Super-replication with transaction costs under model uncertainty for continuous processes
aa r X i v : . [ q -f i n . M F ] F e b Super-replication with transaction costs undermodel uncertainty for continuous processes ∗ Huy N. Chau Masaaki Fukasawa Mikl´os R´asonyiFebruary 5, 2021
Abstract
We formulate a superhedging theorem in the presence of transactioncosts and model uncertainty. Asset prices are assumed continuous anduncertainty is modelled in a parametric setting. Our proof relies on anew topological framework in which no Krein–Smulian type theorem isavailable.
Robust super-replication has a rich literature, starting from the seminal workof [31] about lookback options. In this paper, bounds for option prices and theconnection with the Skorokhod embedding are established. The approach wasemployed further by [9] for barrier options, [16] for double no-touch options,[33] for forward start options, [10] for variance options, among others. We referto the survey of [32] and references therein.The use of optimal transport in robust hedging was developed in [27], [44],[4], [24]. These works provided a dual formulation which transforms the su-perhedging problem into a martingale optimal transportation problem. In acontinuous time setting, [24] proved duality and the existence of a family of sim-ple, piecewise constant super-replication strategies that asymptotically achievethe minimal super-replication cost. In the quasi-sure setting, [22] established atheoretical framework to construct stochastic integral and duality results, while[38], [7] obtained optimal super-replication strategies in discrete time. Variousframeworks are investigated in [17, 8, 34, 2, 14, 1].Under transaction costs, we refer to the discrete-time studies [6, 23]. Aspointed out by [6], it is not easy to come up with a proof of the superhedgingduality using the local method, developed in [7], which uses one-period argumentand then measurable selection techniques to paste the periods together. In[6], the authors introduced a fictitious market without transaction cost using arandomization technique, then the results of [7] were applied. In continuous-time markets with transaction costs, however, the only results we know of arethose of [25]. They establish, under technical conditions, that the pathwise ∗ Huy N. Chau is supported by Center for Mathematical Modeling and Data Science, OsakaUniversity. Masaaki Fukasawa is supported by Osaka University. Mikl´os R´asonyi is supportedby the “Lend¨ulet” grant LP 2015-6 of the Hungarian Academy of Sciences. frictionless mar-kets with continuous price processes, using the “no arbitrage of the first kind”assumption
N A ( P ), where P denotes the set of possible prior probabilities. Asuperhedging theorem is provided under this hypothesis. A crucial step in theirarguments shows that N A ( P ) is equivalent to having N A ( P ) for all individual P ∈ P . In the present paper, somewhat analogously, we assume a “no free lunchwith vanishing risk” condition (stronger than no arbitrage of the first kind) foreach individual model and deduce the superhedging theorem (with transactioncosts) under this assumption. Note that we operate with “martingale” pricesystems while [3] uses “supermartingale deflators”.In the present article we prove a fairly general superhedging theorem. Wetake a parametrization approach, different from the pathwise or quasi-sure set-tings, first used in [13], [40], see also [12]. As far as we know, the present paperis the first to apply to a wide range of continuous-time markets under transac-tion costs and model uncertainty. We now list the main features of the presentpaper.First, we use the new topological framework of [12] for studying hedgingunder model uncertainty, this time in a continuous-time setting. The workingspace is the product topological vector space L ∞ = Q θ ∈ Θ ( L ∞ , w ∗ ) where wedenote the product topology by w ∗ . Here Θ is the set of parameters describinguncertainty and w ∗ is the weak star topology on L ∞ , the space of a.s. equiva-lence classes of bounded random variables. Unlike the local method of [7], thisapproach offers a global method, which is suitable for handling transaction costsin continuous time,Second, for the separation arguments to work, we need the w ∗ -closednessof the set of hedgeable payoffs C in the product space L ∞ . This is typicallyshown relying on the Krein–Smulian theorem. Usual versions of that resultare stated for Fr´echet spaces, see for example [41]. Our product space L ∞ has the predual L θ ∈ Θ L , which is, in general, not Fr´echet, as Θ is typicallyuncountable. Therefore we are not able to apply the Krein–Smulian theoremdirectly. To remedy this, we prove a convex compactness property for the set ofstrategies in a suitable topology. Such a property fails in frictionless markets,it is a particularity of markets with transaction costs. Next we apply Krein–Smulian for finite direct sums of Fr´echet spaces and the w ∗ -closedness of C willbe ensured by convex compactness.Third, we apply the Hahn-Banach theorem in L ∞ to obtain what will becalled consistent price systems in the robust sense. These systems are, in fact,infinite dimensional vectors with finitely many nonzero components such thatdiscounting price processes by them has the same effect as the usual consistentprice systems.The paper is organized as follows. In Section 2 we introduce the marketmodel; consistent price systems in the robust sense; and the main result. Proofsare given in Section 3. Section 4 recalls important facts about topological vectorspaces and establishes the necessary results about convex compactness. Exten-sions of our results to the multi-asset, conic framework of [36] seem straight-forward but they would involve technical complications so we are staying in aone-asset framework for now. 2t seems less obvious to include stock prices with jumps where trading strate-gies are not necessarily right continuous, see [29]. This would necessitate find-ing a topological space in which the set of nice trading strategies is convexlycompact. It would also be interesting to see if the present framework can beadapted to continuous-time frictionless markets, see [12] about what happensin the discrete-time case. Notations . Let I be some index set and X i , i ∈ I be sets. In the productspace X = Q i ∈ I X i , a vector ( f i ) i ∈ I will be denoted by f . If there are anorderings ≥ i given on each X i then we write f ≥ g if f i ≥ i g i for all i ∈ I . If1 ∈ X i for all i then denotes the vector with all coordinates equal to 1 and i denotes the vector with coordinate i equal to 1 and the other coordinates zero.Similarly, when 0 ∈ X i , denotes the vector all of whose coordinates equal 0.The vector space of (equivalence classes of) random variables on (Ω , F , P ) isdenoted by L ( F , P ). As usual, L p ( F , P ) , p ∈ [1 , ∞ ] is the space of p -integrable(resp. bounded) random variables equipped with the standard k · k p norm. Let
T > , F , ( F t ) t ∈ [0 ,T ] , P ) be a filtered probabilityspace, where the filtration is assumed to be right-continuous and F coincideswith the P -completion of the trivial sigma-algebra. For simplicity, we alsoassume F = F T . Consider a financial market model with one risky asset andone risk-free asset (cash) whose price is assumed to be constant one.Let Θ be a (non-empty) set, which is interpreted as the parametrization ofuncertainty. It is not required that Θ is a convex subset of a vector space. Con-sider a family ( S θt ) t ∈ [0 ,T ] , θ ∈ Θ of adapted, positive processes with continuoustrajectories which represent the possible evolutions of the price of the risky assetin consideration.The risky asset is traded under proportional transaction cost λ ∈ (0 , λ ) S θt , (1 − λ ) S θt , respectively.Let H ↑ t , H ↓ t , t ∈ [0 , T ] be predictable processes with non-decreasing left-continuous trajectories. H ↑ t denotes the cumulative amount of transfers fromthe riskless asset to the risky one up to time t and H ↓ t represents the cumulativetransfers in the opposite direction. The set of all such ( H ↑ , H ↓ ) is denoted by V . A useful metric structure can be defined on V , see Subsection 4.3, whichwill be key for later developments.The portfolio position in the risky asset at time t equals φ t := H + H ↑ t − H ↓ t , t ∈ [0 , T ] . Here H is a ( F -measurable, hence deterministic) initial transfer. We alsoassume that φ − := 0.For any real number x ∈ R , we denote x + := max { , x } , x − := max { , − x } .For a pair of initial capital and initial transfer ( z, H ) ∈ R and a strategy H ∈ V , the corresponding liquidation value for parameter θ ∈ θ at time t ∈ [0 , T ]3s defined by W zt ( S θ , H, λ ) := z − H +0 S θ (1 + λ ) + H − S θ (1 − λ ) − Z t (1 + λ ) S θu dH ↑ u + Z t (1 − λ ) S θu dH ↓ u + φ + t (1 − λ ) S θt − φ − t (1 + λ ) S θt . (1) Definition 2.1.
A finite variation process ( H , H ) ∈ R × V is called an x -admissible strategy for the model θ if W t ( S θ , H, λ ) ≥ − x, a.s. for all t ∈ [0 , T ] . (2) Denote by A θx ( λ ) the set of x -admissible strategies for the model θ , set A θ ( λ ) = S x> A θx ( λ ) . The set of admissible strategies for the robust model is defined by A ( λ ) := T θ ∈ Θ A θ ( λ ) . Consider the product spaces L t = Q θ ∈ Θ L ( F t , P ) , L ∞ t = Q θ ∈ Θ L ∞ ( F t , P )with the corresponding product topologies. Other product spaces are definedsimilarly. For notational convenience, we use bold notations for vectors in theseproduct spaces. For t ∈ [0 , T ], we denote by S t := ( S θt ) θ ∈ Θ a vector in L t . For H ∈ A ( λ ), we denote by W t ( H ) = ( W t ( θ, H, λ )) θ ∈ Θ the vector consists of attainable payoffs from the strategy H under model un-certainty. Set L := L T , L ∞ := L ∞ T . L is the non-negative orthant of L .Define K = (cid:8) W T ( H ) : H ∈ A ( λ ) (cid:9) , C = ( K − L ) ∩ L ∞ . For each θ ∈ Θ, the classical no free lunch with vanishing risk condition for S θ , denoted by N F LV R ( λ, θ ), is recalled from [29]. Definition 2.2.
The price process S θ together with the transaction cost λ satisfythe N F LV R ( λ, θ ) condition if for any sequence H n , n ∈ N such that H n ∈ A θ /n ,and W T ( θ, H n , λ ) converges a.s. to some limit W ∈ [0 , ∞ ] a.s. then W = 0 , a.s. The following conditions will be imposed throughout the paper:
Assumption 2.3. (i) There is a sequence of stopping times T n , n ∈ N in-creasing almost surely to infinity such that sup θ ∈ Θ sup t ∈ [0 ,T ] S θt ∧ T n ≤ K n a.s. for some constants K n , n ∈ N .(ii) For all θ ∈ Θ , the stock price S θ satisfies N F LV R ( λ, θ ) for all < λ < . Condition ( i ) is technical, requiring uniform boundedness of the prices up toa sequence of stopping times. Condition ( ii ) assumes that none of the possibleprice processes admits (model-dependent) arbitrage. It holds obviously that C ∩ L ∞ + = { } . Remark 2.4.
To develop an intuition about the current, parametric frameworkof uncertainty we sketch a rather general example. Let Y be some stochasticprocess with values in R m for some m which generates the filtration. We imaginethat Y describes economics factors (dividend yields, interest rates, unemploy-ment rates, statements of firms) which are known (at least we have a reliable4tatistical model for them). The asset price S is assumed to be a nonlinearfunctional of these factors, that is, S θt = F ( θ, Y t ), θ ∈ Θ for some parameter setΘ and a function F : Θ × R m → R + . Here the parameter θ is unknown . Notethat if Y is locally bounded and F is regular enough then a localizing sequence T n for Y satisfies (i) in Assumption 2.3. Example 2.5.
Let the filtration be generated by a Brownian motion W t . Con-sider the possible price processes S θt = F (cid:18)Z t µ θ ( s ) ds + Z t K θ ( t, s ) dW s (cid:19) , t ∈ [0 , T ]where θ runs in an index set Θ, K θ ( t, s ) are kernels of a suitable regularity, µ θ are suitable processes and F : R → (0 , ∞ ) is a function (e.g. exponential). Thisis a family of models which possibly fail the semimartingale property but satisfy N F LV R ( λ, θ ) for every 0 < λ < Definition 2.6.
A consistent price system in the robust sense for S is a pair ( Z , M ) where(i) ≤ Z T ∈ L θ ∈ Θ L , and E [ Z ¯ θT A ] > for some ¯ θ and for some A ∈ F .(ii) M is a local martingale such that X θ ∈ Θ (1 − λ ) Z θt S θt ≤ M t ≤ X θ ∈ Θ (1 + λ ) Z θt S θt , a.s., (3) where Z θt := E [ Z θT |F t ] , t ∈ [0 , T ] for all θ ∈ Θ .The set of such objects is denoted by C (¯ θ, λ, A ) . Let Z ( λ ) := [ A ∈F , ¯ θ ∈ Θ C (¯ θ, λ, A ) . Note that C (¯ θ, λ, A ) is empty when P ( A ) = 0 so we could have restricted theabove definition to sets A with P ( A ) >
0. The quantity Z T may be interpretedas a local martingale density. In the setting without uncertainty, it is usuallyrequired that Z ¯ θT > , a.s. Here, our density Z ¯ θT has to be positive only onsome A with P [ A ] >
0. The new densities might look strange at the firstglance, however, they are comparable to “absolutely continuous local martingalemeasures”, introduced in [20], [21]. The main reason for using this notion is that,Θ being possibly uncountable, no exhaustion argument (as in the Kreps-Yanseparation theorem) can be applied. It will become clear in the proof of Theorem2.8 that the new densities work better than the densities with Z ¯ θT > , a.s. .When Θ is a singleton, it is possible to use the exhaustion argument andwe may assume Z ¯ θT > , a.s. so this definition reduces to the usual definition ofconsistent price systems, see [30], [29]. Remark 2.7.
Let D ⊂ Θ be finite and let Z iT = 0 for all the coordinates i / ∈ D .If, for each θ ∈ D , (1 − λ ) Z θt S θt ≤ M θt ≤ (1 + λ ) Z θt S θt , a.s. (4)5or suitable martingales M θ (that is, the processes ( Z θ , M θ ) determine con-sistent price systems for the individual models S θ , θ ∈ D ) then, defining M t := P θ ∈ D M θt defines a martingale satisfying (3). This shows how “usual”consistent price systems provide objects in C (¯ θ, λ, A ). The important point isthat there are many other elements in C (¯ θ, λ, A ). This is explained in Subsection3.5 of [12] where frictionless toy examples are presented.We state the main result of the paper. Theorem 2.8.
Let Assumption 2.3 be in force. Let G = ( G θ ) θ ∈ Θ ∈ L bebounded or let G ≥ . There exists ( z, H ) ∈ R × A ( λ ) such that W zT ( S θ , H, λ ) ≥ G θ , θ ∈ Θ if and only if z ≥ sup ( Z ,M ) ∈Z ( λ ) E "X θ ∈ Θ Z θT G θ . The proof of Theorem 2.8 is given in Section 3 below. ∗ -closedness of C The following result is classical.
Lemma 3.1.
For any x > , and any θ ∈ Θ , the set {k H k T : H ∈ A θx ( λ ) } isbounded in probability.Proof. We can assume that x = 1 and adopt the argument in [29]. By Assump-tion 2.3 (ii), for each θ ∈ Θ, 0 < b λ < λ , the process S θ satisfies the classicalcondition N F LV R ( b λ, θ ). Let us denote S θ = ( λ − b λ ) inf t ∈ [0 ,T ] S θt > , a.s. Assume that there exist α > H n ∈ A θ ( λ ) , n ∈ N such that forevery n ∈ N , P h S θ k H n k T > n i ≥ α. By Definition 2.1, we have that1 + W t ( S θ , H n , b λ ) ≥ W t ( S θ , H n , b λ ) − W t ( S θ , H n , λ )= Z t (cid:16) (1 + λ ) S θu − (1 + b λ ) S θu (cid:17) dH n, ↑ u + Z t (cid:16) (1 − b λ ) S θu − (1 − λ ) S θu (cid:17) dH n, ↓ u + ( H nt ) + (cid:16) (1 − b λ ) S θ − (1 − λ ) S θu (cid:17) + ( H nt ) − (cid:16) (1 + λ ) S θt − (1 + b λ ) S θt (cid:17) ≥ S θ k H n k t .
6t is clear that ˜ H n = H n /n ∈ A θ ( λ ) and then1 /n + W T ( S θ , ˜ H n , b λ ) ≥ S θ k ˜ H n k T , a.s. (5)and P h S θ k ˜ H n k T > i ≥ α for all n ∈ N . By Lemma 9.8.1 of [19], there existconvex combinations f n ∈ conv { S θ k ˜ H n k T , S θ k ˜ H n +1 k T , ... } such that f n → f, a.s. and P [ f > > . (6)Using the same weights as in the construction of f n , we obtain a sequence ofstrategies b H n such that1 /n + W T ( S θ , b H n , b λ ) ≥ f n , a.s.. Therefore, the random variable f is a F LV R ( b λ, θ ) for S θ , which is a contradic-tion. Remark 3.2. If H is a convex combination of H , H , it may happen thatthere are overlapping regions where H ↑ t H ↓ t = 0, for example when H , ↑ t H , ↓ t = 0 . However, we can always remove these redundant transactions from H and theobtained strategy generates a payoff which is at least as that from H . Withoutfurther notice, from now on we interpret H as the strategy after removingredundancy.Let D be a non-empty finite subset of Θ. We say that a sequence f n , n ∈ N in Q θ ∈ D L Fatou-converges to f if for each θ ∈ D , f θn converges to f θ a.s. and f θn ≥ − x θ , a.s. for some x θ >
0. Define C ( D ) = ( ( W T ( S θ , H, λ ) − h θ ) θ ∈ D : H ∈ \ θ ∈ D A θ ( λ ) , h θ ∈ L , θ ∈ D ) \ Y θ ∈ D L ∞ ( F T , P ) . Proposition 3.3.
The set C ( D ) is Fatou-closed, that is C ( D ) ∩ Q θ ∈ D B ∞ x θ isclosed in the product space Q θ ∈ D L ( F T , P ) , for each ( x θ ) θ ∈ D ∈ R D + .Proof. Let f n , n ∈ N be a sequence in C ( D ) such that for every θ ∈ D , f θn → f θ in probability and f θn ≥ − x θ . We need to find H ∈ T θ ∈ D A θ ( λ ) such that forevery θ ∈ D, W T ( S θ , H, λ ) ≥ f θ , a.s. .By taking a subsequence, we may assume that f θn → f θ , a.s. for all θ ∈ D .By definition, f θn ≤ W T ( S θ , H n , λ ) , ∀ θ ∈ D for some H n ∈ T θ ∈ D A θ ( λ ). Using Theorem 1 of [42], for every θ ∈ D we getthat H n ∈ A θx θ ( λ ) for all n ∈ N . Fix θ ∈ D arbitrarily. Lemma 3.1 implies thatthe set {k H k T , H ∈ A θ x θ ( λ ) } is bounded in probability. Using Lemma B.4 of[29], there are convex combinations ˜ H n, ↑ ∈ conv ( H n,s, ↑ , H n +1 , ↑ , ... ) and a finitevariation process H ↑ such that ˜ H n, ↑ → H ↑ pointwise and k ˜ H n, ↑ k → k H ↑ k pointwise. Similarly, we find another convex combinations, still denoted by˜ H n, ↓ , such that ˜ H n, ↓ → H ↓ pointwise and k ˜ H n, ↓ k → k H ↓ k pointwise.Next we prove that H ∈ T θ ∈ D A θ ( λ ). First, using Lemma 4.3 of [28], weobtain that ˜ H n, ↑ → H ↑ and ˜ H n, ↓ → H ↓ weakly. It follows that for every θ ∈ Θ , t ∈ [0 , T ], W t ( S θ , ˜ H n , λ ) → W t ( S θ , H, λ ) , a.s.. W T ( S θ , H, λ ) ≥ f θ , a.s. for all θ ∈ D . Secondly, for every θ ∈ D , thetriangle inequality and Theorem 1 of [42] again yield that W t ( S θ , ˜ H n , λ ) = − Z t (1 + λ ) S θu d ˜ H n, ↑ u + Z t (1 − λ ) S θu d ˜ H n, ↓ u + ˜ H nt S θt − λ | ˜ H nt | S θt ≥ − x θ , a.s. for 0 ≤ t ≤ T. In other words, H ∈ A θx θ ( λ ) for all θ ∈ D . Remark 3.4.
Fatou-closedness is studied in the quasi-sure approach by [37].In their topological setting, Fatou-closedness, denoted by (
F C ), does not imply“weak star” closedness, denoted by (
W C ). An additional condition, namely P -sensitivity, is required. And it is proved that for a convex and monotoneset, ( W C ) = (
F C ) + P -sensitivity. When the set of priors P is dominated, P -sensitivity is always satisfied, but this is not the case when P is non-dominated.In comparison, our Proposition 3.3 resembles the dominated case (although thelaws of S θ are not necessarily dominated), where the number of uncertain modelsis finite. Nevertheless, our “weak star” closedness, that is w ∗ -closedness, isproved in Proposition 3.5 below by using a different technique. Fatou-closednessis also discussed in the pathwise approach by [15] as a given property ratherthan a proved one.Recall C = (cid:8) ( W T ( H, λ ) − h ) : H ∈ A ( λ ) , h ∈ L (cid:9) \ Y θ ∈ Θ L ∞ ( F T , P ) . Now, we are able to prove
Proposition 3.5.
The convex cone C is w ∗ -closed.Proof. Let f α , α ∈ I be a net in C , i.e. f α ≤ W T ( H α ) , H α ∈ A ( λ ) , such that f α → f in the w ∗ topology for some f ∈ L ∞ . We need to prove that f ∈ C , thatis there exists H ∈ A ( λ ) such that for all θ ∈ Θ, W T ( S θ , H, λ ) ≥ f θ , a.s.. For each θ ∈ Θ, we define H θ = { H ∈ V : H ∈ A θ ( λ ) and W T ( S θ , H, λ ) ≥ f θ , a.s. } . The set H θ is clearly convex for each θ ∈ Θ. In addition, using Lemma 4.3 of[28], we can prove that H θ is closed in V . If we can prove \ θ ∈ Θ H θ = ∅ (7)then the proof is complete. First, we will prove that H D = \ θ ∈ D H θ = ∅ , (8)where D is an arbitrary finite subset of Θ.Proposition 3.3 implies that the set C ( D ) is Fatou-closed. Therefore, usingProposition 4.1, the set C ( D ) is closed in the w ∗ topology of Q θ ∈ D L ∞ ( F T , P ).8ince f θα → f θ in the w ∗ topology for each θ ∈ D , and we obtain that ( f θ ) θ ∈ D ∈ C ( D ), and thus, (8) holds true.Fix θ ∈ Θ arbitrarily. Since x θ = k f θ k < ∞ , Lemma 3.1 shows that theset {k H k T , H ∈ A θ x θ ( λ ) } is bounded in L ( F T , P ), and thus convexly compact,by Proposition 4.5. Since \ θ ∈ Θ H θ = \ D ∈ F in (Θ) H D ∪{ θ } , we conclude that (7) holds. The proof is complete. Corollary 3.6.
For every < λ < , θ ∈ Θ , A ∈ F T , P [ A ] > , the set C ( θ, λ, A ) is nonempty.Proof. Let us fix ¯ θ ∈ Θ arbitrarily. By Proposition 3.5, the convex set C is w ∗ -closed. The compact set 1 A ¯ θ and the closed convex set C are disjoint. Applyingthe Hahn-Banach theorem, there exists Q = ( Z θT ) θ ∈ Θ ∈ L θ ∈ Θ L ( F T , P ) suchthat sup f ∈ C Q ( f ) ≤ α < β ≤ Q (1 A ¯ θ ) . Since ∈ C , it follows that α ≥
0. Since C is a cone, we must have Q ( f ) ≤ , ∀ f ∈ C , (9)and as a consequence, Z θT ≥ , ∀ θ ∈ Θ. Note that E [ Z ¯ θT A ] > . For any stopping times σ ≤ τ ≤ T n and B ∈ F σ , the strategy H = ± B ] σ,τ ] belongs to A ( λ ). Therefore, from (9) we obtain Q (((1 − λ ) S τ − (1 + λ ) S σ )1 B ) ≤ , Q (((1 − λ ) S σ − (1 + λ ) S τ )1 B ) ≤ . (10)Define X τ = X θ ∈ Θ Z θτ (1 − λ ) S θτ , Y σ = X θ ∈ Θ Z θσ (1 + λ ) S θσ . where Z θt = E [ Z θT |F t ]. We compute that E [( X τ − Y σ )1 B ]= E " X θ ∈ Θ Z θτ (1 − λ ) S θτ − X θ ∈ Θ Z θσ (1 + λ ) S θσ ! B = E " X θ ∈ Θ E (cid:2) Z θT |F τ (cid:3) (1 − λ ) S θτ ! B − E " X θ ∈ Θ E (cid:2) Z θT |F σ (cid:3) (1 + λ ) S θσ ! B ≤ , by the tower law of conditional expectation and (10). Similarly, we obtain E [( X σ − Y τ )1 B ] ≤ . Using Lemma 3.8, there is a martingale M n such that on J , T n K ,(1 − λ ) X θ ∈ Θ Z θt S θt ≤ M nt ≤ (1 + λ ) X θ ∈ Θ Z θt S θt , a.s. M n +1 and M n coincide on J , T n K . Therefore, the local martingale M obtained by pastingthe processes M n , n ∈ N together satisfies(1 − λ t ) X θ ∈ Θ Z θt S θt ≤ M t ≤ (1 + λ t ) X θ ∈ Θ Z θt S θt , a.s., t ∈ [0 , T ] . In other words, we obtain an element in C (¯ θ, λ, A ). Remark 3.7.
One may ask: if Assumption 2.3 (ii) is relaxed but a robust nofree lunch condition is imposed, is the set C (or an appropriate enlargement ofit) w ∗ -closed? This would result in a robust version of FTAP but we do notknow yet how to handle this case. We recall Lemma 6.3 of [29].
Lemma 3.8.
Let ( X t ) t ∈ [0 ,T ] and ( Y t ) t ∈ [0 ,T ] be two c`adl`ag bounded processes.The following conditions are equivalent:(i) There exists a c`adl`ag martingale ( M t ) t ∈ [0 ,T ] such that X ≤ M ≤ Y, a.s. (ii) For all stopping times σ, τ such that ≤ σ ≤ τ ≤ T, a.s. , we have E [ X τ |F σ ] ≤ Y σ , and E [ Y τ |F σ ] ≥ X σ , a.s.. Let z and H ∈ A ( λ ) satisfy z + W T ( S θ , H, λ ) ≥ G θ , a.s., θ ∈ Θ . For any ( Z , M ) ∈ Z ( λ ), we have z + E "X θ ∈ Θ Z θT W T ( S θ , H, λ ) ≥ E "X θ ∈ Θ Z θT G θ . (11)Suppose that Z θT has positive values only if θ ∈ D for some D ∈ F in (Θ).Denote by H ↑ ,c , H ↓ ,c the continuous parts of H ↑ , H ↓ , respectively. We definethe process I θt := R t (1 + λ ) S θu dH ↑ u . Using integration by parts, we obtain that d (cid:0) Z θt I θt (cid:1) = I θt − dZ θt + Z θt − (1 + λ ) S θt dH ↑ t + d [ Z θ , I θ ] t = I θt − dZ θt + Z θt − (1 + λ ) S θt dH ↑ ,ct + Z θt − (1 + λ ) S θt ∆ H ↑ t + ∆ Z θt ∆ I θt = I θt − dZ θt + Z θt − (1 + λ ) S θt − dH ↑ ,ct + Z θt (1 + λ ) S θt ∆ H ↑ t , (12)noting that S θ is continuous and I θ is of finite variation. Similarly, we define J θt = R t (1 − λ ) S θu dH ↓ u and compute d (cid:0) Z θt J θt (cid:1) = J θt − dZ θt + Z θt − (1 − λ ) S θt − dH ↓ ,ct + Z θt (1 − λ ) S θt ∆ H ↓ t . (13)10herefore, (12), (13) and the property of M yield X θ ∈ D Z θt W t ( S θ , H, λ ) ≤ X θ ∈ D Z t ( J θu − − I θu − ) dZ θu − Z t M u − dH cu + X u ≤ s ≤ t M u ∆ H u + H t M t = X θ ∈ D Z t ( J θu − − I θu − ) dZ θu + Z t H u − dM u . The RHS of the above inequality is a local martingale and thus a supermartin-gale as W t ( S θ , H, λ ) is uniformly bounded from below. From (11), we have E "X θ ∈ D Z θT G θ ≤ z + E "X θ ∈ D Z θT W T ( S θ , H, λ ) = z. Therefore, z ≥ sup ( Z ,M ) ∈Z ( λ ) E (cid:2)P θ ∈ Θ Z θT G θ (cid:3) . Next, we prove the reverse inequality for the case G θ , θ ∈ Θ are bounded.Let z ∈ R be such that there is no strategy H ∈ A ( λ ) satisfying W zT ( S θ , H, λ ) ≥ G θ , a.s., ∀ θ ∈ Θ . In other words, ( G θ ) θ ∈ Θ − z / ∈ C . Applying the Hahn-Banach theorem, thereexists Q = ( Z θT ) θ ∈ Θ ∈ L θ ∈ Θ L ( F T , P ) such thatsup f ∈ C Q ( f ) ≤ α < β ≤ Q (cid:0) ( G θ ) θ ∈ Θ − z (cid:1) . Since C is a cone containing − L ∞ + , it is necessarily thatsup f ∈ C Q ( f ) = 0 , Q (cid:0) ( G θ ) θ ∈ Θ − z (cid:1) > . We also deduce that Z θT ≥ , a.s., θ ∈ Θ and it is possible to normalize Q suchthat Q ( ) = 1. A similar argument as in the proof of Corollary 3.6 gives themartingale M associated to Z . This means ( Z , M ) ∈ Z ( λ ) and that z < Q (cid:0) ( G θ ) θ ∈ Θ (cid:1) ≤ sup ( Z ,M ) ∈Z ( λ ) E "X θ ∈ Θ Z θT G θ . Finally we investigate the case G θ ≥ , θ ∈ Θ. Let z ∈ R be the numbersuch that z ≥ sup ( Z ,M ) ∈Z ( λ ) E (cid:2)P θ ∈ Θ Z θT G θ ) (cid:3) . Then, for all n ∈ N , we have z ≥ sup ( Z ,M ) ∈Z ( λ ) E "X θ ∈ Θ Z θT G θ ∧ n . The result for bounded G implies for each n ∈ N , there exists H n ∈ A ( λ ) suchthat z + W T ( S θ , H n , λ ) ≥ G θ ∧ n, a.s., ∀ θ ∈ Θ . θ ∈ Θ, Theorem 1 of [42] yields H n ∈ A θz ( λ ) for all n ∈ N . Now, werepeat the argument in Proposition 3.3. For a fixed θ ∈ Θ, the set {k H k T , H ∈A θ z ( λ ) } is bounded in probability by Lemma 3.3. Using Lemma B.4 of [29],there are convex combinations ˜ H n, ↑ , ˜ H n, ↓ and finite variation processes H ↑ , H ↓ such that ˜ H n, ↑ → H ↑ , ˜ H n, ↓ → H ↓ , and k ˜ H n, ↑ k → k H ↑ k , k ˜ H n, ↓ k → k H ↓ k pointwise. From Lemma 4.3 of [28], we obtain ˜ H n, ↑ → H ↑ and ˜ H n, ↓ → H ↓ weakly. It follows that for every θ ∈ Θ , t ∈ [0 , T ], W t ( S θ , ˜ H n , λ ) → W t ( S θ , H, λ ) , a.s.. Since G is bounded from below, H ∈ A ( λ ) and z + W T ( S θ , H, λ ) ≥ G θ , a.s., θ ∈ Θ . The proof is complete.
A bilinear pairing is a triple (
X, Y, h· , ·i ) where X, Y are vector spaces over R and h· , ·i is a bilinear map from X × Y to R . Let ( E, u ) be a topological vector space.Let E ∗ = ( E, u ) ∗ be its dual, i.e., the set of all continuous linear maps from E to R . Then there is a natural bilinear pairing ( E, E ∗ , h· , ·i ). We denote by σ ( E, E ∗ ) the usual weak topology on E and by σ ( E ∗ , E ) the weak-star topologyon E ∗ .Let I be a non-empty set and, for each i ∈ I , let ( X i , τ i ) be a locally convextopological spaces. The topological direct sum of the family ( X i , τ i ), denotedby L i ∈ I ( X i , τ i ), is the locally convex space defined as follows. The vector space L i ∈ I X i is the set of tuples ( x i ) i ∈ I with x i ∈ X i such that x i = 0 for all butfinitely many i . It is equipped with the inductive topology with respect to thecanonical embeddings e i : ( X i , τ i ) → Xx i x = ( x i ) , where x i = x i and x j = 0 whenever j = i , i.e. the strongest locally convextopology on L i ∈ I X i such that all these embeddings are continuous.The product space of of the family ( X i , τ i ), denoted by Q i ∈ I ( X i , τ i ), consistsof the product set Q i ∈ I X i and a topology τ having as its basis the family (Y i ∈ I O i : O i ∈ τ i and O i = X i for all but a finite number of i ) . The topology τ is called the product topology, which is the coarsest topology forwhich all the projections are continuous. Note that the product space defined inthis way is also a topological vector space, see Theorem 5.2 of [11]. Since each( X i , τ i ) is locally convex, Q i ∈ I ( X i , τ i ) is locally convex, too, see Proposition2.1.3 of [5]. If I is uncountable, the product space is not normable.For any index set I , it holds that M i ∈ I ( X i , τ i ) ! ∗ = Y i ∈ I X ∗ i , Y i ∈ I ( X i , τ i ) ! ∗ = M i ∈ I X ∗ i , (14)12ee Corollary 1, page 138 and Theorem 4.3, page 137 of [43].We will be using the pairing h f , g i = X i ∈ I (cid:10) f i , g i (cid:11) i , ∀ f ∈ M i ∈ I X i , g ∈ Y i ∈ I X ∗ i , where h· , ·i i is the natural pairing for X i , X ∗ i . From Corollary 1, page 138 of[43], it holds that σ Y i ∈ I X ∗ i , M i ∈ I X i ! = Y i ∈ I σ ( X ∗ i , X i ) . (15)Let D be a finite index set. In what follows, we will be interested in theduality between E := M γ ∈ D ( L ( F T , P ) , k · k ) , E ∗ = Y γ ∈ D L ∞ ( F T , P ) . (16)We define B ∞ r = { f ∈ L ∞ ( F T , P ) : k f k ∞ ≤ r } , the closed ball of radius r ≥ L ∞ ( F T , P ). The following result is analogous to Proposition 5.2.4 of [19]. Proposition 4.1.
Let C ⊂ E ∗ be a convex set, where E ∗ is defined in (16).The set C is closed in the w ∗ topology if and only if C ∩ Q γ ∈ D B ∞ r is closed in Q γ ∈ D L ( F T , P ) for each r ≥ .Proof. We follow the proof of Proposition 4.4 in [35]. ( ⇒ ) Let f n , n ∈ N be asequence in C ∩ Q γ ∈ Γ B ∞ r such that f n → f in Q γ ∈ D L ( F T , P ). We have toshow that f ∈ C ∩ Q γ ∈ Γ B ∞ r . For each g ∈ L ( F T , P ) and γ ∈ D , the dominatedconvergence theorem implies that lim n →∞ E [ gf γn ] = E [ gf γ ]. Therefore we havethat f ∈ C ∩ Q γ ∈ Γ B ∞ r . ( ⇐ ) Assume that C ∩ Q γ ∈ D B ∞ r is closed in Q γ ∈ D L ( F T , P ). It is easyto check that C ∩ Q γ ∈ D B ∞ r is closed in the Hilbert space Q γ ∈ D ( L , k · k )hence also in its weak topology σ ( Q γ ∈ D L , Q γ ∈ D L ), which is the same as σ ( Q γ ∈ D L ∞ , Q γ ∈ D L )-closedness. Since L ⊂ L , the set C ∩ Q γ ∈ D B ∞ r is also σ ( Q γ ∈ D L ∞ , Q γ ∈ D L )-closed so, by the Krein-Smulian theorem, C is closed inthe weak-star topology. ( L ) N Let L § denote the set of [0 , ∞ ]-valued random variables, equipped with the topol-ogy of convergence in probability. A set A ⊂ L is bounded if sup X ∈ A P ( X ≥ n ) → n → ∞ . Now consider the topological product L := ( L ) N . We call asubset C ⊂ L c-bounded , if π k ( C ) is bounded in L for all coordinate mappings π k , k ∈ N .For any set A we denote by Fin( A ) the family of all non-empty finite subsetsof A . This is a directed set with respect to the partial order induced by inclusion.We reproduce Definition 2.1 of [45]. Definition 4.2.
A convex subset C of some topological vector space is convexlycompact , if for any non-empty set A and any family F a , a ∈ A of closed andconvex subsets of C , one has ∩ a ∈ A F a = ∅ whenever ∀ B ∈ Fin( A ) , ∩ a ∈ B F a = ∅ .
13t was established, independently in both [39] and [45], that every closed andbounded convex subset of L is convexly compact. In this section we will showthe following. Proposition 4.3.
Any c-bounded, convex and closed subset C ⊂ L is convexlycompact. For an element f ∈ L we write f k := π k ( f ), k ∈ N . Let I be a directedset and f i , i ∈ I a net in L . For each i ∈ I , let Γ i denote the set of (finite)convex combinations of the elements { f j : j ≥ i } . The next lemma goes backto Lemma A1.1 of [18]. Its proof closely follows that of Lemma 2.1 in [39], seealso Theorem 3.1 in [45]. Lemma 4.4.
There exist g i ∈ Γ i , i ∈ I such that the nets g ki , i ∈ I converge to g k in probability, for each k ∈ N , where g k ∈ L § .Proof. Set u ( x ) := 1 − e − x , x ∈ [0 , ∞ ] and note that, for given α >
0, there is β > u (cid:18) x + y (cid:19) ≥ u ( x ) + u ( y )2 + β, when | x − y | ≥ α and min( x, y ) ≤ /α. (17)Set, for i ∈ I , s i = sup ( ∞ X k =0 − k Eu ( g k ) : g ∈ Γ i ) . As s i , i ∈ I is a non-increasing net of numbers, it converges to s ∞ := inf i ∈ I s i .Choose a non-decreasing i m , m ∈ N such that s ∞ = lim m →∞ s i m and, for each m , | s ∞ − s i m | ≤ m + 1 (18)and let g i m be such that P ∞ k =0 − k Eu ( g ki m ) ≥ s i m − / ( m + 1).For elements p ∈ I \ { i m , m ∈ N } , there is l = l ( p ) such that s i l +1 ≤ s p ≤ s i l and choose g p ∈ Γ p such that P ∞ k =0 − k Eu ( g kp ) ≥ s i l +1 − / ( l + 1).In order to prove that g ki , i ∈ I is a Cauchy net in L § for each k , we need toestablish that, for each k and each α, ǫ >
0, there is i ( ǫ ) such that, for p, q ≥ i ( ǫ ), P (cid:0) | g kp − g kq | ≥ α, min( g kp , g kq ) ≤ /α (cid:1) ≤ ǫ. Let p, q ≥ i m . Notice that ( g p + g q ) / ∈ Γ i m so ∞ X k =0 − k Eu g kp + g kq ! ≤ s i m . However, by construction, l ( p ) ≥ m so ∞ X k =0 − k Eu (cid:0) g kp (cid:1) ≥ s i l ( p )+1 − l ( p ) + 1 ≥ s i m − m + 1 − ( s i m − s i l ( p )+1 ) , and the latter is ≥ s i m − m +1 , by (18). A similar estimate holds for P ∞ k =0 − k Eu (cid:0) g kq (cid:1) hence we conclude, from (17), that β ∞ X k =0 − k P (cid:0) | g kp − g kq | ≥ α, min( g kp , g kq ) ≤ /α (cid:1) ≤ s i m − s i m m + 1 − s i m m + 1 ≤ m + 1 . k ∈ N , P (cid:0) | g kp − g kq | ≥ α, min( g kp , g kq ) ≤ /α (cid:1) ≤ k +1 β ( m + 1) , which can be made arbitrarily small if m is large enough. The statement isproved. Proof of Proposition 4.3.
Let A be an arbitrary index set and let C a , a ∈ A be closed, convex subsets of C . Assume that for each a ∈ Fin( A ) with a = { a , . . . , a K } we have C a ∩ . . . ∩ C a K = ∅ . Let us pick an element c ( a ) fromthis intersection. Apply Lemma 4.4 to the net c ( a ), a ∈ Fin( A ) to obtainconvex combinations g ( a ) ∈ C a ∩ . . . ∩ C a K such that the net g k ( a ), a ∈ Fin( A )converges to some g k ∈ L § in probability, for each k ∈ N .As C is c-bounded, g k ( a ), a ∈ A are bounded in L , so, actually, g k ∈ L .It follows that g ∈ L and, by the definition of topological products, the net g ( a ), a ∈ Fin( A ) converges to g . However, for each fixed a ∈ A , g ( a ) ∈ C a for each a ∈ Fin( A ) containing a , hence, by the closedness of C a , we also have g ∈ C a .It follows that ∩ a ∈ A C a = ∅ since g is in this intersection. Let V denote the family of non-decreasing, left-continuous functions on [0 , T ]which are 0 at 0. Let r k , k ∈ N be an enumeration of ( Q ∩ [0 , T ]) ∪ { T } with r = T . For f, g ∈ V , define ρ ( f, g ) := ∞ X k =0 − k | f ( r k ) − g ( r k ) | . The series converges since | f ( r k ) − g ( r k ) | ≤ f ( T )+ g ( T ) for each k , and it definesa metric. The corresponding Borel-field is denoted by G .Let V denote the set of pairs H = ( H ↑ , H ↓ ) where H ↑ t , H ↓ t , t ∈ [0 , T ] arepredictable processes such that H ↑ ( ω ) , H ↓ ( ω ) ∈ V for each ω ∈ Ω. Consideredas mappings H ↑ , H ↓ : (Ω , F ) → ( V , G ), they are measurable, by the definitionof the metric ρ . We identify elements of V when they coincide outside a P -nullset. We equip V with the topology coming from the metric ̺ ( H, G ) := E [ ρ ( H ↑ , G ↑ ) ∧
1] + E [ ρ ( H ↓ , G ↓ ) ∧ . Although this metric wasn’t defined, a related convergence structure was intro-duced already in [13].Similarly to Proposition 4.3, we obtain the following convex compactnessresult for subsets of V . Proposition 4.5.
Let C be a convex and closed subset of V . If { H ↑ T + H ↓ T , H ∈ C } is bounded in L then C is convexly compact.Proof. Let A be an arbitrary index set and let C a , a ∈ A be closed, convexsubsets of C . Assume that for any a ∈ Fin( A ) with a = { a , . . . , a K } we15ave C a ∩ . . . C a K = ∅ . We prove that ∩ a ∈ A C a = ∅ . We identify L with( L ) ( Q ∩ [0 ,T ]) ∪{ T } .Denote D = (cid:8) ( H ↑ q , H ↓ q ) q ∈ ( Q ∩ [0 ,T ]) ∪{ T } , H ∈ C (cid:9) ⊂ L and similarly D a = (cid:8) ( H ↑ q , H ↓ q ) q ∈ ( Q ∩ [0 ,T ]) ∪{ T } , H ∈ C a (cid:9) ⊂ D. Clearly, D a , D are convex and closed in L . By hypothesis, the set D is c -bounded. Therefore, Proposition 4.3 implies there exists ( ¯ H ↑ q , ¯ H ↓ q ) q ∈ ( Q ∩ [0 ,T ]) ∪{ T } in ∩ a ∈ A D a = ∅ . Define, for t ∈ [0 , T ] \ Q , H ↑ t = lim q ↑ t,q ∈ Q H ↑ q , H ↓ t = lim q ↑ t,q ∈ Q H ↓ q . Then ( ¯ H ↑ , ¯ H ↓ ) ∈ ∩ a ∈ A C a as required. Corollary 4.6.
Let C be a convex and closed subset of V . If { H ↑ T + H ↓ T , H ∈ C } is bounded in probability then, for each a > , [ − a, a ] × C is convexly compact(as a subset of R × V ). ✷ Proof.
Follows by obvious modifications of the arguments of Subsection 4.2 andProposition 4.5.
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