Robust fundamental theorems of asset pricing in discrete time
aa r X i v : . [ q -f i n . M F ] A ug Robust fundamental theorems of asset pricing indiscrete time ∗ Huy N. ChauAugust 24, 2020
Abstract
This paper is devoted to the study of robust fundamental theorems ofasset pricing in discrete time and finite horizon settings. The new concept“robust pricing system” is introduced to rule out the existence of modelindependent arbitrage opportunities. Superhedging duality and strategyare obtained.
The equivalence between the absence of arbitrage opportunities and the exis-tence of a martingale measure, or the fundamental theorem of asset pricing(FTAP in short), is a core topic to mathematical finance. FTAP results arediscussed in classical models under the assumption that the dynamics of riskyassets are known precisely, see [24], [14], [16], [17]. Nonetheless, this assumptionhas been constantly suspected and it becomes clear that model uncertainty, i.e.the risk of using wrong models, cannot be ignored in practice. Since the seminalwork of Knight (1921) [23], uncertainty modeling has emerged as effective toolsto address this issue.The pathwise approach, pioneered by [19], makes no assumptions aboutthe dynamics of the underlying assets. Instead, the set of all models whichare consistent with the prices of some observed vanilla options is investigated,see also [8], [13], [15], [10], [9]. Pathwise versions of FTAP are given in [32]where one-period market models are considered and in [1] where a superlinearlygrowing option is traded among others.Theory of quasi-sure approach, starting with [29], [33], assumes that thereis a set of priors P instead of a specific reference probability measure. Fornondominated sets of priors, [7] proves a version of FTAP in discrete time, andobtains a family Q of martingale measures such that each P ∈ P is dominatedby a martingale measure in Q which may be nonequivalent to P . Recently, [6]improves that there is a subset P ⊂ P such that each model P ∈ P satisfiesthe classical no arbitrage condition. In a continuous-time setting, [4] shows thatfor each P ∈ P there is an equivalent martingale measure. Under technical ∗ This work is supported by Center for Mathematical Modeling and Data Science, OsakaUniversity. We thank Masaaki Fukasawa, Mikl´os R´asonyi and Martin Schweizer for construc-tive comments on the earlier version of the paper. S θ ,see [2], [5], [26], and recently in [31], [12]. Although this approach has beenemployed for a long time, analogous FTAP results have not been obtained yet.In this work, we attempt to prove such fundamental results. We summarize ourapproach below.The parametrization approach stipulates different dynamics for stock priceson the same probability space instead of a family of laws on the canonicalspace as in the quasi sure approach, reminiscing about the discrepancy betweenthe notions of strong and weak solutions of stochastic differential equation.By relying on the canonical space, randomness in the quasi sure approach isgenerated by the canonical process and less important as the family of laws.In contrast, the parametrization approach may incorporate different sources ofrandomness to each price process and thus it would be flexible enough for othermodeling purposes, see also [12] for other aspects from utility optimization.The first difficulty is that for a given strategy H there is a family of possibleattainable payoffs { H · S θT , θ ∈ Θ } . In classical settings without uncertainty,it is proposed to consider the family of attainable payoffs as a subset of some L p spaces where separation theorems are applied. However, this argument isdesigned for the case when one strategy generates one payoff, and it is not ableto capture such family under uncertainty. We show that the correct functionspace is the product space L = Q θ ∈ Θ L p , which seems to be huge, since Θ isusually an uncountable set. This is the starting point of the present paper.In general, the product space L is not a Banach space, unless | Θ | is finite,and even not a metric space, unless | Θ | is countable. Although having poorstructures, the product space L is typically locally convex and therefore it isenough to apply the Hahn-Banach theorem. If one wishes to use separationarguments on that product space, the closedness of the set of hedgeable claimsneeds to be obtained first. There are crucial differences between sequentialclosedness and topological closedness in L and hence, nets must be used, insteadof sequences, to determine such closures. Continuing this way, we obtain newpricing systems which play the same roles as martingale measures in classicalsettings. The novelty is that the new pricing systems average jointly scenariosand models to rule out model independent arbitrage opportunities.Compared to [7], our approach is a natural extension of the classical frame-work. It is not necessary to assume Ω to be a Polish space and therefore we areable to reach the most generality introduced in literature. Furthermore, we donot work in a “local” fashion where heavy tools from the theory of analytic setsand measurable selections are applied to glue solutions together.The paper is organized as follows. Section 2 introduces the setting. Mainresults are given in Section 3. Some preliminaries and useful results are givenin Section 4. Notations . In the product space X = Q i ∈ I X i , a vector ( f i ) i ∈ I will bedenoted by f . We write f ≥ g if f i ≥ g i for all i ∈ I . In addition, denotes thevector with all coordinates equal to 1 and i denotes the vector with only thecoordinate i equals to 1 and the others are zero.2 The model
Let (Ω , F , ( F t ) t =0 , ,...,T , P ) be a filtered probability space where T ∈ N . Wesuppose that F contains all P -zero sets. Let B ≡ S = ( S , ..., S d ) , d ≥ , be nonnegative risky assets and S t is F t -measurablefor 0 ≤ t ≤ T , S = 1. The increment of a process Y is denoted by ∆ Y t := Y t − Y t − , t = 1 , ..., T. Uncertainty is modeled by parameter θ ∈ Θ. In order toinclude interesting robust models, it is not assumed that Θ is countable.
Assumption 2.1.
The following conditions are imposed throughout the paper:(i) Θ is a closed subset of a separable metric space,(ii) For each ≤ t ≤ T , it holds that lim θ n → θ S θ n t = S θt , a.s. for any sequence ( θ n ) n ∈ N such that θ n → θ in Θ . The space L is equipped with the topology of convergence in probability,induced by the translation-invariant metric d ( f, g ) := E [1 ∧ | f − g | ]. With thisstructure, L is a Fr´echet space, i.e. a locally convex space that is complete.Define the product space L ( F T , P ) := Q Θ L ( F T , P ) with the correspondingproduct topology. For t = 0 , ..., T , define by S t = ( S θt ) θ ∈ Θ a vector in L ( F t , P ).Let A be the set of all predictable processes H t ∈ L ( F t − , P ) , t = 1 , ..., T , i.e.trading strategies. For H ∈ A , we denote H · S t = t X s =1 H s ∆ S s , t = 1 , ..., T. Definition 2.2.
We say that the market satisfies the condition No Robust Ar-bitrage ( N RA ) if for every self-financing strategy H ∈ A , if ∀ θ ∈ Θ , H · S θT ≥ , a.s. then ∀ θ ∈ Θ , H · S θT = 0 , a.s.. (1)The property (1) is rewritten shortly as H · S T ≥ , a.s. then H · S T = 0 , a.s., and it should be noticed that the inequality and equality are θ -wise, see alsoRemark 2.6. Let K = (cid:8) H · S T ∈ L ( F T , P ) , H ∈ A (cid:9) , C = K − L ( F T , P ) , be the set of all attainable and hedgeable payoffs, respectively. With thesenotations, N RA can be formulated as follows K ∩ L ( F T , P ) = { } or equivalently C ∩ L ( F T , P ) = { } . It is easy to see that
N RA reduces to the classical no arbitrage property whenΘ is a singleton.
Remark 2.3.
This robust framework is different from the usual setting withmultiple assets. In a financial market with | Θ | underlying assets, a strategy attime t is a vector ( H θt ) θ ∈ Θ ∈ Q θ ∈ Θ L ( F t − , P ) . In our robust setting, one strategy H t ∈ L ( F t − , P ) is used for all price processes. The set of strategiesfor the robust setting is much smaller, and as a result, the dual space is muchlarger. The discrepancy becomes significant when Θ is uncountable. emark 2.4. Our setting is also different from “large financial” market mod-els, where there are a continuum of securities. For example, in bond markets,zero-coupon bonds are parametrized by their maturities θ which is a continuousparameter. However, only a finite number of bonds are traded at the same time,see [22], [30], [3]. Remark 2.5.
By postulating an appropriate weak topology σ θ on each L p , wedefine the corresponding weak topology on the product space L p = Q θ ∈ Θ ( L p , σ θ ) .Therefore, it is possible to introduce no robust “free lunch” conditions on theproduct space, which seems very useful in continuous time settings. Remark 2.6.
It is not convenient to consider a product measure on the productspace, since a common ω is used for ( S θT ( ω )) θ ∈ Θ instead of ( S θT ( ω θ )) θ ∈ Θ , whichis a different object. In this subsection, we follow the predictable range approach, given in [17]. Theidea is to eliminate redundant strategies.
Lemma 3.1.
Let (Ω , F , P ) be a probability space and E ⊂ L ( F , P ) a subspaceclosed with respect to convergence in probability. We assume that E satisfies thefollowing stability property. If f, g ∈ E and A ∈ F , then f A + g A c ∈ E . Underthese assumptions, there exists an F -measurable mapping P taking value in theorthogonal projection in R d such that f ∈ E if and only if P f = f .Proof. See Lemma 6.2.1 of [17].We define the following closed subspaces of L ( F t − , P ) , ≤ t ≤ T, E θt = { h ∈ L ( F t − , P ) : h ∆ S θt = 0 , a.s. } and E Γ t = T θ ∈ Γ E θt for Γ ⊂ Θ. Each E θt satisfies the assumptions in Lemma 3.1and so E Γ t does. Note that 0 ∈ E Γ t . By Assumption 2.1 (ii), if Γ is a dense subsetof Θ then E Γ t = E Θ t and H Γ t = H Θ t . By Lemma 3.1, E Γ t can be described by amapping P Γ t . We define P Γ ,ct = Id − P Γ t and H Γ t = { f : Ω → R d : f is F t − -measurable and P Γ ,ct f = f } . We say that H ∈ A is in Γ-canonical form if H t ∈ H Γ t , ≤ t ≤ T . And if Γis dense in Θ, we simply say H is in canonical form. The following concernboundedness and convergence results for one step model. Lemma 3.2.
Let Γ be a dense subset of Θ . Let ( H n ) n ∈ N ∈ H t be a sequence incanonical form. It holds that(i) ( H n ) n ∈ N is a.s. bounded if and only if for all θ ∈ Γ , ( H n ∆ S θt ) n ∈ N is a.s.bounded.(i’) Assume in addition that N RA holds. Then ( H n ) n ∈ N is a.s. bounded ifand only if for all θ ∈ Γ , ( H n ∆ S θt ) − n ∈ N is a.s. bounded. ii) ( H n ) n ∈ N converges a.s. if and only if for all θ ∈ Γ , ( H n ∆ S θt ) n ∈ N does.Proof. The “only if” directions are obvious. It suffices to prove the “if” direc-tions.( i ) : Assume that for each θ ∈ Γ, ( H n ∆ S θt ) n ∈ N is a.s. bounded. We prove( H n ) n ∈ N is a.s. bounded, too. If this is not the case, by Proposition 6.3.4 (i) of[17], there is a measurably parameterised subsequence ( L k ) k ∈ N = ( H τ k ) k ∈ N suchthat L k diverges to ∞ on a set B of positive measure. Note that L k , k ∈ N arein canonical form. Let b L k = L k k L k k B ∩k L k k≥ . By passing to another measurablyparameterised subsequence we may assume that b L k → b L , which is of canonicalform and satisfies b L = 1 on B . By assumption, it holds that b L k ∆ S θt → , a.s. forall θ ∈ Γ. Consequently, b L ∆ S θt = 0 , a.s. for all θ ∈ Γ, and thus for all θ ∈ Θ, byAssumption 2.1 (ii), which means that b L ∈ E Θ t . Therefore, b L ∈ E Θ t ∩ H Θ t = { } ,which is a contradiction.( i ′ ) : We proceed as in ( i ), noting that( b L k ∆ S θt ) − = lim k →∞ ( b L k ∆ S θt ) − = 0 , a.s., ∀ θ ∈ Γand thus for all θ ∈ Θ, by Assumption 2.1 (ii). By
N RA , we get that b L ∆ S θt =0 , a.s. for all θ ∈ Θ, which again implies b L = 0, a contradiction.( ii ) : We also prove by contradiction. Assume that ( H n ) n ∈ N does not con-verge a.s.. By ( i ), we may assume that ( H n ) n ∈ N is a.s. bounded. Proposition6.3.3 of [17] implies there is a measurably parameterised subsequence ( H τ k ) k ∈ N converging to H , a.s.. Applying Proposition 6.3.4 (ii) of [17] with f = H ,there is another measurably parameterised subsequence ( H σ k ) k ∈ N converging to b H , a.s. for which P [ H = b H ] >
0. Note also that H , b H are in canonicalform. We have( H − b H )∆ S θt = lim k →∞ H τ k ∆ S θt − lim k →∞ H σ k ∆ S θt = 0 , a.s., ∀ θ ∈ Γ , and hence for all θ ∈ Θ, by Assumption 2.1 (ii), which implies a contradiction.We extend Stricker’s lemma in our setting, noting that the condition
N RA is not used here.
Proposition 3.3.
Let Γ ⊂ Θ be a countable index set. The vector space K Γ = ( T X t =1 H t ∆ S θt ! θ ∈ Γ , H ∈ A ) is closed in Q θ ∈ Γ L ( F T , P ) .Proof. The case T = 1: we use Lemma 4.1 and then Lemma 3.2 (ii). Let ussuppose that assertion holds true for T −
1, and fix the horizon T . By theinductive hypothesis, the set K Γ2 = ( T X t =2 H t ∆ S θt ! θ ∈ Γ , H ∈ A )
5s closed in Q θ ∈ Γ L ( F T , P ).Let H be the set of strategies in canonical form defined as before. Let I be the linear mapping H → Y θ ∈ Γ L ( F T , P ) H ( H ∆ S θ ) θ ∈ Γ . Note that I is continuous and injective. Let F = ( I ) − ( K Γ2 ∩ I ( H )). Since K Γ2 is closed, the set F is a closed subspace of H . It is easy to see that F is stable in the sense of Lemma 3.1. Consequently, there is a F -measurablemapping P so that f ∈ F if and only if P f = f . Define E = { H ∈ H : P H = 0 } . (2)We deduce that elements H in E are in canonical form and the integrals( H ∆ S θ ) θ ∈ Γ are not in K Γ2 . Furthermore, K Γ = ( T X t =1 H t ∆ S θt ! θ ∈ Γ , H ∈ A , H ∈ E ) and the decomposition of elements f ∈ K Γ into f = ( H ∆ S θ ) θ ∈ Γ + f , H ∈ E , f ∈ K Γ2 is unique.Let f n = ( H n, ∆ S θ ) θ ∈ Γ + f ,n , n ∈ N be a sequence in K Γ with H n, ∈ E , f ,n ∈ K Γ2 such that f n → f in Q θ ∈ Γ L ( F T , P ). We prove that f ∈ K Γ . ByLemma 4.1, we find a subsequence, still denoted by n , such that f θn → f θ , a.s. for all θ ∈ Γ. First we will show that ( H n, ) n ∈ N is a.s. bounded. Let A = { ω : lim sup n →∞ k H n, k = ∞} . By Proposition 6.3.4 of [17], there is an F -measurably parameterised subsequence ( τ k ) k ∈ N such that H τ k , → ∞ on A . If P [ A ] >
0, we apply Proposition 6.3.3 of [17] and assume that H τk, k H τk, k → ψ , a.s. on the set A , where ψ = ψ A ∈ E , since E is closed and stable in the senseof Lemma 3.1. Clearly k ψ k = 1 on A . We have that for every θ ∈ Γ, (cid:18) H τ k , k H τ k , k ∆ S θ (cid:19) A + f θ ,τ k k H τ k , k A → , a.s. It follows that f θ ,τk k H τk, k A → − A ψ ∆ S θ . By the closedness of K Γ2 , we obtain( − A ψ ∆ S θ ) θ ∈ Γ ∈ K Γ2 . Since ψ ∈ E , this implies that ψ ∆ S θ = 0 for all θ ∈ Θ and hence ψ = 0. This is a contradiction to k ψ k = 1 on A with P [ A > H n, ) n ∈ N is bounded a.s. and there is an F -measurably parameterisedsequence H τ k , converging a.s. to H . This means that f ,τ k → f − ( H ∆ S θ ) θ ∈ Γ .By the closedness of K Γ2 , we get f − ( H ∆ S θ ) θ ∈ Γ ∈ K Γ2 . The proof is complete.Next, we prove a crucial boundedness property by extending Lemma 3.2 (i’). Proposition 3.4.
Let
N RA hold and Γ be a dense subset of Θ . Let ( H n ) n ∈ N be a sequence of strategies in canonical form such that ( H n · S θT ) − is boundeda.s. for every θ ∈ Γ . Then ( H n ) n ∈ N = ( H n, , ..., H n,T ) n ∈ N is bounded a.s.. roof. We use induction on T . For T = 1, we refer to Lemma 3.2 (i’). Let usassume that the conclusion holds true for T − T . We willprove that ( H n, ) n ∈ N is a.s. bounded. As in the proof of Proposition 3.3, we mayfurther assume that H n, ∈ E , see (2). Let A = { ω : lim sup n →∞ k H n, ( ω ) k = ∞} and let ( τ n ) n ∈ N be an F -measurably parameterized subsequence such that k H τ n , k → ∞ on A and H τ n, k H τ n, k → ψ on A . On A , k ψ k = 1. Put ψ = 0 on A c . Then ψ ∈ E is in canonical form. We compute for each θ ∈ Γ (cid:18) H τ n k H τ n , k · S θ (cid:19) T = H τ n , k H τ n , k ∆ S θ + T X t =2 H τ n ,t k H τ n , k ∆ S θt ! . The first term on the RHS of the above equality is bounded by k ∆ S θ k , whichimplies thatlim sup n →∞ T X t =2 H τ n ,t k H τ n , k ∆ S θt ! − ≤ k ∆ S θ k + lim sup n →∞ (cid:18) H τ n k H τ n , k · S θ (cid:19) − T ≤ k ∆ S θ k . By the induction hypothesis, we have that the sequence˜ H n = (cid:18) , A H τ n , k H τ n , k , ..., A H τ n ,T k H τ n , k (cid:19) is a.s. bounded. Applying Proposition 6.3.3 of [17] to the one-point compactifi-cation K = R d ∪ {∞} , there is an F -measurably parameterized subsequence τ n of τ n such that ˜ H τ n , → H , a.s. , and H is F -measurable. Repeating this argu-ment, we find an F T − -measurably parameterized subsequence τ Tn of τ T − n suchthat ˜ H τ Tn ,T → H T , a.s. , and H T is F T − -measurable. The sequence ˜ H τ Tn , whichmaybe not a sequence of predictable strategies, converges a.s. to the predictablestrategy H = (0 , H , ..., H T ). Hence, ( ˜ H τ Tn · S θ ) T → ( H · S θ ) T , a.s., ∀ θ ∈ Θ.Consequently, ψ ∆ S θ + ( H · S θ ) T = lim n →∞ H τ Tn , k H τ Tn , k ∆ S θ + ˜ H τ Tn · S θT = lim n →∞ A k H τ Tn , k ( H τ Tn · S θ ) T Since ( H n · S θ ) − T is a.s. bounded for every θ ∈ Γ, we obtain that ψ ∆ S θ +( H · S θ ) T ≥ θ ∈ Γ and thus by Assumption 2.1 (ii) and the
N RA condition, we have ∀ θ ∈ Θ ,ψ ∆ S θ + ( H · S θ ) T = 0 , a.s.. Thus, ψ = 0. As a result, P [ A ] = 0 and hence the sequence ( H n, ) n ∈ N is a.s.bounded. Since T X t =2 H n,t ∆ S θt ! − ≤ ( H n · S θ ) − T + k H n, ∆ S θ k , ∀ θ ∈ Γand hence is it is a.s. bounded. The inductive hypothesis shows that thesequence ( H n, , ..., H n,T ) n ∈ N is a.s. bounded.7 .2 The closedness of C First, we prove the following.
Lemma 3.5.
For any countable and dense subset Γ ⊂ Θ , C Γ = (cid:8) ( H · S θT − h θ ) θ ∈ Γ , H ∈ A , h θ ∈ L ( F T , P ) (cid:9) is closed in Q θ ∈ Γ L ( F T , P ) .Proof. We adopt the standard argument in [21]. Since L ( F T , P ) is a metricspace, the product space Q θ ∈ Γ L ( F T , P ) is metrizable, see Proposition 9.3.9 of[25]. Let f θn = H n · S θT − h θn → f θ , in L ( F T , P ) θ ∈ Γ . We prove that ( f θ ) θ ∈ Γ ∈ C Γ . By Lemma 4.1, we may assume that f θn → f θ , a.s., ∀ θ ∈ Γ . We will use an induction over the number of periods. The claimis trivial when there are zero periods. Assuming that the claim holds true forany market with dates { , ..., T } , we will prove the case with dates { , , ..., T } .For a real matrix M , let index ( M ) be the number of rows in M whichvanish identically. Let H be the random ( d × ∞ ) matrix with column vectors H , , H , , ..., H n, ..., i.e. H = H , H , . . . H n, . . .H , H , . . . H n, . . .. . . . . . . . . . . . . . .H d , H d , . . . H dn, . . . . The quantity index ( H ) is a random variable with values in { , , ..., d } . If index ( H ) = 0 , a.s. , we have that H n, = 0 for all n and (6) holds obviously.For the general case, we use induction over i = d, d − , ...,
0, that is if the resultholds true whenever index ( H ) ≥ i, a.s., we prove its validity for i − index ( H ) ≥ i − ∈ { , , ..., d − } . We willconstruct H on a finite partition of Ω as follows. OnΩ = { ω ∈ Ω : lim inf n →∞ k H n, k < ∞} ∈ F we use Lemma 2 of [21] to find F -measurable random indices n k such that onΩ , H n k , converges pointwise to a F -measurable random vector H ∗ . Since thesequence f θn k − H n k , ∆ S θ = T X t =2 H n k ,t ∆ S θt − h θn k converges a.s. to f θ − H ∗ ∆ S θ := ˜ f θ , for all θ ∈ Γ . We apply the inductionassumption to obtain H ∗ , ..., H ∗ T and h θ ≥ , θ ∈ Γ such that˜ f θ = T X t =2 H ∗ t ∆ S θt − h θ , ∀ θ ∈ Γ . Therefore, f θ = H · S θT − h θ on Ω for all θ ∈ Γ . Next we construct H on Ω = { ω : lim inf n →∞ k H n, k = ∞} . Let us define G n, = H n, k H n, k ≤ . F -measurable random indices n k such that G n k , convergespointwise to an F -measurable random vector G with k G k = 1 on Ω . It isobserved that f θn k k H n k , k → , a.s., ∀ θ ∈ Γand thus T X t =2 H n k ,t k H n k , k ∆ S θt − h θn k k H n k , k → − G ∆ S θ , a.s.. By the induction hypothesis, there exist ˜ H , ..., ˜ H T such that T X t =2 ˜ H t ∆ S θt ≥ − G ∆ S θ , on Ω ∈ F . Therefore, we obtain that G ∆ S θ + P Tt =2 ˜ H t ∆ S θt ≥ , a.s. for all θ ∈ Γ andthen for all θ ∈ Θ by Assumption 2.1 (ii). It is necessary that G ∆ S θ + T X t =2 ˜ H t ∆ S θt = 0 , a.s., for all θ ∈ Θ , (3)otherwise, the trading strategy ( G , ˜ H , ..., ˜ H T )1 Ω generates a robust arbitrageopportunity.Since k G k = 1 on Ω , we have that for each ω ∈ Ω , at least one component G j ( ω ) of G ( ω ) is nonzero. DefineΛ = Ω ∩ { G = 0 } , Λ j = (Ω ∩ { G j = 0 } ) \ (Λ ∪ ... ∪ Λ j − ) , j = 2 , ..., d and ¯ H n,t = H n,t − d X i =1 Λ i H jn, G j ( G t =1 + ˜ H t t ≥ ) , t = 1 , ..., T. By (3), we have that ( ¯ H n · S θ ) T = ( H n · S θT ) for all θ ∈ Θ. However, the matrix¯ H has index ( ¯ H ) ≥ i. We now apply the induction hypothesis to obtain H onΩ . Since Ω = Ω ∪ Ω , we have shown that there exist H and h θ ≥ f θ = H · ∆ S θt − h θ for θ ∈ Γ and the proof is complete.
Proposition 3.6.
Let Assumption 2.1 be in forced. Assume that the condition
N RA holds. Then the set C is closed in L ( F T , P ) with respect to the producttopology, that is if ( f α ) α ∈ I be a net in C and f α → f for some f ∈ L ( F T , P ) ,then f ∈ C .Proof. Let f α be a net in C , i.e., f α = H α · S T − h α for some H α ∈ A , h α ∈ L ( F T , P ) and f α → f in the product topology. For every θ ∈ Θ, H α · S θT − h θα → f θ , in L ( F T , P ) . (4)We need to show that f = ( H · S ) T − h for some H ∈ A and h ∈ L ( F T , P ).9 tep 1 (Intersection property) Let
F in (Θ) be the set of all non-empty finitesubsets of Θ. Let D ∈ F in (Θ) be arbitrary and denote H D = { H ∈ A : H is in D -canonical form, H · S θT ≥ f θ , a.s., ∀ θ ∈ D } . It is easy to see that H D is convex and closed with respect to the topology ofconvergence in probability. We will prove a finite intersection property, that is H D = \ θ ∈ D H { θ } = ∅ . (5)By Assumption 2.1 (i), there exists a sequence ( θ n ) n ∈ N ⊂ Θ which is dense inΘ. We use the density of Γ = ( θ n ) n ∈ N ∪ D to prove a stronger statement H Γ = ∅ . (6)The set Γ is also countable and the product space Q Γ L ( F T , P ) is metrizable.From (4) we can find a sequence ( α n ) n ∈ N ⊂ I , which will be denoted by ( n ) n ∈ N without causing any confusion, such that f θn = H n · S θT − h θn → f θ , in L ( F T , P ) ∀ θ ∈ Γ . By Lemma 3.5, there exist H ∈ A , h θ ∈ L ( F T , P ) such that f θ = H · S θT − h θ , for θ ∈ Γ, or equivalently, (6) holds true.
Step 2 (Boundedness of H Γ ) We prove by contradiction that for each t = 0 , ..., T ,the convex set { H t , H ∈ H Γ } is bounded in probability. If this is not the case,there are α > H n ) n ∈ N in H Γ such that for each n ∈ N , P [ k H n,t k ≥ n ] ≥ α − /n. Since H n · S θT is bounded from below by f θ for each θ ∈ Γ, Proposition 3.4implies that H n is a.s. bounded, which is a contradiction. Step 3 (Convex compactness of H Γ ) Let I ′ be an arbitrary set and ( F i ) i ∈ I ′ afamily of closed and convex subsets of H Γ . Assume that ∀ D ∈ F in ( I ′ ) , G D = \ i ∈ D F i = ∅ . (7)We will prove that \ i ∈ I ′ F i = ∅ . Since (7), for each D ∈ F in ( I ′ ) we can choose H D ∈ G D . Consider the net( H + D − H − D ) D ∈ F in ( I ′ ) . By Lemma 2.1 of [28], for every D ∈ F in ( I ), thereexists e H − D ∈ conv { H − E , E ≥ D } such that the net ( e H − D,t ) D ∈ F in ( I ′ ) converges inmeasure to a nonnegative real-valued random variable H − t , for t ∈ { , ..., T − } .It should be emphasized from Step 2 that conv { H − E,t , E ≥ D } is bounded inprobability. Using the same weights as in the construction of e H − D , we obtain e H + D .Again, Lemma 2.1 of [28] and Step 2 imply that there exist b H + D ∈ conv { e H + E , E ≥ D } such that b H + D,t converges to a nonnegative real-valued random variable H + t .Repeating this argument for each t = 0 , ..., T − H t ) t =0 ,...,T − .It is clear that b H D → H and H ∈ T i ∈ I ′ F i , which implies the desired convexcompactness. 10 tep 4 Finally, we have ∅ 6 = \ θ ∈ Θ (cid:0) H { θ } ∩ H Γ (cid:1) ⊂ \ θ ∈ Θ H { θ } , and the proof is complete. Let L ( F T , P ) = Q θ ∈ Θ L ( F T , P ) be the product space, where each L ( F T , P )is equipped with the usual k · k -norm topology. Define also the direct sum L θ ∈ Θ L ∞ ( F T , P ). The duality ( L ( F T , P )) ∗ = L θ ∈ Θ L ∞ ( F T , P ) allows usto identify a linear continuous function L ( F T , P ) → R to a finite vector( Z θ T , ..., Z θ k T ). Such linear continuous function is called strictly positive if Z θ i T ≥ , a.s., i = 1 , ..., k and Z θ i T > , a.s. for some i ∈ { , ..., k } . The notion of robustpricing system below is central of this paper. Definition 3.7.
A robust pricing system for S is a strictly positive linear func-tional Q : L ( F T , P ) → R under which S is a generalized martingale, i.e. Q (1 A t − S t ) = Q (1 A t − S t − ) for all A t − ∈ F t − , ≤ t ≤ T. For each θ ∈ Θ, we denote by Q θ := (cid:8) Q : Q is a robust pricing system for S with Z θT > , a.s. (cid:9) the set of robust pricing systems for the model θ and by Q = S Θ Q θ the setof all robust pricing systems. The following result is a robust version of thecelebrated DalangMortonWillinger theorem, see [14]. Theorem 3.8 (Robust FTAP) . The following are equivalent(i)
N RA holds;(ii) For every θ ∈ Θ , the set Q θ is non-empty.Proof. Up to an equivalent measure change dP dP = c exp( − sup θ ∈ Θ ,t k S θt k ), where c is a normalization constant, we may assume S θt , θ ∈ Θ are in L ( F t , P ).( i ) = ⇒ ( ii ). By Proposition 3.6, the set C is closed in the product space L ( F T , P ) and hence the convex set C ∩ L ( F T , P ) is closed in the productspace L ( F T , P ), too. Fix θ ∈ Θ. Since
N RA , for every x θ + ∈ L ( F T , P ), theclosed and convex set C ∩ L ( F T , P ) and the compact set { x θ + θ } are disjoint.The Hahn-Banach theorem implies that there is Q ∈ (cid:0) L ( F T , P ) (cid:1) ∗ = M θ ∈ Θ L ∞ ( F T , P ) , such that Q ( H · S T − h ) ≤ α, ∀ H ∈ A , h ∈ L ( F T , P ) (8)and Q ( x θ + θ ) >
0. We can identify Q with a finite vector of continuous linearfunctions on L ( F T , P ), that is Q ( f ) = X θ ′ ∈ Θ E [ Z θ ′ T f θ ′ ]11here Z θ ′ T = 0 for all but a finite number of θ ′ . Since L ( F T , P ) is a linearspace, it necessarily holds that α = 0, Q ( H · S T ) = 0 , ∀ H ∈ A , and Z θ ′ T ≥ θ ′ ∈ Θ with P [ Z θT > >
0. Next, an exhaustion argumentis applied to obtain a robust pricing system Q ∗ ∈ Q θ . Let G = ( Q ∈ Coun M θ ∈ Θ L ∞ ( F T , P ) : Q ( f ) ≤ , ∀ f ∈ C , P [ Z θT > > , X θ ′ ∈ Θ E [ Z θ ′ T ] = 1 ) , where L Counθ ∈ Θ L ∞ ( F T , P ) consists of countable sequences in L ∞ ( F T , P ). Notethat G is countably convex, i.e. for a sequence ( Q n ) n ∈ N in G , there exist strictlypositive scalars ( α n ) n ∈ N such that P n ∈ N α n Q n ∈ G . Let c = sup { P [ Z θT > , Q ∈ G} . Choose a sequence ( Q n ) n ∈ N such that P [ Z n,θT ] → c and define Q ∗ = ∞ X n =1 α n Q n for an appropriate sequence of strictly positive scalars ( α n ) n ∈ N . It holds that { Z ∗ ,θT > } = S n { Z n,θT > } and thus P [ Z ∗ ,θT >
0] = c . Finally we show that c = 1. Suppose by contradiction that P [ Z ∗ ,θT = 0] > Z ∗ ,θT =0 θ is anelement in L ( F T , P ). The Hahn-Banach theorem implies that there is Q ∈ G satisfying Q (1 { Z ∗ ,θT > } θ ) > , Q ( f ) ≤ , ∀ f ∈ C . Therefore, for a suitable α ∈ (0 , α Q ∗ +(1 − α ) Q is an elementof G whose support is strictly bigger that Q ∗ , a contradiction. Therefore, Q ∗ isthe required robust pricing system.( ii ) = ⇒ ( i ). Assume there exists a robust arbitrage strategy H ∈ A , thatis H · S θ ′ T ≥ θ ′ ∈ Θ and P [ H · S θT > > θ. By ( ii ), thereexists Q ∈ Q θ with Z θT > , P − a.s. and Q ( H · S T ) > . However, Lemma 4.9 implies that Q ( H · S T ) ≤
0, which is a contradiction.In order to obtain robust pricing systems for each t , we proceed as follows.We define Z t = ( E [ Z θT |F t ]) θ ∈ Θ . (9)Then Z t S t is a martingale under P . Indeed, we compute for any 0 ≤ s ≤ t ≤ T, A s ∈ F s that X θ ∈ Θ E [( Z θt S θt − Z θs S θs )1 A s ] = X θ ∈ Θ E [ Z θt S θt A s ] − X θ ∈ Θ E [ Z θs S θs A s ]= X θ ∈ Θ E [ Z θT S θt A s ] − X θ ∈ Θ E [ Z θT S θs A s ] = 0 . emark 3.9. For each θ ∈ Θ , it is only required that there exists a robustpricing system with strictly positive density Z θT . If the model θ satisfies theclassical no arbitrage condition, then the density Z θT of a martingale measure Q θ for S θ constitutes a robust pricing system Z θT θ , see Example 3.5.1. Theorem 3.10.
Let
N RA hold. Let f be a random variable such that sup θ ∈ Θ sup Q ∈Q θ Q ( f ) < ∞ . Denote the superhedging price of f by π ( f ) := inf { x ∈ R : ∃ H ∈ A such that x + H · S T ≥ f , a.s. } . Then the superhedging duality holds π ( f ) = sup θ ∈ Θ sup Q ∈Q θ Q ( f ) and there exist superhedging strategies H such that π ( f ) + H · S T ≥ f , a.s.. Proof. ( ≥ ) Let x ∈ R be such that x + H · S T ≥ f , a.s. for some H ∈ A . Thenfor an arbitrary Q ∈ Q , it holds that x ≥ Q ( f ) and therefore x ≥ sup Q ∈Q Q ( f ).( ≤ ) Take x < π ( f ). Consequently, we have f − x / ∈ C . Since C is closed,the Hahn-Banach theorem implies there exists Q ∈ L θ ∈ Θ L ∞ ( F T , P ) such that Q ( f ) ≤ f ∈ C and Q ( f − x ) > . Let Q ′ ∈ Q and define Q ′′ = α Q ′ + (1 − α ) Q ∈ Q for suitable α ∈ (0 ,
1) such that Q ′′ ( f − x ) >
0. Thisimplies x ≤ sup Q ∈Q Q ( f ). A contingent claim f is a vector of random variables in L ( F T , P ) such that f ≥ , a.s. . A contingent claim f is called replicable if there exist x ∈ R and H ∈ A such that x + H · S T = f , a.s.. The market is complete if all contingent claims are replicable.
Proposition 3.11.
Let
N RA hold and f be a contingent claim. The followingare equivalent(i) f is replicable.(ii) The mapping Q → R Q Q ( f ) is constant. roof. ( i ) = ⇒ ( ii ). Assume that there exist x ∈ R , H ∈ A such that x + H · S T = f , a.s.. For any θ ∈ Θ and Q ∈ Q θ , the process S is a generalizedmartingale under Q and thus H · S T is also a generalized martingale under Q ,by Lemma 4.9. The conclusion in ( ii ) is then followed by computing Q ( f ) = Q ( x + H · S T ) = x, for every Q ∈ Q .( ii ) = ⇒ ( i ). Let H be a superhedging strategy for f , that is π ( f ) + H · S T − f ≥ , a.s.. If for some θ ∈ Θ, the inequality is strictly positive with strictlypositive probability, then we get for all Q ∈ Q θ < Q ( π ( f ) + H · S T − f ) . Noting that H · S T is a generalized martingale under Q with zero expectationand using ( ii ), we get sup Q ∈Q θ Q ( f ) = sup Q ∈Q Q ( f ) < π ( f ) , which contradicts to Theorem 3.10. Therefore, f is replicable. Theorem 3.12.
Under
N RA , the market is complete if and only if |Q| = 1 .Furthermore, if
N A ( θ ) holds for some θ ∈ Θ , then Θ = { θ } .Proof. Noting Proposition 3.11, we only need to prove the “only if” part. As-sume the market is complete. Then for every A ∈ F T with P [ A ] > θ ∈ Θ,the claim θA is replicable. By Proposition 3.11, we obtain that0 < Q ( θA ) = Q ′ ( θA ) (10)for every Q , Q ′ ∈ Q . This implies that there exists a unique robust pricingsystem Q and | Θ | is at most countable.We consider the case N A ( θ ) holds for some θ ∈ Θ and thus there is amartingale measure Q θ for S θ . Let Q θ = Q θ θ be the corresponding robustpricing system. If Θ is not singleton, for every Q θ ′ ∈ Q θ ′ where θ ′ = θ , we getfrom (10) that 0 < Q θ ′ ( θ ′ A ) = Q θ ( θ ′ A ) = 0 , which is a contradiction. Hence Θ = { θ } and the model θ is complete. We introduce a very simple discrete time market model, and in the sequel, mostof illustrations are carried out in this model. Consider the t -fold Cartesianproduct Ω t = {− , } t . Denote by 2 Ω t the power set of Ω t and Ω = Ω T . Themapping Π t : Ω T → Ω t is defined byΠ t ( ω ) := ( ω , ..., ω t ) , ∀ ω = ( ω , ..., ω T ) ∈ Ω . Set F t = Π − t (2 Ω t ). The measurable space (Ω , Ω ) is equipped with the proba-bility measure P ( { ω } ) = T Y t =1 (cid:18) δ ( { proj t ( ω ) } ) + 12 δ − ( { proj t ( ω ) } ) (cid:19) , δ x : B ( R d ) → { , } is the Dirac measure at xδ x ( B ) = (cid:26) x ∈ B, t is the projection at t from Ω to {− , } ,proj t ( ω , ..., ω T ) = ω t , for ω ∈ Ω . For a pair of parameters θ i = ( µ i , σ i ) ∈ R × R + , we define the real-valued process S θ i as S θ i t = s + t X s =1 ( µ i + σ i proj t ( ω )) , s ∈ R d . We consider the toy example with one-period, i.e., T = 1, and | Θ | = 2. Thedynamics of the risky asset are given by S θ i = 1 + σ i ω + µ i where µ i ∈ R , < σ i for i ∈ { , } . The requirements for a robust pricingsystem Q are (cid:10) Q , ( H ∆ S θ i − h i ) i ∈ Θ (cid:11) ≤ , ∀ H ∈ R , h i ∈ R + , and the normalizing condition. It is easy to observe that Q ∈ R and furthercomputation lead to the following system of equations Q (1)[ σ + µ ] + Q (1)[ σ + µ ] + Q ( − − σ + µ ] + Q ( − − σ + µ ] = 0 , Q (1) + Q (1) + Q ( −
1) + Q ( −
1) = 1 . This system admits a solution ifmin { σ + µ , σ + µ , − σ + µ , − σ + µ } < , (11)max { σ + µ , σ + µ , − σ + µ , − σ + µ } > . (12)We consider the following particular cases.The case µ > σ , > − σ > µ . This is a pathological case where thefirst dynamics increases while the second decreases, and each of them admitsarbitrage opportunities. However, there is no robust arbitrage. Indeed, if H isa robust arbitrage then it should satisfy the following conditions H ( σ ω + µ ) ≥ , H ( σ ω + µ ) ≥ , ∀ ω ∈ Ω , which imply that H = 0 . In other words, the condition
N RA holds true.The case | µ i | < σ i for each i = 1 ,
2. As shown in [31], there exists a uniquemartingale measure for each S θ i , i = 1 , Q θ i ( ω ) = 12 (cid:18) − ω µ i σ i (cid:19) . (13)15e find the solutions to (11), (12). Let 0 ≤ α ≤ β ∈ R be chosen later.Consider Q (1)[ σ + µ ] − Q ( − σ − µ ] = β, Q (1) + Q ( −
1) = α, and Q (1)[ σ + µ ] − Q ( − σ − µ ] = − β, Q (1) + Q ( −
1) = 1 − α. Now we can solve explicitly Q (1) = 12 σ [ β + α ( σ − µ )] , Q ( −
1) = 12 σ [ − β + α ( σ + µ )] , Q (1) = 12 σ [ − β + (1 − α )( σ − µ )] , Q ( −
1) = 12 σ [ β + (1 − α )( σ + µ )] . (14)For each pair ( α, β ) such that0 < β + α ( σ − µ ) < σ , < − β + α ( σ + µ ) < σ , < − β + (1 − α )( σ − µ ) < σ , < β + (1 − α )( σ + µ ) < σ (15)we can construct a robust pricing system. The choice α = 1 , β = 0 correspondsto the robust pricing system ( Q θ ,
0) and the choice α = 0 , β = 0 leads to(0 , Q θ ), where Q θ i , i = 1 , Q θ ,
0) and (0 , Q θ ).In this example (the pathological case is included), robust superhedgingprices can be computed explicitly. For instance, the option giving the stock at T = 1, i.e. ( S θ , S θ ) has the superhedging pricesup Q ∈Q (cid:16) Q S θ + Q S θ (cid:17) = sup α,β satisfy (15) [( β + α ) + ( − β + 1 − α )] = 1 . Remark 3.13.
The parameters α and (1 − α ) are the weights put on the model θ and θ , respectively. The parameter β controls the average of all outcomesunder Q which can be positive (in the classical setting without uncertainty,this should be zero). However, this additional gain for the model θ is exactlycompensated by an opposite gain − β in the model θ so that there is no positivegain on average of all models.In this robust setting, each model θ is considered equally, so α varies on theset of possible values [0 , . In a more realistic setting, traders may choose α consistently with their statistical estimation. For example, they may find thatthe true parameter will stay in a certain interval with high probability, then moreweights are assigned to such interval. .5.2 Example: Dynamic approach may fail for superhedging Let us consider the toy example with T = 2 , | Θ | = 2. The price processes evolveas S it = 1 + σ i ω t , t ∈ { , } , S i = 1 , P − a.s.. Assume further that σ < σ . In this case, each price process admits P as theunique martingale measure. The price x ∈ R at time 0 of the option givingone stock at time 2 is such that x + H ∆ S + H ∆ S ≥ S , P − a.s.,x + H ∆ S + H ∆ S ≥ S , P − a.s.. for some H ∈ R , H ∈ F . It is easy to check that x = 1 is the superhedgingprice together with the strategy H = H = 1. Next, we use the dynamicprogramming approach to compute the price at time 0. First, the price x attime 1 of this option satisfies x + H ∆ S ≥ S , P − a.s.,x + H ∆ S ≥ S , P − a.s.. for some H ∈ F . Taking conditional expectation under P , we obtain x ≥ max i =1 , E [ S i |F ] = max { S , S } , P − a.s., and thus x = (1 + σ )1 ω =1 + (1 − σ )1 ω = − . Note that E [ x ] = 1 + ( σ − σ )and the strategy H = 1 superhedges the option. Secondly, the price at time 0of x is x ′ + H ∆ S ≥ x , P − a.s.,x ′ + H ∆ S ≥ x , P − a.s., or equivalently x ′ + H σ ≥ σ ,x ′ + H σ ≥ σ ,x ′ − H σ ≥ − σ ,x ′ − H σ ≥ − σ . We deduce that x ′ ≥ ( σ − σ ) > x , and this inequality suggests that thedynamic programming approach may be inapplicable for robust superhedging. Before starting our analysis, some no arbitrage conditions under uncertainty arerecalled, see also [6].
Definition 3.14.
The condition sN A (Θ) holds true if the condition
N A ( θ ) holds true for all θ ∈ Θ . The condition wN A (Θ) holds true if there exists some θ ∈ Θ such that the condition N A ( θ ) holds true. | Θ | = 2 and the laws of S θ , S θ are notequivalent. In this example, it may happen that sN A (Θ) and wN A (Θ) fail.We would like to emphasize that the First Fundamental Theorem of [7] is notapplied here since the set { Law( S θ i ) , i = 1 , } is not convex. Indeed, in ournotations, their theorem says that the no robust arbitrage condition holds ifand only if for each i ∈ { , } , there exists a martingale measure Q such thatLaw( S θ i ) << Q << Law( S θ j ) . It happens when j = i , that is S θ i admits no arbitrage opportunities for each i = 1 ,
2, or equivalently, the condition sN A (Θ) holds true. However, such strongrequirement sN A (Θ) is not satisfied as in the pathological case. Furthermore,the results from [7] yield the robust pricing systems of the forms ( Q θ ,
0) and(0 , Q θ ), where Q θ i is a martingale measure for S θ i , i = 1 ,
2, but not all robustpricing systems computed in (14).A similar result is obtained in Theorem 3.4 of [4]: the condition
N A ( P ),see their terminologies, holds if and only if for all P ∈ P , there exists a localmartingale measure. In the multiple priors setting, Theorem 3.8 of [6] showsthat the condition N RA is equivalent to the condition sN A ( D ) for some subset D of Θ. Example 3.5.1 in the present paper cannot be explained by these results.The robust setting in the present paper allows models with arbitrage op-portunities as long as such riskless profits are model dependent. Furthermore,robust pricing systems explain precisely the interaction between models to ruleout model independent arbitrage opportunities by averaging not only possiblescenarios but also possible models. Usually, information from the market, for example option prices, stock prices areused in calibration to select models that fit real data. We show in this sectionthat this fitting procedure helps to reduce the set of robust pricing systems, andhence superhedging prices.Let e ∈ N and g i : Ω → R d , i = 1 , ..., e be traded options which can beonly bought or sold at time t = 0 at the market price g i . We may assume that g i = 0 , i = 1 , ..., e . Assume that the underlying assets for these options are S θ .In this robust setting, the option g i will give the payoff ( g i ( S θT )) θ ∈ Θ at time T , which will be denoted by g i . For a vector a = ( a , ..., a e ) ∈ R e , the optionportfolio from a is given by P ei =1 a i g i . A semi-static strategy ( H, a ) is a pairof H ∈ A and a ∈ R e , and the corresponding wealth at time T is H · S T + e X i =1 a i g i = T X s =1 H s ∆ S s + e X i =1 a i g i . Definition 3.15.
We say that the market satisfies the condition
N RA if forevery self-financing strategy H ∈ A and for every a ∈ R e such that ∀ θ ∈ Θ , H · S T + e X i =1 a i g i ≥ , a.s. then ∀ θ ∈ Θ , H · S T + e X i =1 a i g i = 0 , a.s.. efinition 3.16. A calibrated robust pricing system is a robust pricing systemwhich is consistent with the option prices. We define Q θcal,e = { Q ∈ Q θ : Q ( g i ( S T )) = 0 , i = 1 , ..., e } and Q cal,e = ∪ θ ∈ Θ Q θcal,e . Theorem 3.17.
Under the setting above(a) The following are equivalent:(i)
N RA holds.(ii) ∀ θ ∈ Θ , Q θcal,e = ∅ .(b) Let N RA hold, and f be a random variable. The superhedging price isdefined by π e ( f ) := inf { x ∈ R : ∃ ( H, a ) ∈ A× R e such that x + H · S T + e X i =1 a i g i ≥ f , a.s. } . Then the superhedging duality holds π e ( f ) = sup θ ∈ Θ sup Q ∈Q θcal,e Q ( f ) , and there exits ( H, a ) ∈ A × R e such that π e ( f ) + H · S T + e X i =1 a i g i ( S T ) ≥ f , a.s. (c) Let N RA hold, and f be a random variable. The following are equivalent(i) f is replicable.(ii) The mapping Q Q ( f ) is constant on Q cal,e . Proof.
We proceed by induction as in [7]. For e = 0, the results are true byTheorem 3.8, Theorem 3.10, and Proposition 3.11. Assume that the resultshold true for the market with the stocks and e ≥ g e +1 with the market price g e +10 = 0.Consider (a). Let N RA hold. If g e +1 is replicable in the market consistsof the stocks and available options g , ..., g e , we come back to the case with e options and therefore we may assume that g e +1 is not replicable. If g e +10 ≥ π e ( g e +1 ), then we can construct a robust arbitrage by shorting one unit of g e +1 and using the initial capital π e ( g e +1 ) together with the superhedging strategyfor g e +1 , which exists by the induction hypothesis, to cover g e +1 at time T .Thus, the consistency with RN A implies g e +10 < π e ( g e +1 ). The inductionhypothesis (b) gives g e +10 < π e ( g e +1 ) = sup θ ∈ Θ sup Q ∈Q θcal,e Q ( g e +1 ) , and since g e +1 is not replicable, by the induction hypothesis (c), there is θ + ∈ Θ, Q θ + + ∈ Q θ + cal,e such that g e +10 < Q θ + + ( g e +1 ) < π e ( g e +1 ) . θ − ∈ Θ and Q θ − − ∈ Q θ − cal,e such that − π e ( − g e +1 ) < Q θ − − ( − g e +1 ) < g e +10 < Q θ + + ( g e +1 ) < π e ( g e +1 ) (16)By the induction hypothesis (a), for each θ ∈ Θ there is Q θ ∈ Q θcal,e with Z θ > , a.s. . Choosing appropriate weights λ − , λ + , λ ∈ (0 ,
1) and λ − + λ + + λ = 1,we have that Q ′ := λ − Q θ − − + λ + Q θ + + + λ Q θ ∈ Q θcal,e and Q ′ ( g e +1 ) = 0 . It means that Q ′ ∈ Q θcal,e +1 . Thus we prove (i) implies (ii) in (a). The converseimplication is easy. The proof of (c) is straightforward as well.Next we consider (b). Assume there are x ∈ R , H ∈ A , a ∈ R e +1 such that x + H · S T + P e +1 i =1 a i g i ≥ f , a.s.. We compute easily for every Q ∈ Q cal,e +1 that H · S is a generalized martingale under Q and x = Q ( x + H · S T + P e +1 i =1 a i g i ) ≥ Q ( f ). As a result, we have π e +1 ( f ) ≥ sup θ ∈ Θ sup Q ∈Q θcal,e +1 Q ( f ) . (17)Now we prove the reverse inequality, π e +1 ( f ) ≤ sup θ ∈ Θ sup Q ∈Q θcal,e +1 Q ( f ) . (18)We claim thatthere is a sequence Q n ∈ Q cal,e such that Q n ( g e +1 ) → , Q n ( f ) → π e +1 ( f ).(19)Without loss of generality, we may assume that π e +1 ( f ) = 0. If the claim (19)fails, we get 0 / ∈ { Q ( g e +1 , f ) , Q ∈ Q cal,e } ⊂ R . Using a separation argument,there are α, β ∈ R such that0 > sup Q ∈Q cal,e Q ( α g e +1 + β f ) . (20)By (b) of the inductive hypothesis, it holds thatsup Q ∈Q cal,e Q ( α g e +1 + β f ) = π e ( α g e +1 + β f ) . By definition π e ( ψ ) ≥ π e +1 ( ψ ) for any random variable ψ. Since g e +1 can behedged at price 0, we obtain π e +1 ( α g e +1 + β f ) = π e +1 ( β f ). Therefore,0 > sup Q ∈Q cal,e Q ( α g e +1 + β f ) ≥ π e +1 ( β f ) . Clearly, β = 0. If β > , we obtain π e +1 ( f ) <
0, which contradicts to ourassumption that π e +1 ( f ) = 0. Thus β <
0. Since Q cal,e +1 ⊂ Q cal,e , (20)implies that 0 > Q ′ ( β f ), for Q ′ ∈ Q cal,e +1 . Consequently, Q ′ ( f ) > π e +1 ( f ),contradicting to (17). Therefore, the claim (19) holds true.Since g e +1 is not replicable, there are two robust pricing systems Q θ + + , Q θ − − as in (16). For the sequence Q n as in (19), we can find λ n − , λ n , λ n + ∈ [0 ,
1] suchthat λ n − + λ n + λ n + = 1 and Q ′ n = λ n − Q θ − − + λ n Q n + λ n + Q θ + + ∈ Q cal,e satisfies Q ′ n ( g e +1 ) = 0 , or equivalently, Q ′ n ∈ Q cal,e +1 . By (19), we can choose λ n ± →
0. Therefore, Q ′ n ( f ) →
0, which implies (18). 20
Appendix
Let I be an index set and for each i ∈ I , let ( X i , τ i ) be a topological space. Theproduct space, denoted by Q i ∈ I ( X i , τ i ), consists of the product set Q i ∈ I X i anda 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]. The directsum L i ∈ I X i is defined to be the set of tuples ( x i ) i ∈ I with x i ∈ X i such that x i = 0 for all but finitely many i .If each ( X i , τ i ) is locally convex then Q i ∈ I X i is locally convex, too. If I is uncountable, the product space is not normable. Since the product space isnot first countable, sequential closedness is different from topological closedness.The dual of the product space Q i ∈ I X i is algebraically equal to the direct sumof their duals L i ∈ I X ∗ i .It is known that convergence in probability implies almost sure convergencealong a subsequence. The following lemma extends this result to countableproducts of L spaces. Lemma 4.1.
Let I be a countable set. If a sequence ( f n ) n ∈ N in Q i ∈ I L ( F , P ) converges to f in the product topology, then there exists a subsequence ( n k ) k ∈ N such that f in k → f i , a.s. for all i ∈ I .Proof. We may assume I = N . The metric d ( f , g ) := X i ∈ I i d ( f i , g i )1 + d ( f i , g i )induces the product topology on Q i ∈ I L ( F , P ), where d is the metric inducesthe topology of convergence in probability. Let ( ε k ) k ∈ N be a sequence of positivenumbers decreasing to zero. Since f n → f in d , there exists a subsequence( n k ) k ∈ N such that for all k ∈ N , we have d ( f n k , f ) < ε k k , and hence for every i ∈ I , d ( f in k , f i ) < i ε k k . From this we obtain P [ | f in k − f i | ≥ ε k ] ≤ i k . For every ε >
0, there is K such that ε k ≤ ε for k ≥ K , and thus we computethat ∞ X k = K P [ | f in k − f i | ≥ ε ] ≤ ∞ X k = K P [ | f in k − f i | ≥ ε k ] ≤ i for every i ∈ I . Therefore P ∞ k =1 P [ | f in k − f i | ≥ ε ] is also finite. By the Borel-Cantelli lemma, for every i ∈ I , the set { ω : | f in k ( ω ) − f i ( ω ) | ≥ ε infinitely often } has zero probability for all ε >
0, or equivalently, f in k → f i , a.s. for every i ∈ I . 21 .2 Convex compactness of L A set A ⊂ L is bounded if lim n →∞ sup f ∈ A P [ | f | ≥ n ] = 0 . For any set I wedenote by F in ( I ) the family of all non-empty finite subsets of I . This is adirected set with respect to the partial order induced by inclusion. We recallDefinition 2.1 of [34]. Definition 4.2.
A convex subset C of a topological vector space is convexlycompact if for any non-empty set I and any family ( F i ) i ∈ I of closed and convexsubsets of C , the condition ∀ D ∈ F in ( I ) , \ i ∈ D F i = ∅ implies \ i ∈ I F i = ∅ . The following result gives a characterization for convex compactness in L ,see Theorem 3.1 of [34]. Theorem 4.3.
A closed and convex subset C of L is convexly compact if andonly if it is bounded in probability. Let F be a sigma algebra. Let I be an index set. In this subsection, we workwith the product space L = Q i ∈ I L ( F , P ). Let Q : L → R be a linearfunction such that Q = ( Z θT ) θ ∈ Θ ∈ L Couni ∈ I L ∞ ( F , P ) and E (cid:2)P θ ∈ Θ Z θT (cid:3) = 1. Definition 4.4.
Let
G ⊂ F be two sigma algebras and f = ( f i ) i ∈ I be an F -measurable random variable such that Q ( f ) < ∞ . An G -measurable randomvariable f g is called a generalized conditional expectation of f with respect to G under Q if Q ( f A ) = Q ( f g A ) , ∀ A ∈ G . (21) We denote by Q ( f |G ) the set of all generalized conditional expectations of f . This definition becomes the usual definition for conditional expectation when | I | = 1. However, there are significant differences between the two conceptswhen | I | ≥
2. See Example 4.5 below for the facts that uniqueness and mono-tonicity fail in general.
Example 4.5 (Non-uniqueness) . We consider the toy model with T = 2 and | Θ | = 2 . We also assume that | µ i | < σ i , i = 1 , . A robust pricing system for ( S θ , S θ ) is given by Q ( ω ) = ( Q , Q )( ω ) = 14 (cid:18)(cid:18) − ω µ σ (cid:19) (cid:18) − ω µ σ (cid:19) , (cid:18) − ω µ σ (cid:19) (cid:18) − ω µ σ (cid:19)(cid:19) . Let us consider a conditional expectation of ( S θ , S θ ) given F , that is an F -measurable vector ( X , X ) such that (cid:16) Q S θ + Q S θ (cid:17) ω =1 = (cid:0) Q X + Q X (cid:1) ω =1 , (22) (cid:16) Q S θ + Q S θ (cid:17) ω = − = (cid:0) Q X + Q X (cid:1) ω = − . (23)22 t is easy to see that ( S θ , S θ ) is a solution to the system of equations (22),(23). Other solutions can be found by solving (cid:0) Q Y + Q Y (cid:1) ω =1 = 0 , (24) (cid:0) Q Y + Q Y (cid:1) ω = − = 0 . (25) In the case without uncertainty, for example
Θ = { θ } , there is the uniquesolution Y (1) = Y ( −
1) = 0 to the system of equations (24), (25) and thus S θ is the conditional expectation of S θ . However, under uncertainty, solutionsto (22), (23) are vectors ( S θ + Y , S θ + Y ) where ( Y , Y ) satisfies (24), (25). Some basic properties of generalized conditional expectation are given below.
Proposition 4.6.
The following properties hold true(i) If f g ∈ Q ( f |G ) , then Q ( f g ) = Q ( f ) .(ii) For α , α ∈ R , α Q ( f |G )+ α Q ( f |G ) ⊂ Q ( α f + α f |G ) . The converseinclusion fails.(iii) If f is F -measurable then f ∈ Q ( f |F ) .(iv) If g is G -measurable then gQ ( f |G ) ⊂ Q ( fg |G ) .(v) (Tower property) For H ⊂ G , Q ( f |G|H ) ⊂ Q ( f |H ) .Proof. ( i ): Choosing A = Ω in (21), we obtain the result.( ii ): we can check by definition. To see the failure of the converse inclu-sion, we take Ω = { ω , ω } , Q = ( , ). Taking ( g, − g ) , g ∈ R , we have Q (( g, − g )1 A ) = 0 for every A ∈ G . For g ∈ Q ( f |G ) and g ∈ Q ( f |G ),the vector g + g + ( g, − g ) is in Q ( f + f |G ) but not in Q ( f |G ) + Q ( f |G ).( iii ) and ( iv ): Use definition.( v ): Let f g ∈ Q ( X |G ) be arbitrary and f h ∈ Q ( f g |H ). By definition, we get Q ( f h h ) = Q ( f g h ) for all H -measurable sets h , which are also G -measurable.Therefore Q ( f g h ) = Q ( f h ) and then Q ( f h h ) = Q ( f h ). Definition 4.7.
Let (Ω , F , P ) be a probability space, equipped with a filtration ( F t ) t =0 ,...,T . An adapted process ( M t ) t =0 ,...,T is a generalized martingale under Q if(i) E (cid:2)(cid:12)(cid:12)P θ ∈ Θ Z θt M θt (cid:12)(cid:12)(cid:3) < ∞ , for ≤ t ≤ T ,(ii) Q ( M t A s ) = Q ( M s A s ) , ∀ A s ∈ F s , ≤ s ≤ t ≤ T ,A process M is a generalized supermartingale under Q if (ii) is replaced by Q ( M t A s ) ≤ Q ( M s A s ) , ∀ A s ∈ F s , ≤ s ≤ t ≤ T It is easy to observe that if M is a generalized martingale under Q (resp.generalized supermartingale under Q ) then Q ( M t ) = Q ( M s ) (resp. Q ( M t ) ≤ Q ( M s )) for s ≤ t . Definition 4.8.
An adapted process ( M t ) t =0 ,...,T is a generalized local martin-gale under Q if there is an increasing sequence ( τ n ) n ∈ N of stopping times suchthat P (lim n →∞ τ n = ∞ ) = 1 and that each stopped process M t ∧ τ n τ n > is ageneralized martingale under Q . Lemma 4.9.
Let M ≥ be an adapted process with M = . The followingare equivalent:(i) M is a generalized martingale under Q .(ii) If H is predictable and bounded, then V = H · M is a generalized martin-gale under Q .(iii) If H is predictable, then V = H · M is a generalized local martingale under Q . If in addition that E h(cid:0)P θ ∈ Θ Z θT V θT (cid:1) − i < ∞ , then V is a generalizedmartingale under Q .Proof. ( i ) = ⇒ ( ii ). Assume that sup ≤ t ≤ T − | H t | ≤ c . It is easy to check that V satisfies the property (i) and (ii) in Definition 4.7.( ii ) = ⇒ ( iii ). Define τ n := inf { t : | H t +1 | > n } and H nt := H t t ≤ τ n for n ∈ N . Since H is predictable and finite-valued, ( τ n ) n ∈ N is a sequence ofstopping times increasing a.s. to infinity. By (ii), or each n ∈ N , the process H n · M is a generalized martingale under Q . Noting that V t ∧ τ n = H n · M t , weobtain that V is a generalized local martingale under Q .For convenient notations, we define L t := P θ ∈ Θ Z θt V θt . By assumption, E (cid:2) L − T (cid:3) < ∞ . We show inductively that E (cid:2) L − t (cid:3) < ∞ for t = 1 , ..., T −