refer to the latter as thequasi-sure approach, while the former is usually called the pathwise, or pointwise,approach. In both cases, the model specification may depend on the agent’s marketinformation as well as on her specific modelling assumptions. Changing the sets Ωor P can be seen as a natural way to interpolate between different beliefs. One ofthe principal aims of this paper is to show that both model approaches are equiva-lent in terms of corresponding FTAPs and Superhedging prices.In order to aggregate trading strategies on different level sets Σ ωt in a measurableway, we always assume in this paper that Ω is analytic and P has the followingstructure: Definition 2.1.
A set P ⊆ P ( X ) is said to satisfy the Analytic Product Structurecondition (APS), if P = { P ⊗ · · · ⊗ P T − | P t is F U t -measurable selector of P t } , where the sets P t ( ω ) ⊆ P ( X ) are nonempty, convex andgraph( P t ) = { ( ω, P ) | ω ∈ X t , P ∈ P t ( ω ) } is analytic.This structure facilitates a dynamic programming principle and allows to essentiallypaste together one-step results in order to establish their multistep counterparts.In order to formulate a Fundamental Theorem of Asset pricing we need to define OBUST MODELLING OF FINANCIAL MARKETS 5 the dual objects to trading strategies: the pricing (martingale) measures. Given aset of measures P , following (Bouchard and Nutz, 2015), we define Q P , Φ := { Q ∈ P ( X ) | S is an F U -martingale under Q , ∃ P ∈ P s.t. Q (cid:28) P , E Q [ φ ] = 0 ∀ φ ∈ Φ } , which, in the model-specific case P = { P } , is simply the familiar set of all martingalemeasures equivalent to P . Within the pathwise approach, for a set Ω ⊆ X and afiltration F , we define M f Ω , Φ ( F ) := { Q ∈ P f ( X ) | S is an F -martingale under Q , Q (Ω) = 1 , E Q [ φ ] = 0 ∀ φ ∈ Φ } , where P f ( X ) denotes the finitely supported Borel probability measures on ( X, B ( X )).As a general convention, in this paper we interpret the above sub- and super-scriptsas restrictions on the sets of measures. When we drop some of them it is to indi-cate that these conditions are not imposed, e.g., M Ω ( F ) denotes all F -martingalemeasures supported on Ω. Next letΩ ∗ Φ := { ω ∈ Ω | ∃ Q ∈ M f Ω , Φ ( F ) s.t. Q ( ω ) > } = (cid:91) Q ∈M f Ω , Φ ( F ) supp( Q )with the same convention regarding sub- and super-scripts as above. We also define F M := ( F Mt ) t ∈{ ,...,T } , where F Mt = (cid:92) Q ∈M Ω ( F ) F t ∨ N Q ( F T ) , N Q ( F T ) := { N ⊆ A ∈ F T | Q ( A ) = 0 } and F Mt is the power set of Ω if M Ω ( F ) = ∅ . Remark . Note that F ⊆ F U ⊆ F M holds. All these filtrations generate thesame martingale measures on Ω calibrated to Φ, which we denote by M Ω , Φ .For P ∈ P ( X ), thus N P := N P ( F U ) denotes the collection of its null sets. Like-wise, given a family P ⊂ P ( X ), the collection of its polar sets if given by N P = (cid:84) P ∈ P N P . We say that a property holds P -q.s. if it holds outside a P -polar set.2.2. Notions of Arbitrage.
One of the most important underlying concepts infinancial mathematics is the absence of arbitrage. In the literature on robust pricingand hedging many notions of arbitrage have been proposed to date. We presentthese here together in a unified manned and discuss their relative dependencies.To complement the picture, we establish some novel technical results. These arepostponed to Section 3.2.
Definition 2.3.
Fix a filtration F , a set P , a set S of subsets of X and a set Ω.Recall that semistatic admissible trading strategies are given by ( h, H ) ∈ A Φ ( F ). (Ω) A One-Point Arbitrage (see (Riedel, 2015)) is a strategy ( h, H ) ∈A Φ ( F ) such that h · Φ + H ◦ S T ≥ ω ∈ Ω. OA (Ω) An Open Arbitrage (see (Riedel, 2015)) is a strategy ( h, H ) ∈ A Φ ( F )such that h · Φ + H ◦ S T ≥ SA (Ω) A Strong Arbitrage (see (Acciaio et al., 2016)) is a strategy ( h, H ) ∈A Φ ( F ) such that h · Φ + H ◦ S T > USA (Ω) A Uniformly Strong Arbitrage (see (Davis and Hobson, 2007)) is astrategy ( h, H ) ∈ A Φ ( F ) such that h · Φ + H ◦ S T ≥ ε on Ω for some ε > A( P ) A P -quasi-sure Arbitrage (see (Bouchard and Nutz, 2015)) is a strat-egy ( h, H ) ∈ A Φ ( F ) such that h · Φ + H ◦ S T ≥ P -q.s. and P ( h · Φ+ H ◦ S T > > P ∈ P . If P = { P } a P -quasi-sureArbitrage is called a P -arbitrage and is denoted A ( P ). JAN OB(cid:32)L ´OJ AND JOHANNES WIESEL CA ( P ) A Classical Arbitrage in P (see (Davis and Hobson, 2007)) is a familyof strategies ( h P , H P ) P ∈ P such that, for all P ∈ P , ( h P , H P ) is a P -arbitrage. WA ( P ) A Weak Arbitrage (see (Blanchard and Carassus, 2019)) is a strategy( h, H ) ∈ A Φ ( F ) which is a P -arbitrage for some P ∈ P . IntA ( P ) An Interior Arbitrage (see (Bayraktar et al., 2014)) is a sequence ofstrategies ( h n , H n ) ∈ A Φ ( F ) such that ( h n , H n ) is a P -quasi-sureArbitrage relative to option payoffs given by Φ + sign( h n ) /n for all n large enough. WFLVR (Ω) A Weak Free Lunch With Vanishing Risk (see (Cox and Ob(cid:32)l´oj, 2011),(Cox et al., 2016)) is a sequence of strategies ( h n , H n ) ∈ A Φ ( F )such that there exists a constant c ≥ h, H ) ∈ A Φ ( F ) with h n · Φ + H n ◦ S T ≥ h · Φ + H ◦ S T − c on Ω for all n ∈ N andlim n →∞ ( h n · Φ + H n ◦ S T ) > . locA( P t ( ω ) ) A ( t, ω )-local P -quasi-sure Arbitrage (see (Bartl, 2019)) is a strategy H ∈ R d such that H ∆ S t +1 ( ω ) ≥ P t ( ω )-q.s. (where t ∈ { , . . . , T − } and ω ∈ X ) and there exists P ∈ P t ( ω ) such that P ( H ∆ S t +1 > > A( S ) An Arbitrage de la Classe S (see (Burzoni et al., 2016)) is a strategy( h, H ) ∈ A Φ ( F ) such that h · Φ + H ◦ S T ≥ { ω ∈ Ω | h · Φ + H ◦ S T > } ⊇ Γ for some Γ ∈ S .When we want to stress the role of the filtration we include it as an argument, e.g.,we write, e.g., SA (Ω , F ). When the filtration is not specified it is implicitly takento be F U . We use a prefix N to indicate a negation of any of the above notions,e.g., we say that “ NA ( P ) holds” when there does not exist a P -quasi-sure arbitragestrategy, likewise NUSA (Ω) denotes the absence of a uniformly strong arbitrageon Ω, etc.
Lemma 2.4.
The following relations hold:(1)
USA (Ω) ⇒ SA (Ω) ⇒ OA (Ω) ⇒ (Ω) . (2) SA (Ω) ⇒ WFLVR ( Ω ) . (3) A ( P ) ⇒ A ( P ) for some P ∈ P ⇔ WA ( P ) . (4) WA ( P ) ⇐ CA ( P ) ⇔ A ( P ) for all P ∈ P . (5) A ( P ) ⇒ IntA ( P ) .(6) when Φ = 0 then A ( P ) ⇔ P (cid:16)(cid:83) T − t =0 { ω ∈ X t | locA ( P t ( ω )) holds } (cid:17) > for some P ∈ P .Proof. Items (1) - (4) are immediate. Assertion (6) follows from (Bouchard andNutz, 2015, Lemma 4.6, p.842). For a strategy ( h, H ) ∈ A Φ ( F ) satisfying h · Φ + H ◦ S T ≥ P -q.s.we have, for any ε > h · (Φ + sign( h ) ε ) + H ◦ S T = h · Φ + H ◦ S T + | h | ε ≥ | h | ε ≥ , where | h | = (cid:80) λ ∈ Λ | h λ | . Absence of IntA ( P ) implies that there exists ε > h · Φ + H ◦ S T = −| h | ε P -q.s. , so that h = 0 and absence of A ( P ) follows so (5) holds. (cid:3) OBUST MODELLING OF FINANCIAL MARKETS 7
USA ( X ) was first discussed in (Davis and Hobson, 2007), see also (Cox and Ob(cid:32)l´oj,2011) and (Cox et al., 2016) for a definition of USA (Ω) and
WFLVR (Ω), whereΩ ⊆ X . Note that if we take ( h, H ) = (0 ,
0) in the definition of
WFLVR (Ω)and replace the pathwise inequalities by their P -a.s. counterparts for some fixed P ∈ P ( X ), we recover a discrete version of the NFLVR condition of (Delbaen andSchachermayer, 1994). SA (( R d + ) T ) was used in (Acciaio et al., 2016) in the canonical setup. We refer to(Burzoni et al., 2019a, Theorem 3) for a general FTAP connecting the notion ofStrong and Uniformly Strong Arbitrage under the condition that there exists an op-tion with a strictly convex super-linear payoff in the market. See also (Bartl et al.,2017) for an equivalence result under marginal constraints. In Section 3.2 we discussthe connection between SA (Ω) and USA (Ω) without the above assumptions. A ( S ) is a unifying concept since (Ω), OA (Ω), SA (Ω), USA (Ω) and A ( P ) canall be seen as special cases of A ( S ), see (Burzoni et al., 2016, Section 4.6) for a de-tailed discussion. It was first defined in (Burzoni et al., 2016) in a pathwise setting,see in particular the pathwise Fundamental Theorem of Asset pricing in (Burzoniet al., 2016, Theorem 2 & Section 4). This extends the results obtained in (Riedel,2015) who introduced (Ω) and OA (Ω). OA (Ω) is furthermore defined in thesetup of (Dolinsky and Soner, 2014). A ( P ) was introduced in the quasi-sure setting of (Bouchard and Nutz, 2015), wherethey prove a quasi-sure Fundamental Theorem of Asset pricing and Superhedg-ing Theorem. From Lemma 2.4 above we see that the crucial distinction between CA ( P ) and A ( P ) is the aggregation of arbitrage strategies, which poses a funda-mental technical difficulty overcome in Bouchard and Nutz (2015) by the specific(APS) structure of P . We also note that CA ( P ) was actually referred to as weakarbitrage in (Davis and Hobson, 2007).The notion of interior arbitrage IntA ( P ) was introduced, and called a robust ar-bitrage , by Bayraktar et al. (2014) in the context of transaction costs. Absence of IntA ( P ) is equivalent to absence of A ( P ) not only at the current prices of staticallytraded options Φ but also under all, sufficiently small, perturbations of their prices.This notion was also used in (Hou and Ob(cid:32)l´oj, 2018, Assumption 3.1). It is equivalentto saying that the prices of the options Φ are strictly inside the region of their P -q.s.no-arbitrage prices, thus avoiding the delicate issue of boundary classification. Ingeneral, IntA ( P ) does not imply A ( P ). To see this, take Φ = { ( S T − K ) + } forsome K > S and ∅ (cid:54) = P ⊆ { P ∈ P ( X ) | P ( S T ≤ K ) = 1 } . Then there is no P -q.s.arbitrage, while for every ε > S T − K ) + + ε ≥ ε > RA ( P )holds. Throughout the remainder of this paper, unless otherwise stated, we take
Λ = { , . . . , k } , i.e., we have a finite Φ with k statically traded options. Robust Fundamental Theorem of Asset Pricing.
The first FundamentalTheorem of Asset Pricing characterises absence of arbitrage in terms of existenceof martingale (pricing) measures. In the classical discrete-time setting, this refersto the notion of P -arbitrage. However, in a robust setting, there are many possiblenotions of arbitrage one can consider. If we adopt a strong notion of arbitrage, itsabsence should be equivalent to a weak statement, e.g., M Ω , Φ (cid:54) = ∅ . This is oftendone in the pathwise literature, see (Burzoni et al., 2019a), and leads to a robust(multi-prior) version of the familiar Dalang-Morton-Willinger theorem. Theorem 2.5 (Robust DMW Theorem) . Let P be a set of probability measuressatisfying (APS). Then there exists a universally measurable set of scenarios Ω with P (Ω) = 1 for all P ∈ P and a filtration ˜ F with F ⊆ ˜ F ⊆ F M , such that the followingare equivalent:(1) Q P , Φ (cid:54) = ∅ . JAN OB(cid:32)L ´OJ AND JOHANNES WIESEL (2) P (Ω ∗ Φ ) > for some P ∈ P .(3) M Ω , Φ (cid:54) = ∅ .(4) Ω ∗ Φ (cid:54) = ∅ .(5) NSA ( Ω , ˜ F ) holds.Conversely, for an analytic set Ω there exists a set P satisfying (APS) such that forall ω ∈ Ω there exists P ∈ P with P ( { ω } ) > and such that (1)-(5) are equivalent. The above result follows from Theorem 2.7 below by setting S = { Ω } . To see itsopposite twin we should adopt a weak notion of arbitrage, its absence thus beingequivalent to a strong statement, e.g., for all P ∈ P there exists Q ∈ Q P , Φ such that P (cid:28) Q . This route is most often taken in the quasi-sure literature, see (Bouchardand Nutz, 2015), and leads to the following version of the robust FTAP. Theorem 2.6.
Let P be a set of probability measures satisfying (APS). Then thereexists an analytic set of scenarios Ω with P (Ω) = 1 for all P ∈ P , such that thefollowing are equivalent:(1) N1pA (Ω ∗ Φ ) holds and Ω = Ω ∗ Φ P -q.s.(2) For all P ∈ P there exists Q ∈ Q P , Φ such that P (cid:28) Q .(3) NA ( P , F U ) holds.Conversely, if Ω is an analytic set, then there exists a set P of probability measuressatisfying (APS) such that for all ω ∈ Ω there exists P ∈ P with P ( { ω } ) > andsuch that the following are equivalent:(1) N1pA (Ω) holds and
Ω = Ω ∗ Φ .(2) For all P ∈ P there exists Q ∈ Q P , Φ such that P (cid:28) Q .(3) NA ( P , F U ) holds. Our proof of this theorem, given in Section 4.1, does not rely on the proof of (3) ⇒ (2) given in (Bouchard and Nutz, 2015). Instead we give pathwise arguments. Inparticular, given P ∈ P such that P (Ω \ Ω ∗ Φ ) > P satisfies (APS), it is possible to select Q ∈ Q P , Φ for each P ∈ P such that P (cid:28) Q . Necessarily the support of each P is then concentrated on Ω ∗ Φ .Finally, we give our main abstract result, which establishes a pathwise and prob-abilistic characterisation of the absence of Arbitrage de la Classe S . Its proof ispresented in Section 4.1. As noted above, Arbitrage de la Classe S allows to con-sider many notions of arbitrage at once. Accordingly, the main result below impliesTheorem 2.5 and can be strengthened to imply Theorem 2.6 as will be seen inSection 4. Theorem 2.7.
Assume that P satisfies (APS) and S ⊆ B ( X ) is such that ∃{ C n } n ∈ N ⊆ S s.t. ∀ C ∈ S ∃{ n k } k ∈ N ⊆ N with C nk ↑ C ( k → ∞ ) . (2.1) Then there exists a co-analytic set of scenarios Ω such that P (Ω) = 1 for all P ∈ P and a filtration ˜ F with F ⊆ ˜ F ⊆ F M , such that the following are equivalent:(1) For all C ∈ S with C ⊆ Ω there exists Q ∈ Q P , Φ such that Q ( C ) > .(2) For all C ∈ S with C ⊆ Ω there exists P ∈ P with P (Ω ∗ Φ ∩ C ) > .(3) For all C ∈ S with C ⊆ Ω there exists Q ∈ M Ω , Φ such that Q ( C ) > .(4) { C ∈ B ( X ) | C ⊆ Ω \ Ω ∗ Φ } ∩ S = ∅ .(5) There is no Arbitrage de la Classe S in A Φ (˜ F ) on Ω .Conversely, for an analytic set Ω there exists a set P satisfying (APS), such that forall ω ∈ Ω there exists P ∈ P with P ( { ω } ) > and such that (1)-(5) are equivalent.Remark . Condition (2.1) was first stated in (Burzoni et al., 2016, Cor. 4.30 andthe discussion thereafter). It turns out that for the proof of Theorem 2.7 a weaker
OBUST MODELLING OF FINANCIAL MARKETS 9 condition is sufficient: we only need the properties (cid:91) { C ∈ S | C ∩ (Ω P ) ∗ Φ ∈ N P } ∈ B ( X )(2.2)and (cid:91) { C ∈ S | C ∩ (Ω P ) ∗ Φ ∈ N P } ∩ (Ω P ) ∗ Φ ∈ N P (2.3)to hold, where we refer to Section 4.1 for a formal definition of Ω P . Conditions(2.2) and (2.3) are compatibility conditions on Ω, S , Φ and P . Indeed, they assertthat the (likely uncountable) union of “inefficient” subsets of Ω P \ (Ω P ) ∗ Φ (modulo P -polar sets), stays an “inefficient” subset (modulo P -polar sets). If this conditionis not satisfied for some arbitrary P and S , then there is no reason why a set Ωfor which (2) holds should exist. Take for example a collection P having densitiesand S a set of singletons in X . Then P ( C ) = 0 for any P ∈ P and any C ∈ S so the only Ω which could satisfy (2) is the empty set. We note that when S = { C ⊆ X | C open } then (2.2) is always satisfied and (2.3) is satisfied as soon as X is separable. However, in general, conditions (2.2) and (2.3) may be hard to verify,which is why we provide (2.1) as an easier sufficient condition. Lastly, we remarkthat it is not straightforward to show that S = { C | P ( C ) > P ∈ P } corresponding to NA ( P ) satisfies (2.3), which is why we give a direct proof ofTheorem 2.6 in Section 4.We note that the set Ω can in general not be assumed to be analytic. The impli-cations (1) ⇒ (2) ⇒ (3) ⇒ (4) ⇒ (5) follow directly from the definitions. Apartfrom measurability considerations regarding Ω, the equivalence of (3) , (4) and (5) essentially follows from (Burzoni et al., 2016). Furthermore, given an analytic set Ω,we will simply define P as all the finitely supported probability measures on Ω. Theanalyticity of Ω then implies (APS) of P . We then also have Q P , Φ = M f Ω , Φ andequivalence of (1) and (3) - (5) follows from (Burzoni et al., 2019a). In this context,the essential connection we make is the combination of pathwise and quasi-surecriteria as stated in (2) : for every C ∈ S , the pathwise efficient subset Ω ∗ Φ ∩ C isrequired to be “seen” by at least one measure P in the set P .For a given P , the set Ω in Theorem 2.7 can be explicitly constructed as the con-catenation of the quasi-sure supports of P t ◦ ∆( S t +1 ) − . The main difficulty ofthe proof is then to show the implication (5) ⇒ (1) , where one needs to establishexistence of martingale measures Q ∈ Q P , Φ , which are compatible with Ω and S in the sense of (1) . This, modulo measurable selection arguments, is achieved byfinding an element P ∈ P t ( ω ) such that zero is in the relative interior of the supportof P ◦ ∆( S t +1 ) − . Indeed, let us explain the main idea of the proof of (5) ⇒ (1) based on the following example: assume T = 1, d = 3, Φ = 0 , S = { Ω } and theset Ω ∗ is given by the grey polyhydron in Figure 2. Assume that the support of P ◦ ∆( S ) − for a given measure P ∈ P is given by the blue dot (see Figure 2).Then as 0 ∈ ri(Ω ∗ ), we can find three additional points in Ω ∗ , such that zero is inthe relative interior of the convex hull of the four points. By definition of Ω, thethree additional points are in the support of some measures in P , which we call P , P , P in P . As P is convex, it follows that˜ P := P + P + P + P P as well, as visualised in Figure 3. Since zero is in the relativeinterior of the support of ˜ P , one can now use results from (Rokhlin, 2008) to find amartingale measure Q ∼ ˜ P , in particular P (cid:28) Q .Note that this argument fundamentally relies on the convexity of P t . The analyticproduct structure assumption then grants suitable measurability for the concatena-tion procedure in the multiperiod case. ∆ S ∆ S ∆ S Figure 2.
Construction of a martingale measure Q (cid:29) P for d = 3:the set Ω (grey) and supp( P ◦ (∆ S ) − ) (blue) ∆ S ∆ S ∆ S Figure 3.
Construction of a martingale measure Q (cid:29) P for d = 3:finding a measure ˜ P such that NA( ˜ P ) and ˜ P (cid:29) P holds.2.4. Robust Superhedging Theorem.
In this section we focus on the key resultwhich characterises superhedging prices: the pricing-hedging duality, or the Super-hedging Theorem. As before, we compare pathwise and quasi-sure superhedgingapproaches as extensions of the classical model-specific result, see (F¨ollmer andSchied, 2011, Chapter 5, Theorem 5.30).For a set Ω ⊆ X we denote the pathwise superhedging price on Ω by π Ω ( g ) := inf { x ∈ R | ∃ ( h, H ) ∈ A Φ ( F U ) s.t. x + h · Φ + ( H ◦ S T ) ≥ g on Ω } and denote the P -q.s. superhedging price by π P ( g ) := inf { x ∈ R | ∃ ( h, H ) ∈ A Φ ( F U ) s.t. x + h · Φ + ( H ◦ S T ) ≥ g P -q.s. } . Take an analytic set Ω such that for all P ∈ P we have P (Ω ∗ Φ ) = 1. Using theSuperhedging Theorems of (Bouchard and Nutz, 2015) and (Burzoni et al., 2019a)it is immediate that the following relationships hold for all upper semianalytic g :sup Q ∈M Ω , Φ E Q [ g ] = π Ω ∗ Φ ( g ) ≥ π P ( g ) = sup Q ∈Q P , Φ E Q [ g ] . The above inequality is strict in general. An easy way to see this is to take d = T = S = 1, Φ = 0, g ( S ) = { S =0 } and P = { λ | [0 , } , where λ | [0 , denotes the OBUST MODELLING OF FINANCIAL MARKETS 11
Lebesgue measure on [0 , ∗ = [0 ,
2] and the pathwise superhedgingprice is equal to 1 /
2, while the quasi-sure superhedging price is equal to zero. Infact, to link the super-hedging and pathwise formulations, we have to choose aspecific set Ω P g which depends not only on P but also on g . We determine this setΩ P g by reducing to superhedging under a fixed measure P g as stated in the followingtheorem: Theorem 2.9.
Let P be a set of probability measures satisfying (APS). Assume NA ( P ) holds and let g : X → R be upper semianalytic. Then there exists a measure P g = P g ⊗ · · · ⊗ P gT − and an F U -measurable set Ω P g with P (Ω P g ) = 1 for all P ∈ P ,such that sup Q ∈M Ω P g , Φ E Q [ g ] = π (Ω P g ) ∗ Φ ( g ) = π P g ( g ) = π P ( g ) = sup Q ∈Q P , Φ E Q [ g ] . Conversely, let Ω be an analytic subset of X with Ω ∗ Φ (cid:54) = ∅ and let g : X → R be uppersemianalytic. For any set P ⊆ P ( X ) , which satisfies (APS) and N P = N M f Ω , Φ , wehave sup Q ∈M f Ω , Φ E Q [ g ] = π Ω ∗ Φ ( g ) = π P ( g ) = sup Q ∈Q P , Φ E Q [ g ] . In both cases, the value, if finite, is attained by a superhedging strategy ( h, H ) ∈A Φ ( F U ) . The proof of this result is postponed to Section 4.2. In particular, Theorem 2.9 letsus interpret robust superhedging prices π P ( g ) as classical superreplication prices π P g ( g ) under an “extremal” measure P g . Determining such measures P g is notstraightforward in general. In a one-period case and for a continuous g , we can usethe arguments in the proof of Lemma 4.10 to see that any measure P which attainsthe one-step quasisure support { P ◦ (∆ S T ( ω, · )) − | P ∈ P T − ( ω ) } can be chosen.To extend this result to the multiperiod-case, certain continuity properties of themaps ω (cid:55)→ P t ( ω ) have to be guaranteed: we refer to (Carassus et al., 2019, Prop.3.7) for a sufficient condition.3. Complementary results on superhedging and arbitrage
Extension of Pathwise Superhedging from Ω ∗ to Ω . The preceding re-sults show that quasi-sure and pathwise superhedging are essentially equivalent. As P -q.s. superhedging strategies might be difficult to compute and implement in prac-tice, it might be preferable to work on a prediction set Ω using pathwise arguments.Given that determining Ω ∗ is computationally expensive as well, the quantity ofinterest is then the superhedging price on Ω and not on Ω ∗ seen in the dualityresults in Section 2.4. Thus, we would like to find sufficient conditions under whichthe superhedging strategy associated with π Ω ∗ Φ ( g ) can be extended to Ω withoutany additional cost. The intuition is that as Ω \ Ω ∗ describes non-efficient beliefs,we should be able to superhedge g on this set implementing an arbitrage strategy.It turns out that this intuition is not true in general. Indeed, we run into prob-lems regarding measurability of these arbitrage strategies, which means that thisprocedure only works in special cases.To simplify the analysis, throughout this section only, we assume that Φ = 0 and ω (cid:55)→ S t ( ω ) is continuous. The latter is satisfied, e.g., when ω (cid:55)→ S t ( ω ) is thecoordinate mapping, i.e., S t ( ω ) = ω t . In order to give some intuition and to iden-tify necessary conditions for the sets Ω, Ω ∗ and the function g we first give twocounterexamples: Example . Let d = 1, T = 1 and (Ω , F ) = ( R + \ { } , B ( R + \ { } )). We set S = 2 and S ( ω ) = 2 + ω . Then Ω ∗ = ∅ and trivially π Ω ∗ (1) = inf { x ∈ R | ∃ H ∈ R d such that x + H ◦ S T ≥ ∗ } = −∞ ,π Ω (1) = inf { x ∈ R | ∃ H ∈ R d such that x + H ◦ S T ≥ } = 1 . Thus we have to assume that Ω ∗ ∩ Σ ωt (cid:54) = ∅ in the remainder of this section. We alsonote that here SA (Ω) holds whilst USA (Ω) does not, see Section 3.2.
Example . Let d = 2, T = 1 and Ω = ((2 , ∞ ) × [0 , ∞ )) ∪ ( { } × [0 , F = B (Ω). We set S = (2 ,
2) and S ( ω ) = ω . In particular, ∆ S (Ω) is not a closedset. Note that Ω ∗ = { } × [0 , g ( S , S ) = ∆ S { ∆ S ≤ } + 5(∆ S − { ∆ S > } . It is easy to see that π Ω ∗ ( g ) = 0 and any trading strategy H = ( H ,
1) with H ∈ R is a superhedging strategy. We now claim that we cannot extend superhedging toΩ \ Ω ∗ with initial capital zero. For this we show that even for initial capital one,there exist no superreplication strategies on Ω. Indeed for this we would need1 + H ∆ S + H ∆ S ≥ S −
4) on ∆ S > , ∆ S > , which is equivalent to H ≥ (5 − H )∆ S − S . As H ∈ [4 / , / H → ∞ if ∆ S is arbitrarily close to 0 and ∆ S is sufficiently large. Even if we look at Ω = [2 + ε, ∞ ) × [0 , ∞ ) ∪ { } × [0 ,
7] forsome positive ε , then taking ∆ S arbitrarily large still leads to non-existence ofsuperhedging strategies. In conclusion we will only consider compact sets Ω ∩ Σ ωt inthe rest of this section, on which “arbitrage strategies are effective for superhedging”in a sense defined below. Furthermore, modifying the function g on Ω \ Ω ∗ in theabove example, we can easily construct situations, in which π Ω ( g ) (cid:54) = π Ω ∗ ( g ) fordiscontinuous payoffs g . In conclusion we will also assume that g is continuous inthis section.We can also modify this example so that the ∆ S (Ω) is closed and there is no attain-ment of superreplication strategies for π Ω ( g ). We stress that this is a fundamentaldifference to the case Ω = Ω ∗ , where attainment is always given (see Theorem 2.9).Namely, take Ω = { ( x, y ) ∈ R | x ∈ [2 , ∞ ) , ≤ y ≤ √ x } with the otherelements unchanged. Note that Ω ∗ did not change. Repeating the arguments aboveand looking at ∆ S = 1 /n and ∆ S = 5 + 1 / √ n we find H ≥ n (cid:18)
20 + 5 √ n − − √ n (cid:19) = n √ n → ∞ for n → ∞ .As we have seen in the examples above, in general it is necessary to assume thatΩ ∗ ∩ Σ ωt (cid:54) = ∅ , ω (cid:55)→ g ( ω ) is continuous and that Ω is compact as well as “well suitedfor superhedging by arbitrage strategies”. We first address the second point andshow continuity of the one-step superhedging prices ω (cid:55)→ π t, Ω ∗ ( g )( ω ), which aredefined via a dynamic programming approach: Definition 3.3.
For a Borel-measurable g : X → R we define the one-step super-hedging prices π T, Ω ∗ ( g )( ω ) := g ( ω ) ,π t, Ω ∗ ( g )( ω ) := inf { x ∈ R | ∃ H ∈ R d s.t. x + H ∆ S t +1 ( v ) ≥ π t +1 , Ω ∗ ( g )( v ) ∀ v ∈ Σ ωt ∩ Ω ∗ } , where 0 ≤ t ≤ T − OBUST MODELLING OF FINANCIAL MARKETS 13
A sufficient condition for continuity of ω (cid:55)→ π t, Ω ∗ ( g )( ω ) was identified in (Carassuset al., 2019) and relies on the following assumption: Assumption 3.4.
The sets Σ ωt ∩ Ω ∗ (cid:54) = ∅ and the sets Σ ωt ∩ Ω are compact for all ω ∈ Ω and all ≤ t ≤ T − . Furthermore, for all ≤ t ≤ T − , the correspondence ω (cid:16) S t +1 (Σ ωt ∩ Ω ∗ ) is uniformly continuous from (Ω , d St ) to the subsets of R d endowedwith the Hausdorff distance, and where d St ( ω, ˜ ω ) := max s =0 ,...,t | S s ( ω ) − S s (˜ ω ) | . We refer to (Carassus et al., 2019, Section 3) for a discussion and examples ofsets Ω satisfying Assumption 3.4. The following lemma now follows from a directapplication of (Carassus et al., 2019, Proposition 3.5):
Lemma 3.5.
Let ω (cid:55)→ g ( ω ) be continuous. Under Assumption 3.4 the one-stepsuperhedging prices ω (cid:55)→ π t, Ω ∗ ( g )( ω ) are continuous for all ≤ t ≤ T − . Secondly, Example 3.2 also shows, that it is important to identify the subset ofΣ ωt ∩ Ω, on which “arbitrage strategies are ineffective for superhedging” in thefollowing sense:
Definition 3.6.
We denote by proj ∆ S t +1 (Σ ωt ∩ Ω ∗ ) (∆ S t +1 ( v )) the orthogonal projec-tion of ∆ S t +1 ( v ) onto the linear subspace spanned by ∆ S t +1 (Σ ωt ∩ Ω ∗ ) and definethe set A ωt as the collection of all v ∈ Σ ωt ∩ Ω, for which proj ∆ S t +1 (Σ ωt ∩ Ω ∗ ) (∆ S t +1 ( v ))is not an element of ∆ S t +1 (Σ ωt ∩ Ω ∗ ).For an illustration of the set A ωt see Figure 4. We now state an assumption ensuringcompatibility of A ωt and Σ ωt ∩ Ω ∗ : Assumption 3.7.
For each level set the following is true: if a sequence of points ( v n ) ⊆ A ωt converges to a point v ∈ span (∆ S t +1 (Σ ωt ∩ Ω ∗ )) , then necessarily v ∈ ∆ S t +1 (Σ ωt ∩ Ω ∗ ) . Alas, it turns out that while Assumptions 3.4 and 3.7 are sufficient to establish theequality π Ω ( g ) = π Ω ∗ ( g ) for d = 2, it is not so for d >
2. It is linked with the notionof standard separators introduced in (Burzoni et al., 2019a), which are measurableselectors of pointwise arbitrage strategies. We refer the reader to (Burzoni et al.,2019a, Proof of Lemma 1) and the discussion therein for a detailed definition. Herewe formulate an example, in which the existence of two standard separators togetherwith the measurability constraint on H implies π Ω ( g ) > π Ω ∗ ( g ): Example . Let d = 3, T = 1 and (Ω , F ) = ( R + , B ( R + )). We set ( S , S , S ) =(2 , ,
2) and S ( ω ) = ω ∈ R + \ Q , . − ω if ω ∈ Q ∩ [1 / , ∞ ) , ω ∈ Q ∩ [0 , / ,S ( ω ) = ω if ω ∈ R + \ Q , ω ∈ Q ∩ [1 / , ∞ ) , ω ∈ Q ∩ [0 , / , S ( ω ) = ω ∈ R + \ Q , ω ∈ Q ∩ [1 / , ∞ ) , ω if ω ∈ Q ∩ [0 , / . Then Ω ∗ = R + \ Q and using the notation of (Burzoni et al., 2019a, Proof of Lemma1) the standard separators are given by ξ ,A = (0 , ,
1) and ξ ,A = ( − , , g ( S , S , S ) = (cid:26) (∆ S + | ∆ S | + | ∆ S | ) − if ∆ S ≤ , (∆ S − | ∆ S | − | ∆ S | ) + if ∆ S > , We note that for ∆ S = ∆ S = 0 we have g ( S ) = ∆ S . So in particular tohedge g on Ω ∗ we need initial capital π Ω ∗ ( g ) = 0 and any hedging strategy satisfies H = ( H , , H ) for H , H ∈ R . For any such strategy to also superhedge on Q ∩ [1 / , ∞ ) with initial capital 1 / H has to satisfy in particular1 / H ∆ S − H ≥ S ≤ − , − ∆ S ∆ S S ( v )proj ∆ S ∗ ) (∆ S ( v )) ξ , Ω Legend: A ω ∆ S (Ω ∗ )Lin(∆ S (Ω ∗ )) Figure 4.
Example 3.2 with notation from Definition 3.6.so H ≤ − / H ∈ [1 , / Q ∩ [0 , / / H + H ∆ S ≥ . Taking ∆ S = 0 gives H ≥ − /
4, a contradiction. Thus we will assume that allone-point arbitrages can be reduced to a single standard separator.Note that Example 3.8 can be easily altered to make ∆ S t +1 (Ω ∩ Σ ωt ) compact byadding additional points. For clarity of exposition we have refrained from doing thisbut we conclude that Assumptions 3.4 and 3.7 are not sufficient for d >
2. We haveto add a last assumption, which guarantees measurability of the correspondingUniversal Arbitrage Aggregator. Intuitively it states, that locally, i.e., for every0 ≤ t ≤ T − ω ∈ Ω, there exists at most one arbitragable direction ofthe evolution of assets S , so that the first standard separator is already the UniversalArbitrage Aggregator: Assumption 3.9.
For all ω ∈ Ω and ≤ t ≤ T − we have ξ t +1 , Ω ∩ Σ ωt = H ∗ t ,where H ∗ is the Universal Arbitrage Aggregator of (Burzoni et al., 2019a) for theset Σ ωt ∩ Ω . Theorem 3.10.
Suppose that
Φ = 0 and X (cid:51) ω (cid:55)→ S t ( ω ) is continuous for all ≤ t ≤ T . For an analytic Ω ⊆ X satisfying Assumptions 3.4, 3.7 and 3.9 OBUST MODELLING OF FINANCIAL MARKETS 15 ∆ S ∆ S ∆ S Legend: R + \ QQ ∩ [1 / , ∞ ) Q ∩ [0 , / Figure 5. Ω ⊆ R in Example 3.8 the Superhedging Duality of (Burzoni et al., 2019a) extends from Ω ∗ to Ω for allcontinuous g : X → R , i.e., we have sup Q ∈M f Ω E Q ( g ) = inf { x ∈ R | ∃ H ∈ H ( F U ) such that x + H ◦ S T ≥ g on Ω ∗ } = inf { x ∈ R | ∃ H ∈ H ( F U ) such that x + H ◦ S T ≥ g on Ω } Proof.
As before, we prove the claim by backward induction over t = 0 , . . . , T − ω ∈ Ω. We assume Σ ωt ∩ Ω ∗ (cid:54) = ∅ and (Σ ωt ∩ Ω) \ (Σ ωt ∩ Ω ∗ ) (cid:54) = ∅ , otherwisethe claim is trivial. We first look at the case, where proj ∆ S t +1 (Σ ωt ∩ Ω ∗ ) (∆ S t +1 ( v ))is an element of ∆ S t +1 (Σ ωt +1 ∩ Ω ∗ ), i.e., there exists v (cid:48) ∈ Σ ωt ∩ Ω ∗ such thatproj ∆ S t +1 (Σ ωt ∩ Ω ∗ ) (∆ S t +1 ( v )) = ∆ S t +1 ( v (cid:48) ) . Note that by Assumption 3.9 the stan-dard separator ξ t +1 , Ω ∩ Σ ωt is orthogonal to span(∆ S t +1 (Σ ωt ∩ Ω ∗ )). By definition ofthe superhedging price on Ω there exists an F U t -measurable strategy H t +1 such that π t, Ω ∗ ( g )( v (cid:48) ) + H t +1 ( ω )∆ S t +1 ( v (cid:48) ) ≥ ˆ π t +1 ( g )( v (cid:48) ) for all v (cid:48) ∈ Σ ωt ∩ Ω ∗ , where we can assume without loss of generality that H t +1 ( ω ) ∈ span(∆ S t +1 (Σ ωt ∩ Ω ∗ )). Now we fix v ∈ Σ ωt ∩ Ω and v (cid:48) the corresponding orthogonal projection. Let ε >
0. As π t +1 , Ω ∗ ( g ) is uniformly continuous on Ω ∩ Σ ωt , we can use (Vanderbei,1997, Theorem 1) (in connection with Tietze’s extension theorem to extend thedomain to a convex set) in order to find δ > ε + π t +1 , Ω ∗ ( g )( v (cid:48) ) + S t +1 ( v ) − S t +1 ( v (cid:48) ) δ/ε ξ t +1 , Σ ωt ∩ Ω ( v ) ≥ π t +1 , Ω ∗ ( g )( v ) , where δ is chosen such that for all w, ˜ w ∈ Σ ωt ∩ Ω we have | π t +1 , Ω ∗ ( g )( w ) − π t +1 , Ω ∗ ( g )( ˜ w ) | ≤ ε whenever | S t +1 ( w ) − S t +1 ( ˜ w )) | < δ . Note that ∆ S t +1 ( v ) − ∆ S t +1 ( v (cid:48) ) is orthogonal to H t +1 ( ω ) and ε + π t, Ω ∗ ( g )( g )( v ) + (cid:18) H t +1 ( ω ) + ξ t +1 , Σ ωt ∩ Ω ( v ) δ/ε (cid:19) (∆ S t +1 ( v (cid:48) ) + ∆ S t +1 ( v ) − ∆ S t +1 ( v (cid:48) )) ≥ ε + π t +1 , Ω ∗ ( g )( v (cid:48) ) + S t +1 ( v ) − S t +1 ( v (cid:48) ) δ/ε ξ t +1 , Σ ωt ∩ Ω ( v ) ≥ π t +1 , Ω ∗ ( g )( v ) , Next we use the assumption that A ωt is bounded and has no points of convergence inspan(∆ S t +1 (Σ ωt ∩ Ω ∗ )) outside the set ∆ S t +1 (Σ ωt ∩ Ω ∗ ). In particular the continuous functions π t +1 , Ω ∗ ( g ) and H t +1 ∆ S t +1 are bounded on A ωt . There exists δ > v ∈ ∆ S t +1 (Σ ωt ∩ Ω ∗ ) , ˜ v ∈ A ωt with | v − ˜ v | < δ we still have ε + π t, Ω ∗ ( g )( ω ) + H t +1 ( ω )∆ S t +1 (˜ v ) ≥ π t +1 , Ω ∗ ( g )(˜ v ) . By assumption there exists ˜ δ > v, span(∆ S t +1 (Σ ωt ∩ Ω ∗ ))) > ˜ δ for all˜ v ∈ A ωt with dist(˜ v, ∆ S t +1 (Σ ωt ∩ Ω ∗ )) > δ . Define π max = sup v ∈ A ωt π t +1 , Ω ∗ ( g )( v ) < ∞ and C = inf v ∈ A ωt H t +1 ( ω )∆ S t +1 ( v ) + π t, Ω ∗ ( g )( ω ) > −∞ . Now we note that ε + π t +1 , Ω ∗ ( g )(˜ v ) + H t +1 (˜ v )∆ S t +1 (˜ v ) + | π max | + | C | ˜ δ ξ t +1 , Σ ωt ∩ Ω ( v )∆ S t +1 (˜ v ) ≥ π t +1 , Ω ∗ ( g )(˜ v )for all ˜ v ∈ A ωt . This concludes the proof. (cid:3) Comparison of Strong and Uniformly Strong Arbitrage.
We take nowa closer look at the notions SA (Ω) and USA (Ω) and establish their equivalencein specific market setups. Clearly every Uniformly Strong Arbitrage is a StrongArbitrage. In general the opposite assertion is not true: take for example d = 1, S = 1, S ( ω ) = ω , Ω = (1 , H > S = 1 and an openconvex set Ω such that { } ∩ Ω = ∅ and 1 ∈ ¯Ω admits a Strong Arbitrage butexhibits no Uniformly Strong Arbitrages. On the level of superhedging prices aUniformly Strong Arbitrage on Ω corresponds to π Ω (0) = −∞ . For a financialmarket which exhibits a Strong Arbitrage but no uniformly Strong Arbitrages, thePricing-Hedging duality cannot hold (as there are no martingale measures supportedon Ω) but π Ω (0) = 0. In conclusion, the difference between Uniformly StrongArbitrage and Strong Arbitrage can be seen as a property describing the boundaryof the prediction set Ω and thus manifests itself in the boundary behaviour of thesuperhedging functional S (cid:55)→ inf { x ∈ R | ∃ H ∈ R d s.t. x + H ( S − S ) ≥ } . As it is an upper semicontinuous function, it takes the value zero on the boundaryof Ω, while its lower semicontinuous version takes the value −∞ . Nevertheless thetwo notions agree in specific cases, which we now explore.We assume the canonical setting X = R d + , S ( ω ) = s and set F t = F t for all 0 ≤ t ≤ T . In this section we allow for countably many statically traded options, Λ = N ,but only of European type, Φ = { φ n = φ n ( S T ) | n ∈ N } , with real-valued continuouspayoffs and a common maturity T . We write c = c ( N ) for simplicity. We fix aclosed subset Ω ⊆ ( R d + ) T and recall that martingale measures on Ω calibrated to Φare denoted by M Ω , Φ ( F ). We define | S ( ω ) | := (cid:80) Tt =1 (cid:80) dk =1 | S kt ( ω ) | and denote by C b | S | (Ω) the space of real-valued continuous functions f : Ω (cid:55)→ R such thatsup ω ∈ Ω | f ( ω ) || S ( ω ) | ∨ < ∞ . Finally, we define the calibrated supermartingale measures as SM Ω , Φ ( F ) := { Q ∈ P (Ω) | E Q [ φ n ] ≤ ∀ n ∈ N , E Q [ S t |F t − ] ≤ S t − a.s. ∀ t ≤ T } . The following theorem can be seen as a unification of (Acciaio et al., 2016, Theorem1.3), (Cox and Ob(cid:32)l´oj, 2011, Prop. 2.2, p.6) and (Bartl et al., 2017, Cor. 4.6). We alsorefer to (Burzoni et al., 2019a, Thm. 3), who extend (Acciaio et al., 2016, Thm. 1.3)under the assumption Ω = Ω ∗ and to (Burzoni et al., 2019b, Thm. C.5) for a generaldiscussion in the case Φ = 0. In contrast to the work of Acciaio et al. (2016), we donot need to assume the existence of a convex superlinear payoff g , which might beartificial in some settings, but explicitly enforce tightness of martingale measuresthrough the WFLVR (Ω) condition.
OBUST MODELLING OF FINANCIAL MARKETS 17
Theorem 3.11.
The following hold:(1) SA (Ω) ⇔ USA (Ω) .(2) Assume φ n ∈ C b | S | (Ω) , no short-selling in any of the assets and lim | S T | →∞ ( φ n ( S T )) − | S T | = 0 for all n ∈ N . Then SA (Ω) ⇔ USA (Ω) ⇔ WFLVR (Ω) ⇔ SM Ω , Φ ( F ) = ∅ . (3) As in (2) assume that φ n ∈ C b | S | (Ω) . Furthermore assume that for ev-ery sequence ( ω n ) n ∈ N with lim n →∞ | S ( ω n ) | = ∞ , there exists a sequence ( h k , H k ) k ∈ N of trading strategies, a constant C > and a sequence ( p k ) k ∈ N such that • lim k →∞ lim n →∞ ( h k · Φ+ H k ◦ S T )( ω n ) | S ( ω n ) | ∨ > and • | h k · Φ + H k · S T | ≤ C ( | S | ∨ on Ω for all k ∈ N , • lim k →∞ ( h k · Φ + H k ◦ S T )( ω ) = − lim k →∞ p k for all ω ∈ Ω .Then SA (Ω) ⇔ USA (Ω) ⇒ WFLVR (Ω) ⇔ M Ω , Φ ( F ) = ∅ , but in general WFLVR (Ω) does not imply SA (Ω) .Remark . (1) The case | Φ | < ∞ is covered in (Bouchard and Nutz, 2015;Burzoni et al., 2019a)), while the case | Φ | = ∞ is not. The basic idea inboth works is to inductively construct a martingale measure calibrated toa finite number of options.(2) Contrary to the case | Φ | < ∞ (see (Burzoni et al., 2019a, proof of Theorem1, p.1050)), the set M Ω , Φ ( F ) might not necessarily contain any finitelysupported martingale measures.(3) An example showing that WFLVR (Ω) does not imply SA (Ω) is given in(Cox and Ob(cid:32)l´oj, 2011, Prop. 2.2).(4) A special but important case of (3) is T = 1 , d = 1 and φ n ( S ) = ( S − n ) + − p n , where p n ≥
0. In this case we can set H k = 0, h k = e k for all k ∈ N , where e k is the k th unit vector and note thatlim k →∞ lim n →∞ ( h k · Φ + H k ( S − S ))( ω n ) | S ( ω n ) | ∨ k →∞ lim n →∞ ( S ( ω n ) − k ) + − p k | S ( ω n ) | = 1 > k →∞ ( S ( ω ) − k ) + − p k = − lim k →∞ p k , in particular all three conditions in (3) are satisfied. Proof.
For simplicity of exposition we only give the proof for T = 1. This conveysthe important ideas, while the multiperiod case extends these via a dynamic pro-gramming approach and can found in (Wiesel, 2020).Regarding (1) , clearly USA (Ω) ⇒ SA (Ω), so we show SA (Ω) ⇒ USA (Ω). Let( h, H ) ∈ R k × R d be a Strong Arbitrage. We show that it is actually a UniformlyStrong Arbitrage. For x ∈ R d + we denote by | x | := (cid:80) di =1 x i the (cid:96) -norm of x anddefine the compact set K = [0 , s + 2 | s | ] × [0 , s + 2 | s | ] × · · · × [0 , s d + 2 | s | ].Then, as S (cid:55)→ h · Φ( S ) + H ( S − S ) is continuous and positive on a compact setΩ ∩ K , there exists ε > h · Φ( S ) + H · ( S − S ) ≥ ε on K ∩ Ω . Scaling ( h, H ) suitably we can without loss of generality assume take ε = 2 | s | .Let e = (1 , . . . ,
1) be the row unit vector in R d . Then h · Φ( S ) + ( H + e ) · ( S − S ) ≥ | s | − | s | = | s | on K ∩ Ω . (3.1) Furthermore on Ω \ K we have h · Φ( S ) + ( H + e ) · ( S − S ) ≥ e · ( S − S ) ≥ | s | − | s | = | s | . Now we show (2) . Clearly the relation
USA (Ω) ⇒ WFLVR (Ω) holds and by (1) also SA (Ω) ⇔ USA (Ω). Further,
WFLVR (Ω) readily implies SM Ω , Φ ( F ) = ∅ since otherwise if Q ∈ SM Ω , Φ ( F ) then, by Fatou’s lemma,0 ≥ lim inf n →∞ E Q [ h n · Φ + H n ( S − S )] ≥ E Q [lim inf n →∞ h n · Φ + H n ( S − S )] > , a contradiction. Next we show NUSA (Ω) ⇒ ( SM Ω , Φ ( F ) (cid:54) = ∅ ) following closely theargument in (Acciaio et al., 2016, proof of Prop. 2.3 and Theorem 1.3, pp. 240-242).We denote by c +00 the subset of all non-negative sequences in c . We define the set K := (cid:8) h · Φ( S ) + H ( S − S ) (cid:12)(cid:12) ( h, H ) ∈ c +00 × R d + (cid:9) ⊆ C b | S | (Ω) . Note that K is convex and non-empty. Furthermore denote the positive cone of C b | S | (Ω) by C ++ (Ω) = (cid:26) f ∈ C b | S | (Ω) (cid:12)(cid:12)(cid:12)(cid:12) inf ω ∈ Ω f ( ω ) | S ( ω ) | ∨ > (cid:27) . By NUSA (Ω) we have K ∩ C ++ (Ω) = ∅ . An application of Hahn-Banach theoremyields existence of a positive measure µ = µ r + µ s such that (cid:90) Ω f | S | ∨ dµ > f ∈ C ++ (Ω) , (cid:90) Ω f | S | ∨ dµ ≤ f ∈ K. We now aim to show that the normalised measure Q given by d Q := 1 | S | ∨ (cid:18)(cid:90) | S | ∨ dµ r (cid:19) − dµ r is an element of SM Ω , Φ . For this let us first assume that µ r = 0. Then (cid:90) Ω e ( S − S ) | S | ∨ dµ = (cid:90) Ω | S | − | S | | S | ∨ dµ s = (cid:90) Ω dµ s > µ is positive, which is a contradiction. As (cid:82) Ω ( φ n ( S )) − | S | ∨ dµ s = 0, we conclude (cid:90) Ω φ n ( S ) | S | ∨ dµ r ≤ (cid:90) Ω φ n ( S ) | S | ∨ dµ ≤ n ∈ N . Furthermore (cid:90) Ω S − S | S | ∨ dµ r = 0and thus NUSA (Ω) ⇒ SM Ω , Φ (cid:54) = ∅ follows.Lastly we show (3) . For this we follow the same construction as in (2) . In particularredefining K := (cid:26) h · Φ( S ) + H ( S − S ) (cid:12)(cid:12)(cid:12)(cid:12) ( h, H ) ∈ c × R d (cid:27) ⊆ C b | S | (Ω) . we note that again by NUSA (Ω) we have K ∩ C ++ (Ω) = ∅ . Thus all that is left toshow is µ s = 0. Let us assume towards a contradiction µ s (cid:54) = 0 and take ( h k , H k ) k ∈ N such that lim k →∞ (cid:90) Ω h k · Φ( S ) + H k ( S − S ) | S | ∨ dµ s > . (3.2) OBUST MODELLING OF FINANCIAL MARKETS 19
Then by symmetry of K and the same reasoning as in (2) we have (cid:90) Ω h k · Φ( S ) + H k ( S − S ) | S | ∨ dµ = 0 for all k ∈ N . (3.3)Using (3.2) and (3.3)lim k →∞ (cid:90) Ω h k · Φ( S ) + H k ( S − S ) | S | ∨ dµ r = − lim k →∞ (cid:90) Ω h k · Φ( S ) + H k ( S − S ) | S | ∨ dµ s < . (3.4)Note that for a sequence ( p k ) n ∈ N withlim k →∞ h k · Φ( S ) + H k ( S − S ) = − lim k →∞ p k for all ω ∈ Ω we need to have by no
WFLVR (Ω) that lim k →∞ p k = 0, so the LHSof (3.4) is equal to zero, a contradiction. (cid:3) Technical results and proofs
Proof of Theorems 2.6 and 2.7.
We start with the following technicalobservation:
Proposition 4.1.
Let Ω be analytic. Then the FTAP of (Bouchard and Nutz,2015) implies: N1pA (Ω , F U ) ⇔ Ω = Ω ∗ Φ Proof.
Set ˆ P := P f (Ω). To apply the FTAP of (Bouchard and Nutz, 2015) we onlyneed to show that ˆ P t ( ω ) := P f (proj t +1 (Ω ∩ Σ ωt )) has analytic graph: we thereforefix n ∈ N and consider the Borel measurable functionΣ : X n → X n × (cid:0) R d + (cid:1) ( t +1) n ( ω , . . . ω n ) (cid:55)→ ( ω , . . . , ω n , S t ( ω ) , . . . , S t ( ω n ))and note that the image Σ(Ω n ) is analytic, since Ω is analytic and the image ofan analytic set under a Borel measurable map as well as the Cartesian product ofanalytic sets is analytic (see (Bertsekas and Shreve, 1978, Prop. 7.38 & 7.40, p.165)). Next we consider the continuous function F : X n × (cid:0) R d + (cid:1) ( t +1) → X n × (cid:0) R d + (cid:1) ( t +1) n ( ω , . . . ω n , x ) (cid:55)→ ( ω , . . . , ω n , x, . . . , x ) . Note that F (cid:16) X n × (cid:0) R d + (cid:1) ( t +1) (cid:17) ∩ Σ(Ω n )is analytic and as projections of analytic sets are analytic A n := { ( ω, ˜ ω , . . . , ˜ ω n ) | ω ∈ Ω , ˜ ω i ∈ proj t +1 (Ω ∩ Σ ωt ) , i = 1 , . . . , n } is analytic as well. Let ∆ n ⊆ R n denote the simplex. Since the functions G : A n × ∆ n → Ω × P ( X n ) × ∆ n ( ω, ˜ ω , . . . ˜ ω n , λ , . . . , λ n ) (cid:55)→ ( ω, δ ˜ ω , . . . , δ ˜ ω n , λ , . . . , λ n )and H : Ω × P ( X n ) × ∆ n → Ω × P ( X )( ω, δ ˜ ω , . . . , δ ˜ ω n , λ , . . . , λ n ) (cid:55)→ (cid:32) ω, n (cid:88) i =1 δ ˜ ω i λ i (cid:33) are continuous, it follows that graph (cid:16) ˆ P t (cid:17) = (cid:83) n ∈ N H ( G ( A n × ∆ n )) is analytic.Take now ω ∈ Ω and P ∈ P f (Ω) such that P ( { ω } ) >
0. By the FTAP of (Bouchardand Nutz, 2015) there exists Q ∈ M such that Q (cid:28) ˜ P for some ˜ P ∈ P f (Ω), E Q [ φ j ] = 0 for all j = 1 , . . . , k and P (cid:28) Q . In particular Q ∈ M f Ω and Q ( { ω } ) > ∗ Φ and fix P ∈ ˆ P such that supp( P ) = { ω , . . . , ω n } forsome n ∈ N . We can find Q , . . . , Q n ∈ M f Ω such that Q i ( { ω i } ) > i = 1 , . . . , n .Then Q := 1 /n (cid:80) ni =1 Q i ∈ M f Ω , Φ and Q ( { ω i } ) > i = 1 , . . . , n , i.e., P (cid:28) Q . (cid:3) We now give a complete proof of the quasi-sure FTAP in (Bouchard and Nutz,2015) using results from (Burzoni et al., 2019a). We first look at the case Φ = 0and start with an auxiliary lemma:
Lemma 4.2.
Let t ∈ { , . . . , T } and Ω ⊆ X t be analytic. Then the conditionalstandard separator of (Burzoni et al., 2019a) denoted by ξ t, Ω is F U t − -measurable.Proof. We shortly recall arguments from (Burzoni et al., 2019a)[proof of Lemma 1]:let us define the multifunction ψ t, Ω : ω ∈ X (cid:16) { ∆ S t (˜ ω ) | ˜ ω ∈ Σ ωt − ∩ Ω } ⊆ R d . Then ψ t, Ω is an F U t − -measurable multifunction. Indeed, for O ⊆ R d open we have { ω ∈ X | ψ t, Ω ( ω ) ∩ O (cid:54) = ∅} = S − t − ( S t − ((∆ S t ) − ( O ) ∩ Ω)) . As ∆ S t is Borel measurable (∆ S t ) − ( O ) ∈ F t . Also as intersections, projectionsand preimages of analytic sets are analytic (see (Bertsekas and Shreve, 1978, Prop.7.35 & Prop. 7.40)), we find that { ω ∈ X | ψ t, Ω ( ω ) ∩ O (cid:54) = ∅} is analytic and inparticular F U t − -measurable. Let S d be the unit sphere in R d , then by preservationof measurability also the multifunction ψ ∗ t, Ω ( ω ) := { H ∈ S d | H · y ≥ y ∈ ψ t, Ω ( ω ) } is F U t − -measurable and closed-valued. Let { ξ nt, Ω } n ∈ N be its F U t − -measurable Cas-taing representation. The conditional standard separator is then defined as ξ t, Ω = ∞ (cid:88) n =1 n ξ nt, Ω . (cid:3) Remark . We recall that this separator has the property that it aggregates allone-dimensional One-point Arbitrages on Σ ωt − ∩ Ω in the sense that { ω ∈ X | ξ ( ω ) · ∆ S t ( ω ) > } ⊆ { ω ∈ X | ξ t, Ω ( ω ) · ∆ S t ( ω ) > } for every measurable selector ξ of ψ ∗ t, Ω . Proof of Theorem 2.6 for
Φ = 0 . We start by proving the first part of Theorem 2.6,i.e., we are given a set of measures P satisfying ( AP S ) and we need to constructΩ = Ω P such that (1)-(3) are equivalent. We define for ω ∈ X t − ˜ χ F t − ( ω ) = (cid:92) { A ⊆ R d closed | P (∆ S t ( ω, · ) ∈ A ) = 1 ∀ P ∈ P t − ( ω ) } . Then ˜ χ F t − is closed valued and P (∆ S t ( ω, · ) ∈ ˜ χ F t − ( ω )) = 1 for all P ∈ P t − ( ω )and all ω ∈ X t − . Evidently˜ χ F t − ( ω ) = { x ∈ R d | ∀ ε > P (∆ S t ( ω, · ) ∈ B ( x, ε )) > P ∈ P t − ( ω ) } = (cid:91) P ∈ P t − ( ω ) supp( P ◦ ∆ S t ( ω, · ) − ) . Also it follows from (Bouchard and Nutz, 2015, Lemma 4.3, page 840), that ˜ χ F t − is analytically measurable. We quickly repeat their argument: let us define l : X t − × P ( X ) → P ( R d ) l ( ω, P ) = P ◦ ∆ S t ( ω, · ) − . OBUST MODELLING OF FINANCIAL MARKETS 21
Then l is Borel measurable. Next we consider R : X t − (cid:16) P ( R d ) R ( ω ) := l ( ω, P t − ( ω )) = { P ◦ ∆ S t ( ω, · ) − | P ∈ P t − ( ω ) } . Since its graph is analytic, it follows that for O ⊆ R d open { ω ∈ X t − | ˜ χ F t − ( ω ) ∩ O (cid:54) = ∅} = { ω ∈ X t − | R ( O ) > R ∈ R ( ω ) } = proj X t − { ( ω, R ) ∈ graph( R ) | R ( O ) > } is analytic as R (cid:55)→ R ( O ) is Borel.We also note that for ε > x (cid:55)→ R ( B ε ( x )) is continuous, so ( x, R ) (cid:55)→ R ( B ε ( x )) is Borel andgraph( ˜ χ F t − ) = { ( ω, x ) ∈ ( X t − × R d ) | x ∈ ˜ χ F t − ( ω ) } = (cid:92) ε ∈ Q + proj X t − × R d (cid:0) { ( ω, R, x ) ∈ (graph( R ) × R d ) | R ( B ε ( x )) > } (cid:1) is analytic. Now we define U = { ω ∈ X t | ∆ S t ( ω ) ∈ ˜ χ F t − ( ω ) } . Then U = proj X t (graph(∆ S t ) ∩ graph( ˜ χ F t − ))is analytic and by Fubini’s theorem P ( U ) = 1 holds for all P ∈ P . We now setΩ P = T (cid:92) t =1 (cid:110) ω ∈ X t | ∆ S t ( ω ) ∈ ˜ χ F t − ( ω ) (cid:111) , which is again analytic and P (Ω P ) = 1 for all P ∈ P .Having defined Ω P we can now begin to prove equivalence of (1)-(3) . If (2) holdsthen (3) follows immediately by a contradiction argument, so we now show themore involved implications (3) ⇒ (1) and (1) ⇒ (2) . Let us start with the proof of (3) ⇒ (1) : we assume that there exists ˆ P ∈ P such that ˆ P (cid:0) Ω P \ (Ω P ) ∗ (cid:1) >
0. Wewant to find H ∈ H ( F U ) and ˜ P ∈ P such that H ◦ S T ≥ P -q.s and ˜ P ( H ◦ S T > >
0. For this we take t = T − P (cid:0) { ω ∈ proj T − (Ω P ) | there is a One-point Arbitrage on Σ ωT − ∩ Ω P } (cid:1) > . Let us now fix ω ∈ { proj T − (Ω P ) | there is a One-point Arbitrage on Σ ωT − ∩ Ω P } .Denote by ξ T, Ω P the F U T − -measurable standard separator of Lemma 4.2. Now wedefine for each P ∈ P T − ( ω ) the push-forward of P as P ∆ S T ( ω, · ) ( A ) = P (∆ S T ( ω, · ) ∈ A ) , where A ∈ B ( R d ). We note that by definition P ∆ S T ( ω, · ) (cid:16) ˜ χ F T − ( ω ) (cid:17) = 1holds for all P ∈ P T − ( ω ). With a slight abuse of notation we recall the set B ( ω ) := { ω (cid:48) ∈ proj T (Σ ωT − ∩ Ω P ) | ξ T, Ω P ( ω ) · ∆ S T ( ω, ω (cid:48) ) > } from (Burzoni et al., 2019a, proof of Lemma 1, Step 1) and note that for all P ∈ P T − ( ω ) P (cid:0) { ω (cid:48) ∈ proj T (Σ ωT − ∩ Ω P ) | ξ T, Ω P ( ω ) · ∆ S T ( ω, ω (cid:48) ) > } (cid:1) = P ∆ S T ( ω, · ) ( { x ∈ R d | ξ T, Ω P ( ω ) · x > } )follows. Clearly the set { x ∈ R d | ξ T, Ω P ( ω ) · x > } is open in R d , thus by definitionof ˜ χ F T − ( ω ) there is a ˜ P ∈ P T − ( ω ) such that˜ P ∆ S T ( ω, · ) ( { x ∈ R d | ξ T, Ω P ( ω ) · x > } ) > or there are no One-point Arbitrages on Σ ωT − ∩ Ω P . To finish the proof of (3) ⇒ (1) we need to select ˜ P in a measurable way and this follows by standard arguments:Define the correspondence Ψ : R d × X T − (cid:16) P ( X ) byΨ( H, ω ) = { P ∈ P T − ( ω ) | E P [ H · ∆ S T ( ω, · )] + > } . This function has analytic graph by arguments in (Nutz, 2016, proof of Lemma3.4, p.11), so we can employ the Jankov-von-Neumann theorem (cf. (Bertsekas andShreve, 1978, Proposition 7.49, page 182)) to find a universally measurable kernel P (cid:48) T − : R d × X T − → P ( X )such that P (cid:48) T − ( H, ω ) ∈ P T − ( ω ) for all ( H, ω ) ∈ R d × X T − and P T − ( H, ω ) ∈ Ψ( H, ω ) on { Ψ( H, ω ) (cid:54) = ∅} . Then also the kernel ω (cid:55)→ ˜ P T − ( ω ) := P (cid:48) T − ( ξ T, Ω P ( ω ) , ω )is universally measurable. Defining ˜ P := ˆ P | X T − ⊗ ˜ P T − , which is the productmeasure formed from the restriction of ˆ P to X T − and ˜ P T − gives ˜ P ( ξ T, Ω P · ∆ S T > >
0. This proves (3) ⇒ (1) by backward induction.Lastly we show (1) ⇒ (2) : let us assume P ((Ω P ) ∗ ) = 1 for all P ∈ P . Note thatby the arguments given in the proof of (3) ⇒ (1) this means that T (cid:91) t =1 { proj t − (Ω P ) | there is a One-point Arbitrage on Σ ωt − ∩ Ω P } is a P -polar set, so in particular 0 ∈ ri( ˜ χ F t − ( ω )) for all t = 1 , . . . , T and P -q.e. ω ∈ X . Here ri( ˜ χ F t − ( ω )) denotes the relative interior of the convex hull of˜ χ F t − ( ω ). Let ˆ P ∈ P be fixed. We define for an arbitrary P ∈ P and ω ∈ X t − thesupport of P t − ( ω ) ◦ ∆ S − t ( ω, · ) conditioned on F t − as χ P F t − ( ω ) = { x ∈ R d | P t − ( ω )(∆ S t ( ω, · ) ∈ B ε ( x )) > ε > } . Using selection arguments which are explained below, we can now find measurableselectors P (0 , , . . . , P (0 ,d ) , P (1 , , . . . , P ( T − ,d ) such that P ( t, ( ω ) , . . . , P ( t,d ) ( ω ) ∈ P t ( ω )and P (0 , , . . . , P ( T − ,d ) fulfil the following property: define˜ P t ( ω ) = 1 d + 1 (cid:32) ˆ P t ( ω ) + d (cid:88) i =1 P ( t,i ) ( ω ) (cid:33) for t = 0 , . . . , T − ω ∈ X t . Then for ˜ P = ˜ P ⊗ · · · ⊗ ˜ P T − we have0 ∈ ri (cid:16) χ ˜ P F t − (cid:17) ˜ P -a.s. for all 1 ≤ t ≤ T, where ri (cid:16) χ ˜ P F t − (cid:17) denotes the relative interior of the convex hull of χ ˜ P F t − .We note that since P t ( ω ) is convex, we have ˜ P t ( ω ) ∈ P t ( ω ) for ω ∈ X t and bydefinition ˆ P (cid:28) ˜ P holds. Now it follows from (Rokhlin, 2008, Theorem 1, page 1),that there exists a martingale measure Q equivalent to ˜ P . The fact that ˜ P ∈ P implies Q ∈ Q P , which shows the claim.We now present the measurable selection argument: we fix t ∈ { , . . . , T } . Notethat for all ω ∈ proj t − ((Ω P ) ∗ ) we conclude 0 ∈ ri(∆ S t ( ω, Σ ωt − ∩ (Ω P ) ∗ )) bydefinition of (Ω P ) ∗ , which implies by (Bonnice and Reay, 1969, Theorem D, p.1)that there exist P , . . . , P d ∈ P t − ( ω ), which might not be pairwise distinct, s.t.0 ∈ ri (cid:32) supp (cid:32) ˆ P t − ( ω ) + P + · · · + P d d + 1 ◦ ∆ S t ( ω, · ) − (cid:33)(cid:33) . OBUST MODELLING OF FINANCIAL MARKETS 23
Note that ω (cid:55)→ ˆ P t − ( ω ) is universally measurable. We define the correspondence ρ : P ( X ) d +1 (cid:16) R d by ρ : ( P , P , . . . , P d ) = supp (cid:18) P + P + · · · + P d d + 1 ◦ ∆ S t ( ω, · ) − (cid:19) . Note that for O ⊆ R d open we have { ( P , P , . . . , P d ) | ρ ( P , P , . . . , P d ) ∩ O (cid:54) = ∅} = d (cid:91) i =0 { ( P , P , . . . , P d ) | P i ◦ ∆ S t ( ω, · ) − ( O ) > } . Since P (cid:55)→ P ( O ) is Borel measurable, we conclude that ρ is weakly measurable. Letus denote by S d the unit sphere in R d . By preservation of measurability (cf. (Rock-afellar and Wets, 2009, Exercise 14.12, page 653)) it follows that the correspondenceΨ : P ( X ) d +1 (cid:16) R d Ψ( P , P , . . . , P d ) = { H ∈ S d | H · y ≥ y ∈ ρ ( P , P , . . . , P d ) } is weakly measurable. Then also the correspondence ˜Ψ : P ( X ) d +1 (cid:16) R d ˜Ψ( P , P , . . . , P d ) = { H ∈ S d | H · y ≤ y ∈ ρ ( P , P , . . . , P d ) }∩ Ψ( P , P , . . . , P d )is weakly measurable and closed-valued. Let V be a countable base of R d . The set { ( P , P , . . . , P d ) | ˜Ψ( ω, P , . . . , P d ) = Ψ( P , P , . . . , P d ) } = (cid:92) O : O ∈ V ( { ( P , P , . . . , P d ) | Ψ ∩ O (cid:54) = ∅} ∩ { ( P , P , . . . , P d ) | ˜Ψ ∩ O (cid:54) = ∅}∪ { ( P , P , . . . , P d ) | Ψ ∩ O = ∅} ∩ { ( P , P , . . . , P d ) | ˜Ψ ∩ O = ∅} )is Borel measurable. Note that for an arbitrary convex set C ⊆ R d the relationship0 ∈ ri( C ) ⇔ ( ∀ H ∈ S d s.t. H · x ≥ ∀ x ∈ C ⇒ H · x = 0 ∀ x ∈ C )holds. Let A := { ( P , P , . . . , P d ) | ∈ ri (cid:0) ρ ( P , P , . . . P d ) (cid:1) for i = 1 , . . . , d } . Then from the above arguments it follows that A is Borel and in particular theset-valued mapping A ( ω, P ) := { ( P , . . . , P d ) | ∈ ri (cid:0) ρ ( P , P , . . . P d ) (cid:1) , P i ∈ P t − ( ω ) for i = 1 , . . . , d } has analytic graph. We can now employ the Jankov-von-Neumann theorem (cf.(Bertsekas and Shreve, 1978), Proposition 7.49, page 182) to find universally mea-surable kernels P it − : X t − → P ( X ) such that for every ω ∈ X t − we have P it − ( ω ) ∈ P t − ( ω ) and0 ∈ ri (cid:16) ρ ( ˆ P t − ( ω ) , P t − , . . . P dt − ) (cid:17) . This concludes the proof of (1) ⇒ (2) .The second part of Theorem 2.6 follows immediately from Proposition 4.1. (cid:3) Before continuing the proof of Theorem 2.6 let us first give a short remark on themeasurability of the arbitrage strategies involved in the proof of above:
Remark . By the FTAP of (Burzoni et al., 2019a) there exists a filtration ˜ F with F ⊆ ˜ F ⊆ F M such that there is no Strong Arbitrage in H (˜ F ) on Ω P . Moreconcretely there exists an H (˜ F )- and thus H ( F M )-measurable arbitrage aggrega-tor H ∗ . So in particular if P (Ω P \ (Ω P ) ∗ ) > P ∈ P , then H ∗ is an H (˜ F )-measurable P -q.s arbitrage. In general the inclusion ˜ F ⊆ F U does not hold.This is why we need to construct a new F U -measurable arbitrage strategy, whichcaptures the arbitrages essential for P . More generally, in this paper we manage to avoid using projectively measurable sets, which were essential for the arguments in(Burzoni et al., 2019a). In fact, all our trading strategies are universally measurablewithout invoking the axiom of projective determinacy.Furthermore, we hope that by constructing an explicit arbitrage strategy in theproof of (3) ⇒ (1) we can clarify the proof of (Burzoni et al., 2016), Theorem 4.23,pp. 42-46 (in particular (Burzoni et al., 2016)[A.3]) by offering a similar to theabove (but much simpler) reasoning for the case P = { P } . Introducing a measur-able separator ξ it is apparent that j z in (Burzoni et al., 2016, p.44) can always bechosen equal to one in our setting. Also the resulting strategy H P therein can bechosen universally measurable.To prove the first part of Theorem 2.6 for the case Φ (cid:54) = 0 we recall the followingnotion from (Burzoni et al., 2019a): Definition 4.5 ((Burzoni et al., 2019a), Def. 4) . A pathspace partition scheme R ( α ∗ , H ∗ ) of Ω is a collection of trading strategies H , ..., H β ∈ H ( F U ), α , . . . , α β ∈ R k and arbitrage aggregators ˜ H , . . . , ˜ H β for some 1 ≤ β ≤ k such that(1) the vectors α i , 1 ≤ i ≤ β are linearly independent,(2) for any i ≤ β α i · Φ + H i ◦ S T ≥ A ∗ i − , where A = Ω, A i := { α i · Φ + H i ◦ S T = 0 } ∩ A ∗ i − ,(3) for any i = 0 , . . . , β , ˜ H i is an Arbitrage Aggregator for A i ,(4) if β < k , then either A β = ∅ or for any α ∈ R k linearly independent from α , . . . , α β there does not exist H such that α · Φ + ( H ◦ S T ) ≥ A ∗ β . Definition 4.6 ((Burzoni et al., 2019a), Def. 5) . A pathspace partition scheme R ( α ∗ , H ∗ ) is successful if A ∗ β (cid:54) = ∅ .We quote the following results: Lemma 4.7 ((Burzoni et al., 2019a), Lemma 5) . For any R ( α ∗ , H ∗ ) , A ∗ i = Ω ∗{ α j · Φ | j ≤ i } .Moreover, if R ( α ∗ , H ∗ ) is successful, then A ∗ β = Ω ∗ Φ . Lemma 4.8 ((Burzoni et al., 2019a), Proof of Theorem 1 for Φ (cid:54) = 0) . A pathspacepartition scheme R ( α ∗ , H ∗ ) is successful if and only if Ω ∗ Φ (cid:54) = ∅ . We now complete the first part of the proof of Theorem 2.6 for the case Φ (cid:54) = 0:
Proof of Theorem 2.6 for Φ (cid:54) = 0 . The existence of Ω P and (1) ⇒ No Strong Ar-bitrage in A Φ (˜ F ) on Ω P , (2) ⇒ (3) follow exactly as before. We now argue that (1) ⇒ (2) holds in the spirit of (Bouchard and Nutz, 2015, Theorem 5.1, p. 850), byinduction over the number e of options available for static trading. In particular wecan assume without loss of generality that there exists a random variable ϕ ≥ | φ j | ≤ ϕ for all j = 1 , . . . , k and consider the set Q ϕ = { Q ∈ Q P | E Q [ ϕ ] < ∞} in order to avoid integrability issues. So let us assume there are e ≥ φ , . . . , φ e , for which (1) ⇒ (2) holds. We introduce an additional op-tion g = φ e +1 and assume P (cid:16) (Ω P ) ∗{ φ ,...,φ e +1 } (cid:17) = 1 for all P ∈ P . Then clearly P (cid:16) (Ω P ) ∗{ φ ,...,φ e } (cid:17) = 1 for all P ∈ P and by the induction hypothesis there is noarbitrage in the market with options { φ , . . . , φ e } available for static trading. Let P ∈ P . Then by exactly the same arguments as in (Bouchard and Nutz, 2015, proofof Theorem 5.1(a)) we can use convexity of Q ϕ and Theorem 2.9 to find a measure Q ∈ Q ϕ , such that P (cid:28) Q and Q ∈ Q P , { φ ,...,φ e +1 } , so (2) holds.Lastly it remains to show (3) ⇒ (1) . Let us thus assume there exists ˆ P ∈ P suchthat ˆ P (Ω P \ (Ω P ) ∗ Φ ) >
0. We want to find ( h, H ) ∈ A Φ ( F U ) and ˜ P ∈ P such that OBUST MODELLING OF FINANCIAL MARKETS 25 h · Φ + H ◦ S T ≥ P -q.s and ˜ P ( h · Φ + H ◦ S T > >
0. We use the properties ofa pathspace partition scheme R ( α ∗ , H ∗ ) recalled above. We define m = min( k ∈ { , . . . , β } | ˜ P ( A k \ A ∗ k ) > P ∈ P )˜ m = min( k ∈ { , . . . , β } | ˜ P ( A ∗ k − \ A k ) > P ∈ P ) , where A = Ω P . If ˜ m ≤ m then we select the strategy ( α ˜ m , H ˜ m ) ∈ A Φ ( F U ) whichsatisfies H ˜ m ◦ S T + α ˜ m · Φ ≥ A ∗ ˜ m − . We note that P ( A ∗ ˜ m − ) = 1 for all P ∈ P by definition of m, ˜ m and { H ˜ m ◦ S T + α ˜ m · Φ > } = A ∗ ˜ m − \ A ˜ m , so that˜ P ( H ˜ m ◦ S T + α ˜ m · Φ > > P ∈ P . If ˜ m > m , then P ( A m ) = 1 forall P ∈ P , ˜ P ( A m \ A ∗ m ) > P ∈ P , so we can argue as in the proof ofProposition 2.6 for Φ = 0 (3) ⇒ (1) using a standard separator and measurableselection of a measure in P .As before, the second part of Theorem 2.6 follows immediately from Proposition4.1. This concludes the proof. (cid:3) Proof of Theorem 2.7.
We recall the analytic set Ω P from the proof of Theorem 2.6for Φ = 0 and the sets { C n } n ∈ N from (2.1). Now we define B := (cid:91) n ∈ N { C n | P ( C n ∩ (Ω P ) ∗ Φ ) = 0 for all P ∈ P } ∈ B ( X ) . We claim that (2.1) implies that B ∩ (Ω P ) ∗ Φ = (cid:91) { C ∈ S | P ( C ∩ (Ω P ) ∗ Φ ) = 0 for all P ∈ P } ∩ (Ω P ) ∗ Φ = (cid:91) { C ∈ S | C ∩ (Ω P ) ∗ Φ ∈ N P } ∩ (Ω P ) ∗ Φ . Indeed, clearly B ⊆ (cid:83) { C ∈ S | C ∩ (Ω P ) ∗ Φ ∈ N P } . Now assume towards acontradiction that there exists ω ∈ (cid:16)(cid:91) { C ∈ S | C ∩ (Ω P ) ∗ Φ ∈ N P } ∩ (Ω P ) ∗ Φ (cid:17) \ ( B ∩ (Ω P ) ∗ Φ ) . In particular ω ∈ C for some C ∈ S such that C ∩ (Ω P ) ∗ Φ ∈ N P . By By (2.1) thereexists n ∈ N such that C n ⊆ C and ω ∈ C n . This implies ω ∈ B and thus showsthe claim.Let us now first assume that Φ = 0 and setΩ := Ω P \ ((Ω P ) ∗ ∩ B ) ∈ F U . (4.1)By assumption we have P (Ω) = 1 for all P ∈ P . By definition of the () ∗ operationΩ ∗ = ((Ω P ) ∗ \ B ) ∗ Φ = (Ω P \ B ) ∗ follows. To see the above equality, take a martingale measure Q ∈ M Ω , Φ andassume that Q (Ω \ (Ω P \ B )) >
0. As Ω \ (Ω P \ B ) = Ω P \ (Ω P ) ∗ ∩ B we concludethat Q (Ω P \ (Ω P ) ∗ ) >
0. Since any calibrated martingale measure supported on asubset of Ω is in M Ω P this leads to a contradiction to the definition of (Ω P ) ∗ . Also,Ω P \ B = Ω P ∩ B c is the intersection of two analytic sets, so we conclude that Ω ∗ is analytic. Lastly, by definition of Ω P we conclude Ω ∗ = (Ω P ) ∗ P -q.s..The implications (1) ⇒ (2) ⇒ (3) ⇒ (4) ⇒ (5) follow directly from the definition.Thus we only need to show (5) ⇒ (1) . Let us fix C ∈ S such that C ⊆ Ω. NoArbitrage de la Classe S on Ω implies that Ω ∗ ∩ C (cid:54) = ∅ . From (4.1) we thusconclude that P ((Ω P ) ∗ ∩ C ) > P ∈ P . As Ω ∗ = (Ω P ) ∗ P -q.s. thisimplies P (Ω ∗ ∩ C ) >
0. Using a construction similar to the proof of Proposition 2.6for the case Φ = 0, we can find a measure ˜ P ∈ P such that ˜ P ( C ) > P ( ·| Ω ∗ ). By (Rokhlin, 2008, Theorem 1),we conclude that there exists a martingale measure Q ∈ Q P equivalent to ˜ P ( ·| Ω ∗ ),in particular Q ( C ) >
0. The case Φ (cid:54) = 0 can now be treated similarly: indeed,we define Ω P , Φ as in the proof of Theorem 2.6 for Φ = 0, but now including the statically traded options Φ in the definition of the quasi-sure support and followthe same arguments as above. This concludes the proof. (cid:3) Proof of Theorem 2.9.
We first show that the quasi-sure superhedging the-orem of (Bouchard and Nutz, 2015) implies the second part of Theorem 2.9.
Proposition 4.9.
Let Ω be an analytic subset of X and Ω ∗ Φ (cid:54) = ∅ . Let the set P satisfy (APS) and N P = N M f Ω , Φ Then N Q P , Φ = N M f Ω , Φ and for an uppersemianalytic function g : X → R sup Q ∈M f Ω , Φ E Q [ g ] = π Ω ∗ Φ ( g ) = π P ( g ) = sup Q ∈Q P , Φ E Q [ g ] . (4.2) Proof.
That Q P , Φ and M f Ω , Φ have the same polar sets follows by the definition ofΩ ∗ Φ and (Burzoni et al., 2019a, Lemma 2). We now show (4.2): consider P Ω := P f (Ω ∗ Φ ) . Note that there is no M f Ω , Φ -q.s. arbitrage iff there is no P Ω -q.s arbitrage.We now show that Ω ∗ Φ is analytic if Ω is analytic. Recall the set P Z, Φ from Lemma5.4 of (Burzoni et al., 2017), page 13 defined by P Z, Φ := (cid:26) P ∈ P f ( X ) | ∃ Q ∈ M fX, Φ such that d Q d P = c ( P )1 + Z (cid:27) , where Z = max i =1 ,...,d max t =0 ,...,T S it and c ( P ) = ( E P [1 + Z ] − ) − . (Burzoni et al.,2017) show that the set { ( ω, P ) | ω ∈ X ∗ , P ∈ P ω } is analytic, where P ω = { P ∈ P Z, Φ | P ( { ω } ) > } . Note that { ( ω, P ) | ω ∈ X ∗ , P ∈ P ω } ∩ (cid:0) Ω × P f (Ω) (cid:1) is analytic and the projection of the above set to the first coordinate is exactly Ω ∗ Φ ,which shows that Ω ∗ Φ is analytic. We note ω (cid:55)→ P Ω t ( ω ) = P f (proj t +1 (Σ ωt ∩ Ω ∗ Φ ))has analytic graph by exactly the same argument as in the proof of Proposition4.1 replacing Ω by Ω ∗ Φ . The result now follows from the Superhedging Theorem of(Bouchard and Nutz, 2015) and the definition of M f Ω , Φ . (cid:3) We now show that the classical P -a.s. one-step superhedging duality can be deducedby means of pathwise reasoning: Lemma 4.10.
Let t ∈ { , . . . , T − } and g : X t +1 → R be F U t +1 -measurable.Let P ∈ P ( X ) and fix ω ∈ X t such that NA ( P ) holds for the one-period model ( S t ( ω ) , S t +1 ( ω, · )) . Then sup Q ∼ P , Q ∈M X E Q [ g ( ω, · )] = inf { x ∈ R | ∃ H ∈ R d s.t. x + H ∆ S t +1 ( ω, · ) ≥ g ( ω, · ) P -a.s. } . Proof. As g is F U t +1 -measurable, by (Bertsekas and Shreve, 1978, Lemma 7.27,p.173) there exists a Borel-measurable function ˜ g : ( R d ) t +1 → R such that g ( ω ) =˜ g ( S t +1 ( ω )) for P -a.e. ω ∈ X t +1 . Assume first that S t +1 (cid:55)→ ˜ g ( S t ( ω ) , S t +1 )is continuous. Define χ P := supp( P ◦ ∆ S t +1 ( ω, · ) − ). Then as NA ( P ) holds χ P = ( χ P ) ∗ and thus by (Burzoni et al., 2019a, Theorem 2) and continuity of OBUST MODELLING OF FINANCIAL MARKETS 27 S t +1 (cid:55)→ ˜ g ( S t ( ω ) , S t +1 ) as well as S t +1 (cid:55)→ H ( S t +1 − S t ( ω ))sup Q ∼ P , Q ∈M X E Q [ g ( ω, · )] ≤ inf { x ∈ R | ∃ H ∈ R d s.t. x + H ∆ S t +1 ( ω, · ) ≥ g ( ω, · ) P -a.s. } = inf { x ∈ R | ∃ H ∈ R d s.t. x + H ∆ S t +1 ( ω, · ) ≥ ˜ g ( S t ( ω ) , · ) on χ P } = sup Q ∈M fχ P E Q [˜ g ( S t ( ω ) , · )] ≤ sup Q ∼ P ◦ ∆ S t +1 ( ω, · ) − , Q ∈M R d E Q [˜ g ( S t ( ω ) , · )]= sup Q ∼ P , Q ∈M X E Q [ g ( ω, · )] . If S t +1 (cid:55)→ ˜ g ( S t ( ω ) , S t +1 ) is Borel-measurable, then by Lusin’s theorem (see (Cohn,2013, Theorem 7.4.3, p.227)) there exists an increasing sequence of compact sets( K n ) n ∈ N such that K n ⊆ χ P , P ◦ ∆ S t +1 ( ω, · ) − ( K cn ) ≤ /n and ˜ g ( S t ( ω ) , · ) | K n is continuous. In particular there exists n ∈ N such that for all n ≥ n we have K n = ( K n ) ∗ . By the above argumentinf { x ∈ R | ∃ H ∈ R d s.t. x + H ∆ S t +1 ( ω, · ) ≥ ˜ g ( S t ( ω ) , · ) on K n } (4.3) = sup Q ∼ P ◦ ∆ S t +1 ( ω, · ) − Q ∈M Kn E Q [ g ( ω, · )]holds for n ≥ n . The claim now follows by taking suprema in n ∈ N on both sidesof (4.3):inf { x ∈ R | ∃ H ∈ R d s.t. x + H ∆ S t +1 ( ω, · ) ≥ g ( ω, · ) P -a.s. } = sup n ∈ N inf { x ∈ R | ∃ H ∈ R d s.t. x + H ∆ S t +1 ( ω, · ) ≥ ˜ g ( S t ( ω ) , · ) on K n } = sup n ∈ N sup Q ∼ P ◦ ∆ S t +1 ( ω, · ) − Q ∈M Kn E Q [˜ g ( S t ( ω ) , · )] ≤ sup Q ∼ P ◦ ∆ S t +1 ( ω, · ) − Q ∈M R d E Q [˜ g ( S t ( ω ) , · )] ≤ inf { x ∈ R | ∃ H ∈ R d s.t. x + H ∆ S t +1 ( ω, · ) ≥ g ( ω, · ) P -a.s. } . (cid:3) Using this one-step duality result under fixed P and (APS) of P we now prove thefirst part of Theorem 2.9, which is restated in the following proposition: Proposition 4.11.
Let NA ( P ) hold and let g : X → R be upper semianalytic.Then there exists a measure P g = P g ⊗ · · · ⊗ P gT − and an F U -measurable set Ω P g with P (Ω P g ) = 1 for all P ∈ P , such that π P ( g ) = π ˆ P ( g ) = π (Ω P g ) ∗ Φ ( g ) = sup Q ∈M Ω P g , Φ E Q [ g ] = sup Q ∈Q P , Φ E Q [ g ] . Proof.
We note that by NA ( P ) and Theorem 2.6 the difference Ω P \ (Ω P ) ∗ Φ is P -polar. We first take Φ = 0. Recall the definition of the one-step functionals givenin (Bouchard and Nutz, 2015, Lemma 4.10, p. 846) E T ( g )( ω ) = g ( ω ) E t ( g )( ω ) = sup Q ∈Q t ( ω ) E Q [ E t +1 ( g )( ω, · )] , t = 0 , . . . , T − . By (APS) and upper semianalyticity of g, every E t ( g ) is upper semianalytic. Weshow recursively that for every t = 0 , . . . , T − P -q.e. ω ∈ X t there existsa measure P ∈ P ( X ) such that NA ( P ) holds andsup Q ∈Q t ( ω ) E Q [ E t +1 ( g )( ω, · )] = sup Q ∼ P , Q ∈M X E Q [ E t +1 ( g )( ω, · )] . Note that by measurable selection arguments and construction of Ω P we concludethat for P -q.e. ω ∈ X t the properties NA ( P t ( ω )) and P (proj t +1 (Ω P ∩ Σ ωt )) = 1hold for all P ∈ P t ( ω ). We now fix t ∈ { , . . . , T − } and ω ∈ X t such that NA ( P t ( ω )) and P (proj t +1 (Ω P ∩ Σ ωt )) = 1 for all P ∈ P t ( ω ) holds. Note that thereexists a sequence ( P n ) n ∈ N such that P n ∈ P t ( ω ) for all n ∈ N andsup Q (cid:28) P n , Q ∈M X E Q [ E t +1 ( g )( ω, · )] ↑ sup Q ∈Q t ( ω ) E Q [ E t +1 ( g )( ω, · )] ( n → ∞ ) . We see from the proof of Theorem 2.6 for Φ = 0 in Section 4.1 that under NA ( P t ( ω ))and for a fixed P ∈ P t ( ω ), we can always find ˜ P ∈ P t ( ω ) such that P (cid:28) ˜ P and NA ( ˜ P ) holds. Thus we can assume without loss of generality that NA ( P n )holds for all n ∈ N . Define ˆ P n := (cid:80) nk =1 − k / (1 − − n ) P k ∈ P t ( ω ) as well as P gt ( ω ) := (cid:80) ∞ k =1 − k P k and note that NA ( ˆ P n ) as well as NA ( P gt ( ω )) hold for all n ∈ N . Furthermore E t ( g )( ω ) = sup n ∈ N sup Q (cid:28) P n , Q ∈M X E Q [ E t +1 ( g )( ω, · )] ≤ sup n ∈ N sup Q ∼ ˆ P n , Q ∈M X E Q [ E t +1 ( g )( ω, · )] ≤ sup Q ∼ P gt ( ω ) , Q ∈M X E Q [ E t +1 ( g )( ω, · )]= inf { x ∈ R | ∃ H ∈ R d s.t. x + H ∆ S t +1 ( ω, · ) ≥ E t +1 ( g )( ω, · )) P gt ( ω )-a.s. } , where the last equality follows from Lemma 4.10. Define π P gt ( ω ) t ( g ) = inf { x ∈ R | ∃ H ∈ R d s.t. x + H ∆ S t +1 ( ω, · ) ≥ E t +1 ( g )( ω, · ) P gt ( ω )-a.s. } π P t ( ω ) ( g ) = inf { x ∈ R | ∃ H ∈ R d s.t. x + H ∆ S t +1 ( ω, · ) ≥ E t +1 ( g )( ω, · ) P t ( ω )-q.s. } . Clearly π P gt ( ω ) ( g ) ≤ π P t ( ω ) ( g ). Now assume towards a contradiction that the in-equality is strict and set ε := π P t ( ω ) ( g ) − π P gt ( ω ) ( g ) >
0. Furthermore note that fora sequence of compact sets ( K n ) n ∈ N such that K n ↑ R d we have π P gt ( ω ) | Kn ( g ) := inf { x ∈ R | ∃ H ∈ R d s.t. x + H ∆ S t +1 ≥ E t +1 ( g ) P gt ( ω )( ·| ∆ S t +1 ( ω, · ) ∈ K n )-a.s. }↑ π P gt ( ω ) ( g ) ( n → ∞ ) ,π P t ( ω ) | Kn ( g ) := inf { x ∈ R | ∃ H ∈ R d s.t. x + H ∆ S t +1 ≥ E t +1 ( g ) P t ( ω ) | K n -q.s. }↑ π P t ( ω ) ( g ) ( n → ∞ ) , where P t ( ω ) | K n := { P ( ·| ∆ S t +1 ( ω, · ) ∈ K n ) | P ∈ P t ( ω ) , P (∆ S t +1 ( ω, · ) ∈ K n ) > } . Choose n ∈ N large enough, such that π P t ( ω ) | Kn ( g ) − π P gt ( ω ) | Kn ( g ) > ε/ . Denoteby H K n the closed set of H ∈ R d such that π P gt ( ω ) | Kn ( g ) + ε/ H ∆ S t +1 ( ω, · ) ≥ E t +1 ( g )( ω, · )) P gt ( ω )( ·| ∆ S t +1 ( ω, · ) ∈ K n )-a.s.Then for every H ∈ H K n there exists P Hn ∈ P t ( ω ) such that P Hn ( { π P gt ( ω ) | Kn ( g ) + ε/ H ∆ S t +1 ( ω, · ) < E t +1 ( g )( ω, · )) } ∩ { ∆ S t +1 ( ω, · ) ∈ K n } ) > . Note that there exists a countable sequence ( H kn ) k ∈ N , which is dense in H K n . Inparticular for every H ∈ R d such that π P gt ( ω ) | Kn ( g ) + ε/ H ∆ S t +1 ( ω, · ) ≥ E t +1 ( g )( ω, · )) P gt ( ω )( ·| ∆ S t +1 ( ω, · ) ∈ K n )-a.s.there exists k ∈ N such that π P gt ( ω ) | Kn ( g ) + ε/ H kn ∆ S t +1 ( ω, · ) ≥ E t +1 ( g )( ω, · )) P gt ( ω )( ·| ∆ S t +1 ( ω, · ) ∈ K n )-a.s.Set now P n = (cid:80) ∞ k =1 − k P H kn n ∈ P ( X ) and note that for all n ∈ N large enough π ( P gt ( ω )+ P n ) | Kn ( g ) − π P gt ( ω ) | Kn ( g ) ≥ ε/ . OBUST MODELLING OF FINANCIAL MARKETS 29
Taking K n ↑ R d we have in particular E t ( g )( ω ) ≤ sup Q ∼ P gt ( ω ) , Q ∈M X E Q [ g ] < lim n →∞ sup Q ∼ ( P gt ( ω )+ P n ) | Kn , Q ∈M X E Q [ g ] ≤ E t ( g )( ω ) , a contradiction. Thus E t ( g )( ω ) = π P t ( ω ) ( g ) = π P gt ( ω ) ( g ) . As 0 ∈ ri(supp(( P gt ( ω ))) a natural universally measurable candidate for a superhedg-ing strategy ω (cid:55)→ H t +1 ( ω ) is the right derivative lim ε ∈ Q , ε ↓ ( E εe i t ( g )( ω ) −E t ( g )( ω )) /ε where E εe i t ( g )( ω ) is the superhedging price for the Borel-measurable stock ( S t + εe i ), i = 1 , . . . , d (cid:48) ≤ d instead of S t . This is a pointwise limit of differences of upper sem-inanalytic functions and thus universally measurable. For ω ∈ X t such that thisquantity does not exist, we set H t +1 ( ω ) = 0. Furthermore in order to show thatthe map ω (cid:55)→ P gt ( ω ) can be chosen to be universally measurable we first note thatin (Bouchard and Nutz, 2015, Lemma 4.8, p.843) the set { ( Q , P ) ∈ P (proj t +1 (Σ ωt )) × P (proj t +1 (Σ ωt )) | E Q [∆ S t +1 ( ω, · )] = 0 , P ∈ P t ( ω ) , Q (cid:28) P } is analytic. Thus we can apply the Jankov-von-Neumann selection theorem (see(Bertsekas and Shreve, 1978, Proposition 7.50, p.184)) to find 1 /n -optimisers ( Q nt ( ω ) , P nt ( ω ))for E t ( g )( ω ) and the claim follows. The case Φ (cid:54) = 0 can be handled by induction asin the proof of Theorem 2.6 for Φ (cid:54) = 0.In conclusion we have found a strategy ( h, H ) ∈ A Φ ( F U ) such thatsup Q ∈Q P , Φ E Q [ g ] + h · Φ + ( H ◦ S T ) ≥ g P -q.s.We now defineΩ P g = Ω P ∩ (cid:40) ω ∈ X (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) sup Q ∈Q P , Φ E Q [ g ] + h · Φ( ω ) + ( H ◦ S T )( ω ) ≥ g ( ω ) (cid:41) ∈ F U . This concludes the proof. (cid:3)
Remark . By NA ( P ) Proposition 4.11 implies for g = 00 = inf { x ∈ R | ∃ ( h, H ) ∈ A Φ ( F U ) such that x + h · Φ + ( H ◦ S T ) ≥ P -q.s. } = inf ˜ g ∈ E inf { x ∈ R | ∃ ( h, H ) ∈ A Φ ( F U ) such that x + h · Φ + ( H ◦ S T ) ≥ ˜ g on (Ω P ˜ g ) ∗ Φ } = inf ˜ g ∈ E sup Q ∈M Ω P ˜ g , Φ E Q [˜ g ] , where we define E = { ˜ g : X → ( −∞ , F U -measurable | ˜ g = 0 P -q.s. } . In particular for every ˜ g ∈ E there exists Q ∈ M X, Φ such that E Q [˜ g ] = 0. A similarresult was obtained by (Burzoni et al., 2019b) in a more general setup. Aggregatingthe martingale measures corresponding to all ˜ g (and thus to all P -polar sets) toachieve a result comparable to (Bouchard and Nutz, 2015) in a setup without using(APS) of P remains an open problem. Data Availability Statement:
Data sharing is not applicable to this article as nonew data were created or analyzed in this study.
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