Institute of Statistics and Decision Support Systems, University of Vienna Brünnerstrasse 72, A-1210 Wien, Austria

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1 MARKET FREE LUNCH AND LARGE FINANCIAL MARKETS Irene Klein Institute of Statistics and Decision Support Systems, University of Vienna Brünnerstrasse 72, A-1210 Wien, Austria Abstract. The main result of the paper is a version of the Fundamental Theorem of Asset Pricing (FTAP) for large financial markets based on an asymptotic concept of no market free lunch for monotone concave preferences. The proof uses methods from the theory of Orlicz spaces. Moreover various notions of no asymptotic arbitrage are characterized in terms of no asymptotic market free lunch, the difference lies in the set of utilities. In particular it is shown directly that no asymptotic market free lunch with respect to monotone concave utilities is equivalent to no asymptotic free luch. In principle the paper can be seen as the large financial market analogue of Frittelli (2004) and Klein (2005). 1. Introduction In a semimartingale model of a securities market Frittelli (2004) introduced the notion no market free lunch (NMFL) which is a condition of no arbitrage type induced by the preferences of all agents in the market. Preferences are given by expected utility, where the expectation is taken with respect to an equivalent probability measure, which determines together with the utility function an agent. Frittelli gave characterizations of the well-known notions no arbitrage (NA) and no free lunch with vanishing risk (NFLVR) in terms of NMFL, the difference between the characterizations depends on the class of utility functions. Moreover he provided a new version of the Fundamental Theorem of Asset Pricing (FTAP), that is, he gave a direct proof of the equivalence between NMFL with respect to monotone concave utility functions and the existence of a (local/sigma) martingale measure. The proof is not based on any other version of the FTAP. A large financial market is a sequence of traditional market models, each of them based on a finite number of assets. Under the assumption that there isn t any kind of arbitrage on any of the finite markets Kabanov and Kramkov (1994) introduced the 1991 Mathematics Subject Classification. 46A20, 46E30, 46N10, 60G44, 60H05,. Key words and phrases. market free lunch, large financial market, contiguity of measures, asymptotic free lunch, fundamental theorem of asset pricing, Orlicz space. 1 Typeset by AMS-TEX

2 2 I. KLEIN notions no asymptotic arbitrage of first (NAA1) and second kind (NAA2). These notions give one sided versions of the FTAP for large financial markets in the sense that they are equivalent to the existence of a sequence of equivalent martingale measures with a contiguity property with respect to the sequence of the original probability measures in one direction, compare Kabanov and Kramkov (1994, 1998) and Klein and Schachermayer (1996a and 1996b). However, the direct analogue to the existence of an equivalent martingale measure for the case of a large financial market seems to be the existence of a sequence of martingale measures which is contiguous with respect to the sequence of the original probabilities and vice versa (that is, bicontiguous). So, it was a natural to ask whether the symmetric property of bicontiguity has an economic interpretation. The general answer to this question was given in Klein (2000) where the notion no asymptotic free lunch (NAFL) was introduced, which is the large financial market analogue of the concept no free lunch (NFL) of Kreps (1981). This lead to a general version of the FTAP for large financial markets. However, the condition NAFL is rather technical and, unfortunately, cannot be replaced by a weaker condition of this type in general. For continuous processes the bicontiguity property is equivalent to no asymptotic free lunch with bounded risk (NAFLBR) which has an intuitive interpretation. In the present paper I introduce the notion no asymptotic market free lunch (NAMFL) which is the translation of the concept no market free lunch of Frittelli (2004) to large financial markets. It turns out that NAMFL with respect to monotone concave preferences also is equivalent to the existence of a bicontiguous sequence of martingale measures, which is the main result of this paper. This is another version of the FTAP for large financial markets in full generality with the additional property that the notion NAMFL has a clear economic interpretation via preferences of investors. The proof is based on some ideas of Frittelli (2004) but is more involved than the proof of the analogous result there. One has to introduce a uniform concept of no market free lunch with respect to monotone concave preferences (which turns out to be equivalent to the non uniform concept) and use the theory of Orlicz spaces and a quantitative version of the Halmos Savage Theorem. The proof shows the closeness between the theory of monotone concave utilities and Orlicz spaces. This connection obviously also holds in the small market case: with the use of Orlicz spaces one can prove directly that NMFL with respect to monotone concave preferences is equivalent to the concept NFL. This was done in Klein (2005). Besides the FTAP result various no asymptotic arbitrage conditions from the literature are characterized in terms of no asymptotic market free lunch. The difference lies in the choice of the set of utilities defining the market free lunch. By the FTAP result the characterization of NAFL in terms of no market free lunch is already clear: it is equivalent to NAMFL with respect to monotone concave preferences. I will give a direct proof of this result as well (which in fact is the step justifying that we can replace the NAMFL condition by a uniform version). This can be seen as the analogue result to Klein (2005) for large financial markets. When speaking about large financial markets one has to remark on the history of the theory. The idea of asymptotic arbitrage appeared first in Ross (1976) and in the important note of Huberman (1982). The theory of large financial markets

3 MARKET FREE LUNCH AND LARGE MARKETS 3 can be seen as a modern version of arbitrage pricing theory (APT). However, the classic APT does not touch the problem of the existence of an equivalent martingale measure, so it does not provide a version of the FTAP for a large financial market. Let s conclude the introduction by a short overview how the paper is organized: in Chapter 2 I introduce some notations and deal with the small market case, that is, I assemble some results on market free lunch for a traditional market model. I follow Frittelli (2004) and Klein (2005). In Chapter 3 I introduce large financial markets and define an asymptotic concept of no market free lunch. In Chapter 4 I state and prove a version of the FTAP for large financial markets based on no asymptotic market free lunch. In Chapter 5 I give characterizations of various no asymptotic arbitrage properties in terms of no asymptotic market free lunch. In Chapter 6 I show that NAMFL with respect to monotone concave preferences does indeed hold uniformly and, moreover, is equivalent to NAFL. Finally, in the appendix some properties of Orlicz spaces, Young functions and the Mackey topology are assembled as well as some facts on contiguity of sequences of probabilities, a convex duality result of Bellini and Frittelli (2002) based on Rockefellar (1971) and a quantitative version of the Halmos Savage Theorem of Klein and Schachermayer (1996b). 2. Notations and No Market Free Lunch on a Small Market Let (Ω, F, (F t ) t [0, ), P ) be a filtered probability space where the filtration satisifies the usual assumptions. Let P be the class of probability measures on (Ω, F) equivalent to P. Whenever it is clear which P is meant, the notation L p is used for L p (Ω, F, P ) (where p = 0, p = 1 or p = ), whenever the dependence on a certain measure R is stressed, the notation L p (R). Recall the setting of Kabanov (1997). Define S to be the space of semimartingales X = (X t ) t=0 with X 0 = 0, which is a Frechet space with the quasi norm D(X) = sup{ n 1 2 n E[1 H X n ] : H predictable, H 1}. As usual H X t is the stochastic integral of H with respect to X on the interval ]0, t]. Fix in S a closed convex subset X 1 of processes X 1 which contains 0 and satisfies the following condition: if X, Y X 1 and H, G 0 are bounded predictable processes, HG = 0 and Z = H X + G Y 1, then Z X 1. Let X = λ>0 λx 1. Define convex sets K 1 = {X : X X 1 } and K = {X : X X }. Remark 2.1. Note that we can take as X 1 the set of all stochastic integrals H S with respect to one fixed d dimensional semimartingale S based on (Ω, F, (F t ) t R+, P ) such that H S 1. The space X 1 is closed by the Theorem of Mémin (1980). This setting describes a financial market with d assets, where the price processes are given by S. Define C = (K L 0 +) L. In the light of Remark 2.1 K can be interpreted as the cone of all replicable claims, and C is the cone of all claims in L that can be superreplicated, that is, claims in C are dominated by elements of K. Define the set of separating measures M = {Q P : K L 1 (Q) and E Q [f] 0 for all f K}.

4 4 I. KLEIN If S in Remark 2.1 is bounded (locally bounded) then M consists of all P absolutely continuous probability measures such that S is a martingale (local martingale). In general, for unbounded S, M is the set of P absolutely continuous probabilities such that the admissible stochastic integrals are supermartingales. Recall the well-known conditions of no arbitrage (NA), no free lunch with vanishing risk (NFLVR) and no free lunch (NFL): (NA) C L + = {0} (NFLVR) C L + = {0} (NFL) C L + = {0}, where C is the L norm closure of C and C the closure of C with respect to the weak star topology. The Fundamental Theorem of Asset Pricing (FTAP) says that the absence of an appropriate no-arbitrage condition is equivalent to M P. If S is (locally) bounded this means that there exists an equivalent (local) martingale measure. In the unbounded case this implies that the set M σ of equivalent sigma martingale measures is non empty and that M is dense in M σ for the variation topology, Delbaen and Schachermayer (1998). For finite Ω (Harrison and Pliska, 1981) and for general Ω and a finite number of trading dates (Dalang,Morton and Willinger, 1990) NA is sufficient. For general Ω and continuous time the FTAP was proved in full generality by Delbaen and Schachermayer (1994, 1998) using the stronger condition NFLVR. Moreover, it is clear that the even stronger condition NFL of Kreps (1981) is also equivalent to M P. No Market Free Lunch. Frittelli (2004) introduced a notion of market free lunch that depends on the preferences of the investors in the market. Let U be a class of utility functions u : R R { } and let D(u) = {x R : u(x) > }. The preference of a certain agent (determined by some R P and u U) can be represented by expected utility: f 1 f 2 E R [u(f 1 )] E R [u(f 2 )]. A market free lunch with respect to U is the following: there is w L ++ such that for all P P and all u U: sup f C E P [u(f w)] u(0), where L ++ = {w L : P (w 0) = 1 and P (w > 0) > 0}. This means that all agents ( P, u) in the market regard the same claim w as a free lunch. The supremum above is computed on {f C : E P [u (f w)] < + } and if this set is empty the supremum is. Definition 2.2. There is no market free lunch with respect to U, NMFL(U), if for all w L ++ there is P P and u U such that sup f C E P [u(f w)] < u(0). The notion NMFL(U) depends on the class U. Frittelli (2004) used the following sets of utility functions.

5 MARKET FREE LUNCH AND LARGE MARKETS 5 Definition 2.3. U 0 = {u : R R { } such that u is non-decreasing on R} U 1 = {u U 0 : u is left-continuous in 0 interior(d(u))} U 2 = {u U 0 : u is finite valued and concave on R} It is obvious that NMFL(U 2 ) NMFL(U 1 ) NMFL(U 0 ) as U 2 U 1 U 0. Frittelli gave characterizations of (NA) and (NFLVR) in terms of no market free lunch and proved a version of the FTAP via NMFL(U 2 ). Theorem 2.4 (Frittelli, 2004). a) NMFL(U 0 ) NA. b) NMFL(U 1 ) NFLVR. c) NMFL(U 2 ) M P (FTAP). By Delbaen and Schachermayer (1994, 1998), who proved that M P NFLVR, it is clear that, moreover, NMFL(U 1 ) NMFL(U 2 ). It will become clear later on that this is not the case for the analogue notions in large financial markets. Indeed, we will see in Chapter 6 that NAMFL(U 1 ) no asymptotic arbitrage of first kind and in Chapter 4 that NAMFL(U 2 ) there is a sequence of measures Q n M n P n that is bicontiguous with respect to (P n ). From the literature we know that then NAMFL(U 1 ) cannot be equivalent to NAMFL(U 2 ), compare Kabanov and Kramov (1994, 1998), Klein (2000), Klein and Schachermayer (1996a, 1996b). In Klein (2005) it was shown directly that NMFL(U 2 ) is equivalent to NFL. Lemma 2.6 below is a version of this result that is slightly stronger, as it also includes the equivalence to a uniform version of NMFL(U 2 ). Define for each ε > 0 the set (2.1) D ε = {w L (P ) : 0 w 1 and E P [w] ε}. It is clear that P (w ε 2 ) ε 2 for each w Dε. Definition 2.5. There is uniformly no market free lunch for U 2, NMFL (U 2 ), if for each ε > 0 there exists δ > 0, u U 2 and P P such that for all w D ε sup E P [u(f w)] < u(0) δ. f C Let us discuss the difference between MFL(U 2 ) and MFL (U 2 ). If there is a MFL (U 2 ) then all agents agree on the fact that a strictly positive profit can be made (described by the set D ε ). They do not agree on the way how they can reach the profit. That is, there does not have to be one h L ++ representing the free lunch but for each agent there is some w D ε that is a free lunch (and this means that with positive probability the strictly positive part of elements of D ε is at least ε 2 for all agents). On the other hand if there is a MFL(U 2 ) the agents do not only agree that a profit can be made but on the special choice h L ++ which is the free lunch.

6 6 I. KLEIN The following lemma shows that NMFL(U 2 ) does indeed hold uniformly (that is, NMFL(U 2 ) is equivalent to NMFL (U 2 )) and, moreover, both notions are equivalent to NFL. This is called uniform as the choice of u U and P P in the definition of NMFL does only depend on ε but not on the special choice of w D ε. Note that the δ in the definition has nothing to do with the uniformity, it is just a technical detail which will be useful in the large financial market case and which is put here because of the symmetry of the definitions for small and large financial markets. Lemma 2.6. The following conditions are equivalent (i) NMFL(U 2 ) (ii) NMFL (U 2 ) (iii) NFL. Proof. (ii) (i) is obvious, (i) (iii) is as in the proof of Proposition 3.2 in Klein (2005) and (iii) (ii) is an easy adaption of the proof of in Proposition 3.2 of Klein (2005) (which says NFL NMFL(U 2 )) using the fact that the unit ball of L is weak star compact. The uniform version of NMFL(U 2 ) was not necessary in the proof of Frittellis version of the FTAP (Theorem 2.4 (c)), but if one wants to prove an analogous result for a large financial market it will turn out that exactly such kind of uniformity is needed. 3. Large Financial Markets and No Market Free Lunch In a large financial market a sequence of small market models is considered, that is, a sequence of semimartingales (S n ) where S n is based on (Ω n, F n, (F n t ), P n ). The interpretation of the superscript n in expressions such as K n, C n, M n, P n, D ε,n etc. is then obvious. Throughout the paper the assumption (3.1) M n P n, for all n N is made. This ensures that NA, NFLVR and NFL hold on all the small markets and, moreover, NMFL(U) holds for U = U 0, U 1, U 2 on each small market. The following notation is used: (Q n ) (P n ) means that the sequence of probability measures (Q n ) is contiguous with respect to the sequence of probability measures (P n ) and vice versa. Contiguity is used as a concept of absolute continuity of probability measures in a uniform way for sequences of probability measures. For a precise statement see Lemma 7.10 of part D of the appendix. Though any form of no arbitrage holds on any of the small markets, there is still the possibility of various approximations of an arbitrage profit by trading on the sequence of small markets, compare Chapters 5 and 6 and Kabanov and Kramkov (1994, 1998), Klein and Schachermayer (1996a and 1996b), Klein (2000 and 2003). Let us now define an asymptotic concept of no market free lunch for the large financial market which is inspired by Definition 2.2:

7 MARKET FREE LUNCH AND LARGE MARKETS 7 Definition 3.1. There is no asymptotic market free lunch for U, NAMFL(U), if for every ɛ > 0 and every sequence w n D ε,n there is u U and a sequence P n P n with ( P n ) (P n ) such that lim sup n sup f n C n E P n[u(f n w n )] < u(0). So, if there is an asymptotic market free lunch then all agents (represented by a utility u U and a sequence of measures ( P n )) regard the same sequence w n D α,n (for a fixed α > 0) as a kind of an asymptotic free lunch. Indeed, in the limit they all are able to hedge the claim according to their beliefs and preferences (at least on a subsequence of markets), as lim sup n sup f n C n E P n[u(f n w n )] u(0) for all P n P n with ( P n ) (P n ) and all u U. The α is fixed, so the profit does not disappear in the limit: Remark 3.2. For a sequence (w n ) such that w n D α,n for all n N the following holds: if (P n ) ( P n ) then there is β > 0 such that w n D β ( P n ) for all n N. Indeed, it is clear that P n (w n α 2 ) α 2 for each wn D α,n. By Lemma 7.10 there is δ > 0 such that P n (w n α 2 ) δ, for all n, and therefore E P n[wn ] α 2 δ =: β. This shows that w n D β ( P n ). Note that in the large financial market case the sequence of sets D ε,n plays the role of L ++ of Definition 2.2; this ensures that the strictly positive part of the functions does not disappear for n. 4. A FTAP for Large Financial Markets based on No Asymptotic Market Free Lunch A FTAP for a large financial market is an equivalence result of a notion of no asymptotic arbitrage type and the existence of a sequence of measures (Q n ) such that Q n M n P n and moreover (Q n ) is uniformly equivalent with respect to the sequence (P n ), that is, (Q n ) (P n ) (this is the analogue of an equivalent martingale measure for the large financial market, compare Lemma 7.10). In the case of (locally) bounded S n this means that there exists a sequence (Q n ) of equivalent (local) martingale measures such that (Q n ) (P n ). In the unbounded case it implies the existence of a sequence (Q n ) of equivalent sigma martingale measures such that (Q n ) (P n ) (this is an easy consequence of the fact that Mσ n is dense in M n for the variation topology, for all n). In the case of continuous processes the economically intuitive notion no asymptotic free lunch with bounded risk (NAFLBR), for the definition compare Chapter 6, is necessary and sufficient, Klein (2003). However, in the general (that is, non-continuous) case NAFLBR is not sufficient. One needs the stronger concept no asymptotic free lunch (NAFL), which is the generalization to large financial markets of the concept NFL by Kreps and was introduced in Klein (2000). The definition of NAFL is recalled in Chapter 5. NAFL gives a version of the FTAP for large financial markets in full generality. Unfortunately the concept NAFL is of a very technical and unintuitive nature, it involves generalized sequences and subtle descriptions of Mackey neighbourhoods of 0. Therefore it is of interest if there is a notion that gives the same general result but which can be interpreted in an easier way. It turns out that one can prove a theorem in full generality using NAMFL (U 2 ), a uniform version of NAMFL(U 2 ). This

8 8 I. KLEIN is the large financial market analogue of Frittelli s result. Moreover NAMFL (U 2 ) is fairly intuitive, economically reasonable and less technical than NAFL. But why is it admissible to use the uniform version of no asymptotic market free lunch? One can argue that the notions NMFL(U 2 ) and NMFL (U 2 ) are equivalent on a small market, so it seems plausible to use a uniform version of no asymptotic market free lunch in order to prove a FTAP for large financial markets. In Chapter 5 the gap will be closed, I will show there that the use of the uniform notion is indeed justified as NAMFL (U 2 ) is equivalent to NAMFL(U 2 ). Moreover this gives an analogue result to Lemma 2.6 for large financial markets, as it turns out that both notions are equivalent to no asymptotic free lunch (NAFL). Observe that Definition 4.1 below is a uniform version of NAMFL(U 2 ) and it is the exact analogue of NMFL (U 2 ) (compare Definition 2.5) for a sequence of markets. In the following some facts about Orlicz spaces L F (P ) and Young functions are used. The set of all Young functions is denoted by Y. All relevant definitions and results are assembled in the Appendix (Chapter 7). Definition 4.1. There is uniformly no asymptotic market free lunch with respect to U 2 on the large financial market, NAMFL (U 2 ), if for every ɛ > 0 there is δ > 0, u U 2 and a sequence P n P n with ( P n ) (P n ) such that for every sequence w n D ε,n lim sup sup E P n[u(f n w n )] < u(0) δ n f n C n Similarly as in the small market case, this is called uniformly as the u and the ( P n ) in the NAMFL condition depends only on ε and not on the special choice of a sequence w n D ε,n ). On the other hand, in the NAMFL condition for each sequence w n D ε,n, there is some u U 2 and some sequence ( P n ). It will become clear that, in contrast to the small market case, the uniformity is crucial when one is dealing with large financial markets. In the proof we will have to use that the u and ( P n ) only depend on ε. Theorem 4.2 (A version of the FTAP for large financial markets). NAMFL (U 2 ) there exists a sequence of measures Q n M n P n with (Q n ) (P n ). Proof of Theorem 4.2. ( ) Let (Q n ) (P n ), fix ε > 0 and take u(x) = x. Take an arbitrary w n D ε,n. Then E P n[w n ] ε and, as w n D ε,n and (P n ) (Q n ), E Q n[w n ] > δ for some δ > 0 that depends only on ε (compare Remark 3.2). So, as Q n M n, E Q n[f n ] 0 and hence NAMFL (U 2 ). sup E Q n[u(f n w n )] = sup E Q n[f n ] E Q n[w n ] < δ, f n C n f n C n ( ) Let ε > 0 be fixed. By NAMFL (U 2 ) there is δ > 0, ũ U 2 and a sequence P n P n with ( P n ) (P n ) such that for every sequence A = (A n ) with A n F n and P n (A n ) ε (that is, 1I A n D ε,n ) it holds that lim sup sup E P n[ũ(f n 1I A n)] < ũ(0) 3δ. n f n C n

9 MARKET FREE LUNCH AND LARGE MARKETS 9 Without loss of generality let ũ(0) = 0 (just replace ũ(x) by ũ(x) ũ(0) and the same holds). By Lemma 7.4 for δ > 0 as above there is u U 2 such that u(x) ũ(x) + δ and there is a Young function F such that u is defined as u F in (7.2). It is clear that for this u (4.1) lim sup n sup E P n[u(f n 1I A n)] < u(0) 2δ = 2δ 0 = u(+ ). f n C n So the assumptions of Theorem 7.14 are fulfilled for every n N ε,a for G = C n, N = M n, w = 1I A n and u. This implies that there exists Q An M n and λ An > 0 such that, for all n N ε,a, (4.2) E P n[ λ An dq An d P n where v(y) = sup x R (u(x) xy). 1I A n + v(λ An dq A n d P )] = sup E n P n[u(f n 1I A n)] < δ, f n C n Claim 1: There is N < such that N ε,a N for all sequences A as above. Indeed, otherwise for each k there is a sequence (A n k ) such that for some n k k (4.3) sup f n k C n k E n P k [u(f n k 1I n A k )] δ. k Define a new sequence B k = A n k k, then still P n k (B k ) ε. For the sequence (B k ) (4.3) gives a contradiction to (4.2) verifying Claim 1. From now on fix the N as above. Claim 2: There exists λ 0 > 0 and λ 1 > 0 such that for all sequences (A n ) as above λ 0 λ A n λ 1, for all n N. In particular λ 0 = δ. By definition clearly v(y) u(0) = 0 for all y. Hence, as Q An (A n ) 1 and by (4.2) and therefore λ A n > δ =: λ 0. λ A n λ A nq An (A n ) + v(λ An dq An ) < δ, As for the second assertion, note that the definition of u implies that v is a Young v(y) function (Lemma 7.5). By Lemma 7.3 lim y y = + and therefore, a fortiori, lim y (v(y) y) = +. As 0 Q An (A n ) 1, by Jensen s inequality and by (4.2): d P n λ An + v(λ An ) = λ An + v(e P n[λ An dq An ]) d P n λ An Q An (A n ) + E P n[v(λ An dq An )] < δ. As v(λ An ) λ An < δ there exists λ 1 > 0 (depending only on v and therefore on ε but not on the sequence A) such that λ An λ 1, this proves Claim 2. d P n

10 10 I. KLEIN So far I have shown, that (4.4) λ An Q An (A n ) + E P n[v(λ An dq An )] < δ, for all n N, and that δ λ An λ 1, where v(y), δ > 0, λ 1 > 0 and N only depend on ε but not on the special choice of the sequence A n with P n (A n ) ε. (4.4) implies that there exists γ > 0 (depending only on ε) such that Q An (A n ) > γ for all n N. Indeed, as v 0 δ > λ An Q An (A n ) + E P n[v(λ An dq An which implies that Q An (A n ) > δ λ An δ λ 1 =: γ. d P n d P n )] λ An Q An (A n ), Define now G(y) = v(δy) λ 1 δ. By Lemma 7.5 v Y and so, clearly, G Y as well. Claim 3: dqan d P K G ( P n ) = {f L n G ( P n ) : f G 1}, for each sequence (A n ) as above. G Y and so strictly increasing. Hence, as δ λ An and by (4.4) (4.5) E P n[g( dqan d P n )] 1 So dqan d P n λ E 1 δ P dq A n n[v(λan d P n n )] < δ+λa Q A n (A n ) λ 1 δ δ+λ1 λ 1 δ = 1, K G ( P n ) for each n N and each sequence (A n ) as above. This proves Claim 3. Let us recapitulate what we found. For fixed ε > 0 there is a sequence ( P n ) (with ( P n ) (P n ) and P n P n ) and γ > 0 and N N such that for all n N the following holds: for all A n F n with P n (A n ) ε there is Q An M n with (a) Q An (A n ) > γ and (b) Q An P n, where P n = M n {Q P n dq : d P K G ( P n )}. Note that P n is n a convex set of P n absolutely continuous probability measures. Apply Proposition 7.15 for each n N and for the convex set P n of P n absolutely continuous probability measures. This gives for each ε > 0 a sequence (Q ε,n ) n N such that Q ε,n (B n ) µ := ε2 γ 2 for each B n with P n (B n ) 4ε. Moreover Q ε,n P n (note that by Lemma 7.12 this implies that (Q ε,n ) n N ( P n ) n N ) for each ε). Define, for ε = 2 j, j = 1, 2,..., and for n N, Q n = j=1 2 j Q 2 j,n. Claim 4: The sequence (Q n ) n N satisfies: 1) Q n M n P n 2) (Q n ) n N ( P n ) n N and hence (Q n ) n N (P n ) n N. The proof can be found in Klein (2000), for the convenience of the reader it is recalled in the Appendix (Lemma 7.13). Now, for each 1 n N 1, take an arbitrary Q n M n P n. As Q n P n for n = 1,..., N 1 these Q n can be used to complete the sequence and it still holds that (Q n ) n 1 (P n ) n 1. This finishes the proof.

11 MARKET FREE LUNCH AND LARGE MARKETS Characterizations of Various No Asymptotic Arbitrage Conditions in Terms of No Market Free Lunch The results of this chapter are in the style of Frittelli (2004), compare Theorem 2.4 (a) and (b). Let us briefly recall some concepts of asymptotic arbitrage that appear in the literature. Definition 5.1. On the large financial market there is (1) an asymptotic arbitrage of first kind (AA1) if there are c k > 0 with c k 0 and a sequence ξ k c k K n k 1 such that lim k P n k (ξ k L k ) > 0, where L k > 0 tends to infinity, (2) an asymptotic arbitrage of second kind (AA2) if there is a sequence ξ k K n k 1 and α > 0, such that lim k P n (ξ k α) = 1, (3) a strong asymptotic arbitrage (SAA) if there are c k > 0 with c k 0 and a sequence ξ k c k K n k 1 and α > 0, such that lim n P n k (ξ k α) = 1. A large financial market satisfies no asymptotic arbitrage of first kind (NAA1), no asymptotic arbitrage of second kind (NAA2), no strong asymptotic arbitrage (NSAA) if there does not exist a sequence as in (1), (2), (3), respectively. There are various results relating the notions NAA1 and NAA2 to the existence of a sequence of martingale measures (Q n ) which is contiguous with respect to (P n ) in one direction. Indeed, NAA1 (P n ) is contiguous with respect to the sequence of upper envelopes of the measures Q n M n P n whereas NAA2 the sequence of lower envelopes of the measures Q n M n P n is contiguous with respect to (P n ), see Kabanov and Kramkov (1998) or in a different formulation Klein and Schachermayer (1996a, 1996b). These results can be considered as one sided versions of the FTAP. Definition 5.2. There is an asymptotic free lunch with bounded risk (AFLBR) on a large financial market if there is α > 0 and a sequence ξ k K n k 1 such that (i) P n k (ξ k α) α for all k N and (ii) lim k P n k (ξ k < ε) = 0 for all ε > 0. The large financial market satisfies no asymptotic free lunch with bounded risk (NAFLBR) if there is no such sequence as above. As already mentioned in Chapter 4 in the case of continuous processes NAFLBR is necessary and sufficient for the existence of a sequence of measures (Q n ) with Q n M n P n and (Q n ) (P n ). In general NAFLBR is not sufficient for this condition, compare Chapter 6 for the stronger notion no asymptotic free lunch (NAFL). Remark 5.3. If there is a sequence ξ k K n k realizing AA1, AA2, SAA, AFLBR, respectively, then there is a sequence f k C n k with the respective properties as ξ k. Indeed, note that g k,l = ξ k l ξ k in probability and g k,l = ξ k (ξ k g k,l ) C n k. Define some more sets of utility functions, compare also Definition 2.3.

12 12 I. KLEIN Definition 5.4. U 0 = {u U 0 : 0 interior(d(u))} U 3 = {u U 0 : u is concave and finite-valued on [ 1, )} U 4 = {u U 0 : u( 1) > } U 5 = {u : R R { }} The following relations hold. Note that in the list below there is no result on NAMFL(U 2 ), as this is of a more technical nature. The whole Chapter 6 will be devoted to this. Theorem 5.5. a) NAMFL(U 0 ) NSAA. b) NAMFL(U 1 ) NAA1. c) NAMFL(U 3 ) NAFLBR. d) NAMFL(U 4 ) NAA2. e) Neither NSAA nor NAA2 imply NAMFL(U 5 ) which is the weakest possible form of NAMFL(U). Proof of Theorem 5.5. a) ( ) Suppose there is a SAA. By definition and by Remark 5.3 this implies that there is f k C n k with f k c k and there is α > 0 such that lim k P n k (A k ) = 1, where A k = {f k α}. Define w k = α1i Ak, then w k D α 2,n k for k large enough. Let u U 0, then there exists ε > 0 such that u(x) > for all x > ε. So for k large enough (such that c k < ε) it follows that, for all ( P n ) (P n ) with P n M n P n, sup E n P k [u( f k w k )] E n P k [u(f k w k )] f k C n k E P n k [u(f k 1I A c k )] u( c k ) P n k (A c k) + u(0) P n k (A k ), where A c k = Ωn k \ A k. Passing to the limit this gives a contradiction to NAMFL(U 0 ) as ( P n ) (P n ) implies that P n k (A k ) 1. (b) ( ) Suppose there is an AA1, then there are f k C n k with f k c k for c k 0. Moreover there is α > 0 such that P n k (A k ) α for all k large enough, where A k = {f n k α}. Define w k = α1i Ak, then it is clear that w k D α2,n k. Let u U 1. Because of the left continuity at 0, for each k, there is δ k > 0 such that u(x) > u(0) 1 k for δ k x 0. Choose a subsequence again denoted by k such that f k w k δ k 1I A c k, whence f k w k δ k. Define f k = f k (f k w k ) + C n k. Then 0 f k w k δ k. Therefore u( f k w k ) u(0) 1 k. This gives a contradiction to NAMFL(U 1). ( ) Suppose there is an AMFL(U 1 ). Define u k U 1 by { for x < 1 k u k (x) = 0 for x 1 k.

13 MARKET FREE LUNCH AND LARGE MARKETS 13 Then there is w k D α,n k and f k C n k such that E P n k [u k (f k w k )] 2 k. This implies that P n k (f k w k 1 k ) = 1. For k large enough that 1 k < α 4 it is clear that P n k (f k α 4 ) P n k (w k α 2 ) α 2, giving an AA1, take for instance f k = kf k C n k. (c) Suppose there is an AFLBR, then there are f k C n k such that f k 1. Moreover there is α > 0 such that P n k (f k α) α and lim k P n k (f k < ε) = 0, for all ε > 0. Take any u U 3, then there exists δ n > 0 such that u(x) > u(0) 1 n for δ n x 0. Choose δ k 0 and a subsequence of f k again denoted by k such that P n k (A k ) = 1 ε k, where A k = {f k δ k } and lim k ε k = 0. Let w k = α1i {f k α} and define f k = f k (f k w k ) + C n k. Then, for every ( P n ) (P n ), E P n k [u( f k w k )] = E P n k [u( (f k w k ) )] (u(0) 1 k ) P n k (A k ) + u( 1) P n k (A c k). Passing to the limit for k, this is a contradiction to NAMFL(U 3 ) as P n k (A c k ) 0 for every ( P n ) (P n ). ( ) Suppose now there is an AMFL(U 3 ). Define u k U 3 by for x < 1 u k (x) = e k2x + 1 for 1 x < 0 0 for x 0. By assumption there are w k D α,n k and f k C n k with E P n k [u(f k w k )] 2 k. This implies that P n k (f k 1) = 1 for all k. Claim: lim k P n k (A k ) = 0, where A k = {f k w k < 1 k }. Suppose to the contrary that there exists β > 0 such that for a subsequence P n k (A k ) β. Then E P n k [u k (f k w k )] = E P n k [u k (f k w k )1I Ak ] + E P n k [u k (f k w k )1I A c k ] < ( e k + 1)β, which is a contradiction for k large enough. But this gives an AFLBR. Indeed, P n k (f k 1 k ) P n k (A c k ) 1 for k. Moreover, for k large enough, P n k (f k α 4 ) P n k (w k α 2 ) P n k (A k ) α 4. (d) ( ) Suppose there is an AA2. This implies that there is α > 0 and f k C n k such that f k 1 and lim k P n k (A k ) = 1 where A k = {f k α}. Define w k = α1i Ak. Then, w k D α 2,n k for all k large enough. For all ( P n ) (P n ) and for all u U 4, E P n k [u(f k w k )] E P n k [u((f k w k )1I A c k )] u( 1) P n k (A c k) + u(0) P n k (A k ). Passing to the limit for k gives contradiction to NAMFL(U 4 ) as P n k (A k ) 1 for all ( P n ) (P n ).

14 14 I. KLEIN (e) Consider the following easy example. Let 1 > α > 0 and define for each n a random variable f n on a suitable base (Ω n, F n, P n ) by { 1 on A n, where P n (A n ) = α w n = 0 on B n, where P n (B n ) = 1 α The process (S n t ) t {0,1} is defined as S n 0 = 0 and S n 1 = w n, F n 0 is trivial and F n 1 = σ(w n ). It is obvious that this large financial market satisfies NAA2 and NSAA, as there cannot be a subsequence and β > 0 such that lim P n (w n β) = 1. On the other hand there is an AMFL(U 5 ) (and hence an AMFL(U) for every subclass U U 5 ). Indeed, obviously, w n = (S n 1 S n 0 ) C n and, moreover w n D α,n. Hence, for every u U 5 and every ( P n ) (P n ), sup E P n[u(f n w n )] E P n[u(w n w n )] = u(0). f n C n 6. A Characterization of No Asymptotic Free Lunch in Terms of No Market Free Lunch/ No Asymptotic Market Free Lunch with Respect to U 2 Holds Uniformly Recall the definition of NAFL of Klein (2000). Let F be a Young function and define K F (P n ) = {f L F (P n ) : f F 1} compare (7.3). By Remark 7.8 K F (P n ) is a weakly compact subset of L 1 (P n ). As in (7.4) define the polar V F (P n ) = (K F (P n )) = {g L (P n ) : E P n[gh] 1 for all h K F (P n )}. The notations K F,n and V F,n are used for K F (P n ) and V F (P n ), respectively, if it is clear which measure P n is meant. Definition 6.1. There is NAFL if for each ε > 0 there exist F Y such that C n (D ε,n + V F,n ) = for all n N, where D ε,n is defined as in (2.1). Definition 6.1 says that the set C n is, for each ε > 0, separated from D ε,n by some Mackey neighbourhood V F,n of 0. See part C of the Appendix for an explanation why a fundamental system for the Mackey neighbourhoods of 0 (in L (P n )) can be given by the sets V F,n. Therefore NAFL is a direct translation of NFL to a sequence of probability spaces. Indeed, if NAFL holds, it is not possible to approximate a strictly positive gain by elements of the sequence of sets (C n ) in a Mackey (or, equivalently, weak star) sense. Note that in Klein (2000) the weakly compact subsets of L 1 were described in a slightly different way. However, a quick check of all proofs in Klein (2000) shows that the sets K F,n can be used instead of the sets K ϕ,n which are used there. I will now prove the analogue of Lemma 2.6 for large financial markets. This result shows that in Chapter 4 it is justified to use the uniform version of the no market free lunch condition with respect to U 2. Moreover it gives a characterization in terms of no market free lunch of NAFL.

15 MARKET FREE LUNCH AND LARGE MARKETS 15 Proposition 6.2. The following conditions are equivalent (i) NAMFL(U 2 ) (ii) NAMFL (U 2 ) (iii) NAFL. It is now clear that: Corollary 6.3. NAMFL(U 2 ) there exists a sequence of Q n M n P n such that (Q n ) (P n ). The following lemma shows that the condition NAFL does not depend on the choice of a sequence ( P n ) such that P n P n and ( P n ) (P n ). Lemma 6.4. Suppose that (f F ) F Y with f F C n(f ) is an asymptotic free lunch with respect to (P n ). Then (f F ) F Y is an asymptotic free lunch with respect to any ( P n ) (P n ) where P n P n. Proof. By assumption there is α > 0 such that for all F Y there is f F C n(f ) which can be written as f F = w F +g F, where w F D α (P n(f ) ) and g F V F (P n(f ) ). I will show that (f F ) is an asymptotic free lunch for every sequence of probability measures ( P n ) such that ( P n ) (P n ). Indeed, as (P n ) ( P n ) there exists β > 0 such that w F D β ( P n ) (Remark 3.2). As ( P n ) (P n ) Lemma 7.12 implies that there exists ψ Y such that, for all n, d P n dp n K ψ (P n ). Take now an arbitrary F Y. Let G be the complementary Young function of F. Moreover let ϕ be the complementary Young function of ψ. Define G(y) = (ϕ 2 G)(4y), it is clear that G Y. Let F be the complementary Young function of G. Claim: the g F V F (P n(f ) ), which is given by AFL for the Young function F, is in V F,n(F ) ( P n(f ) ). Set n( F ) := n(f ), then this readily implies AFL for the sequence ( P n ). Indeed, let n = n(f ). By Lemma 7.9 V F (P n ) 2K G (P n ) L (P n ). As g F V F (P n ): This gives G(2 g F ) Lϕ(P n ) 1 2. implies E P n[ϕ(2 G(2 g F ))] = E P n[g( gf 2 )] 1. n d P Moreover, as dp n K ψ (P n ), Lemma 7.2 (iii) E P n[ G(2 g F )] = E P n[ d P n G(2 g F dp )] 2 d P n n dp n Lψ (P n ) G(2 g F ) Lϕ(P n ) 1. But this means that g F 1 2 K G( P n ) L ( P n ) V F,n ( P n ), again by Lemma 7.9. Proof of Proposition 6.2. It is clear that (ii) (i). (iii) (ii) Suppose there is an AMFL (U 2 ). There is ε > 0 such that for all δ > 0, for all u U 2 and for all ( P n ) (P n ) (so in particular for the original sequence (P n )) there exists a sequence w n D ε,n such that lim sup n sup f n C n E P n[u(f n w n )] u(0) δ.

16 16 I. KLEIN I will show that there is AFL. Indeed, let G Y arbitrary and let K G (P n ) as usual, compare (7.3). Let F be the complementary Young function of G and define { F ( 2x) for x 0 u F (x) = 0 for x > 0. By AMFL (U 2 ) for δ = 1 2, for uf U 2 and for the sequence (P n ) there is n = n(f, δ) and f n C n and w n D ε,n such that E P n[u F (f n w n )] 1. Set g n = (f n w n ), then clearly E P n[f (2g n )] 1 and therefore g n F 1 2. By Lemma 7.2 (iii), for all h K G (P n ), E P n[g n h] E P n[ g n h ] 2 g n F h G 1, which means that g n (K G,n ) = V G,n. This implies that there is an asymptotic free lunch. Indeed, define f n = f n (f n w n ) + C n, then f n = w n g n and so f n (D ε,n + V G,n ) C n. This can be done for every G Y, which means AFL. (i) (iii) Suppose there is an AFL, that is, there is α > 0 such that for each F Y there is f F = w F + g F C n(f ), where w F D α,n(f ) and g F V F,n(F ). By Lemma 6.4 this is an AFL for an arbitrary sequence ( P n ) (P n ) (from now on fix ( P n ) and assume that the sets D α,n and V F,n are defined with respect to P n ). Take now any ũ U 2, w.l.o.g assume that ũ(0) = 0. By Lemma 7.4, for all k, there is u k U 2 with u k (x) ũ(x) + 2 k and there is F k Y such that u k := u F k defined as in (7.2). Moreover, because of the definition of C n(f ) which allows to subtract positive elements of L, one can assume that the g F V F,n (where n = n(f )) satisfy that g F 0. It is clear that F k (x) := kf k (2x) Y as well. Let G k be the complementary Young function of F k. Take the g k := g Gk V Gk,n(G k) given by the AFL. As, by Lemma 7.9, V Gk,n k 2K F k,n k the following holds: E n P k [ F k ( gk 2 )] = E P n k [kf k (2 gk 2 )] 1. This gives E P n k [F k ( g k )] 1 k and hence E P n k [u k (g k )] 1 k. So, for ũ uk 2 k, lim sup k E n P k [ũ(f k w k )] lim sup(e n P k [u k (f k w k )] 2 k ) = k lim sup(e n P k [u k (g k )] 2 k ) lim ( 1 k k k 2 k ) = 0 = ũ(0). This gives an AMFL(U 2 ). Indeed, for every P n and every ũ U 2 with ũ(0) = 0, lim sup n sup E P n[ũ(f n w n )] 0 = ũ(0). f n C n

17 MARKET FREE LUNCH AND LARGE MARKETS Appendix A. Orlicz spaces and Young functions. I follow the lines of of Kusuoka (1993). Definition 7.1. F : [0, ) [0, ) is a Young function if (i) F is continuously differentiable, (ii) F (0) = F (0) = 0, (iii) F is strictly increasing and lim t F (t) =. Note that the definition of a Young function is not always as given in Definition 7.1, the differentiability is not required in general, see Rao and Ren (1991). However, as I follow closely the lines of Kusuoka (1993), I stick to his definition of a Young function which is as given above. Moreover it is convenient to use the differentiability in some places. The class of all Young functions is denoted by Y. For F Y there is a complementary Young function G Y defined as follows G(y) = y 0 (F ) 1 (t)dt, y 0, where (F ) 1 is the inverse function of F. By definition it is clear that the complementary Young function of the complementary Young function G of F is again the original F. The complementary Young function satisfies (7.1) G(y) = max(xy F (x)). x 0 For each F Y let L F (P ) = {f L 0 (Ω, F, P ) : E P [F (a f )] < for some a > 0}. Define a norm. F on L F by f F = inf{a > 0 : E[F ( 1 a f )] 1}. The space L F is called Orlicz space. Let L 0 F (P ) = {f L 0 (Ω, F, P ) : E P [F (a f )] < for all a > 0}. Lemma 7.2. Let F Y and G its complementary Young function. Then (i) L F is a Banach space with norm. F. (ii) If g L G then Φ : L 0 F R given by Φ(f) = E[fg] is a continuous linear functional and 1 2 g G Φ (LF ) 2 g G. (iii) If f L F and g L G the following relation holds E[ fg ] 2 f F g G.

18 18 I. KLEIN B. Some Properties of Young Functions and the Connection to U 2. Lemma 7.3. Let F Y. Then the function F (x) x F (x) F (x) lim x 0 x = 0 and lim x x = +. is strictly increasing. Moreover Proof. Let 0 x 1 < x 2 and define λ = x1 x 2 < 1. The strict convexity of F and the fact that F (0) = 0 imply that F (λx 2 ) < λf (x 2 ), which shows the monotonicity of F (x) x. Moreover, by de l Hospital, F (x) F lim x 0 x = lim (x) x 0 1 = F F (x) F (0) = 0 and lim x x = lim (x) x 1 =. Let F Y. Define a function u F U 2 as follows { F ( x) for x 0 (7.2) u F (x) = 0 for x > 0 Lemma 7.4. Let u U 2 with u(0) = 0. For each ε > 0 there is F ε Y such that u F ε(x) ε u(x) for all x R, where u F ε is defined as in (7.2). Proof. The only problem occurs if the left derivative at 0 of u is strictly positive. Let δ be such that u( δ) ε. Define u ε as above and such that u ε (x) u(x) for x < δ (it is clear that there is F ε Y such that this works.) Moreover, one can choose u ε such that 0 u ε (x) u(x) on the interval [ δ, 0]. So, u ε (x) u(x) u( δ) ε for x [ δ, 0]. It is clear that u ε U 2. Lemma 7.5. Let F Y. If u F is defined as in (7.3) then the convex dual v(y) = sup(u F (x) xy) y 0, x R is in Y. In particular, v is the complementary Young function of F. Proof. By the definition of u F v(y) = sup(sup( xy), sup( F ( x) xy)) x>0 x 0 = sup( F ( x) xy) x 0 = sup( F (x) + xy) x 0 This is the definition of the complementary Young function, compare (7.1). C. Orlicz Spaces and the Mackey Topology. Define the unit ball K F (P ) of the Orlicz space L F (P ): (7.3) K F (P ) = {f L F (P ) : f F 1}.

19 MARKET FREE LUNCH AND LARGE MARKETS 19 Lemma 7.6. The set K F (P ) is closed in L 1 (P ) and uniformly integrable. In particular, sup h K F (P ) E[ h 1I { h κ} ] Proof. For h K F and κ > 0 h κ F (κ). E[ h 1I { h κ} ] = E[ F ( h ) F ( h )1I { h κ}] κ F (κ) E[F ( h )] κ F (κ), as by Lemma 7.3 F (y) y is strictly increasing. Moreover lim y F (y) y = + and so lim sup κ h K F (P ) E P [ h 1I { h κ} ] lim κ κ F (κ) = 0, this shows uniform integrability. To show closedness in L 1 take h n K F with h n h in L 1, then a subsequence of F ( h n ) (still denoted by n) converges to F ( h ) a.s. By Fatou E[F ( h )] = E[lim F ( h n )] lim inf E[F ( h n )] 1, as F ( h n ) 0. This shows that h K F. Moreover the following result holds, compare Kusuoka (1993). Lemma 7.7. Let A be a uniformly integrable subset of L 1. Then there is a Young function F such that A K F. So, the uniformly integrable subsets of L 1 can be described completely with the help of the sets K F, F Y. Define now the polar of K F : (7.4) V F (P ) = (K F ) = {g L (P ) : E P [gh] 1 for all h K F (P )}. Remark 7.8. Note that K F is a weakly compact convex balanced subset of L 1. Therefore K F = (K F ) = (V F ) by the Bipolar Theorem, compare Horvath (1966). Indeed, convexity is clear and so L 1 closedness implies closedness for the topology σ(l 1 (P ), L (P )) (i.e. the weak topology on L 1 ). Together with uniform integrability this gives compactness for the weak topology, see for example Woytaszczyk (1991). It is clear that K F is balanced. The following description of the Mackey topology of L is used in Chapter 6. The sets V F, for all F Y, give a fundamental system for all Mackey neighbourhoods of 0. Indeed, V F is the polar of the weakly compact subset K F (P ) of L 1 (P ). On the other hand, by Lemma 7.7 it is clear that for each uniformly integrable set in L 1 (P ) there is G Y such that U K G. So the polars of all K F (P ) form a fundamental system of neighbourhoods for the Mackey topology (which is the topology of uniform convergence on all weakly compact sets of L 1 (P )). For the details on the Mackey topology compare Horvath (1966). The following lemma gives a helpful description of the sets V F.

20 20 I. KLEIN Lemma 7.9. Let F Y and G be the complementary Young function of F. Then 1 2 KG,n L (P n ) V F,n 2K G,n L (P n ). Proof. As, by Lemma 7.2, E P n[ hg ] 2 h F g G the first inequality is clear. As for the second inequality let g V F,n. As g L (P n ) it is clear that g defines a continuous linear functional Φ on L 0 F (P n ) via Φ( ) = E P n[g ]. Moreover, as E[gh] 1 for all h K F (P n ), it is clear that Φ (LF (P n )) 1. And so g G 2 Φ (LF (P n )) 2 again by Lemma 7.2. This gives that g 2K G,n L (P n ). D. Contiguity of Sequences of Probability Measures. Let P n and Q n be probability measures on a probability space (Ω n, F n ). The concept of contiguity can be interpreted as uniform absolute continuity. For a precise statement compare Lemma 7.10 below, the easy proof can be found for example in Klein (2000). Lemma Let Q n P n, for all n. Then (Q n ) (P n ) is equivalent to the following condition: for all ε > 0 there is δ > 0 such that, for all n N, Q n (A n ) < ε for all A n F n with P n (A n ) < δ. Moreover, contiguity can be related to uniform integrability. The following result can be found in a more general form in Jacod Shiryaev. Lemma Assume that, for all n N, Q n P n. Then (Q n ) (P n ) if and only if ( dqn dp n P n ) is uniformly integrable, that is, lim sup E P n κ n [ ] dq n dp 1I n dq n = 0. { dp n >κ} Together with Lemma 7.7 this gives the following characterization of contiguity in terms of Young functions/orlicz spaces: Lemma Let Q n P n, for all n. Then (Q n ) (P n ) if and only if there is F Y such that dqn dp n K F (P n ), for all n, where K F (P n ) is defined as in (7.3). Proof. The sets K F (P n ) satisfy the uniform integrability condition: lim sup sup κ n h n K F (P n ) E P n[h n 1I {h n κ}] = 0, compare Lemma 7.6. The rest is clear by Lemma 7.11 above and Lemma 7.7. Lemma Suppose that for each ε > 0 there exists a sequence (Q n,ε ) with Q n,ε M n such that (i) there is δ > 0 such that for A n F n with P n (A n ) ε it holds that Q n,ε (A n ) δ, for all n, and (ii) (Q n,ε ) (P n ).

21 MARKET FREE LUNCH AND LARGE MARKETS 21 Let Q n = j=1 2 j Q n,2 j. Then Q n M n P n, for all n, and moreover (Q n ) (P n ). Proof. Observe that Q n M n P n, since { dqn dp n = 0} j=1 { j dqn,2 dp n < δ j }, whence (i) implies that, for all n, P n [ dqn dp = 0] = 0. n Let us now prove that (Q n ) n 1 (P n ) n 1. By Lemma 7.10 one has to show that for all γ > 0 there exists µ > 0 such that, for all n, Q n (A n ) < γ for all A n F n with P n (A n ) < µ. Let γ > 0 be fixed and N N large enough to guarantee that j=n+1 2 j < γ 2. By (ii) (Q n,2 j ) n 1 (P n ) n 1 for j = 1, 2,..., N, whence for each j there exists µ j > 0 such that for all n: whenever P n (A n ) < µ j it follows that Q n,2 j (A n ) < γ 2. Let now µ = min j N µ j and A n F n be such that P n (A n ) < µ. Then N Q n (A n ) = 2 j Q n,2 j (A n ) + 2 j Q n,2 j (A n ) < γ 2 + γ 2 = γ. j=1 j=n+1 To prove that (P n ) n 1 (Q n ) n 1 observe that (ii) implies that, for all j N, there exists µ j such that for all n: Q n,2 j (A n ) < µ j implies that P n (A n ) < 2 j. Let γ > 0 be fixed and choose N N such that 2 (N 1) < γ. Define µ = 2 2N µ N. Let now A n F n such that Q n (A n ) < µ. Then P n [A n ] = P n [A n < 2 N + P n [ A n E. Two Useful Results. { dq n,2 N dp n < µ N }] { }] + P [A n n dq n,2 N dp µ n N { dq n dp n 2 N µ N }] < 2 N + 2N µ N Q n (A n ) < γ. The following result of Bellini Frittelli (2002, Thm. 2.1 and Cor. 2.2) is based on Rockefellar (1971). Theorem u : R R is non decreasing and concave on R (i.e. u U 2 ). Let P P and G a convex cone with L + G L. Let w L such that sup{e P [u(f + w)] : f G} < u(+ ). Then N = {Q P : dq d P G L 1 ( P )} and sup{e P [u(f + w)] : f G} = min min E Q N λ>0 P [λ dq dq w + v(λ )], d P d P where v(y) = sup x R (u(x) xy), G = {ξ ba(ω, F, P ) : ξ(f) 0 f G} is the polar of G, and ba(ω, F, P ) is the space of all bounded finitely additive measures on (Ω, F) absolutely continuous with respect to P. In Chapter 4 Proposition 1.3 of Klein and Schachermayer (1996) is applied. For the convenience of the reader it is recalled here. Proposition 7.15 (Quantitative Version of the Halmos-Savage Theorem). For fixed ε > 0 and δ > 0 the following statement is true: let M be a convex set of P - absolutely continuous probability measures such that for all sets A F with P(A) > ε there is Q M with Q(A) > δ. Then there is Q 0 M such that Q 0 (A) > ε2 δ 2 for all A F with P (A) > 4ε.