Asymptotic arbitrage in non-complete large financial markets ASYMPTOTIC ARBITRAGE IN NON-COMPLETE LARGE FINANCIAL MARKETSЮ. M. Кабанов и Д. О. Крамков ввели понятие «больших финансовых рын ков». Вместо того, чтобы рассматривать -как это обычно делается в финан совой математике -некоторый случайный процесс S цен акций, заданный на фильтрованном вероятностном пространстве (Q,!F, (^)1 фильтрованных вероятностных пространств. Та кая модель оправдывается тем, что инвестор может делать вклады не на одной, а на нескольких фондовых биржах (в модели Кабанова и Крамкова -на счетном числе).Привычное понятие арбитража тогда можно интерпретировать с помощью понятий асимптотического арбитража, где важно различать между собой два ро да асимптотического арбитража, введенные Кабановым и Крамковым. В случае, когда для каждого п £ N рынок полон (т.е. существует единственная «локально мартингальная» мера Q n для процесса S" на Т п , эквивалентная Р"), Кабанов и Крамков показали, что контигуальность (Р")п>1 относительно (Q n )n^l (со ответственно, наоборот) эквивалентна отсутствию асимптотического арбитража первого (соответственно, второго) рода.В настоящей статье мы распространяем этот результат на случай неполного рынка, когда для каждого п £ N множество эквивалентных локально мартингаль ных мер непусто, но не обязательно одноэлементно. Возникает вопрос, можно ли перенести на этот случай теорему Кабанова и Крамкова, выбирая подходящую последовательность (Q n )n^l эквивалентных локально мартингальных мер.Оказывается, что в части, характеризующей асимптотический арбитраж пер вого рода, теорема может быть непосредственно перенесена на этот случай, однако в части, характеризующей асимптотический арбитраж второго рода, необходимы некоторые изменения. Мы также строим пример, показывающий, что этих изме нений нельзя избежать. Ключевыеслова и фразы: арбитраж, асимптотический арбитраж, контигу альность мер, эквивалентная мартингальная мера, «бесплатный завтрак», «бес платный завтрак с исчезающе малым риском», большие финансовые рынки. Introduction.In this paper we deal with arbitrage possibilities in a large financial market, a concept originally introduced by Kabanov and Kramkov, see [10]. Following these authors we define a large financial market to be a sequence of filtered probability spaces (Q n , T n , (^r")i£R+, P"). On each of these "small" spaces we can trade in d(n) securities, whose price processes are denoted by an R^^-valued [T? )-adapted semimartingale.We suppose that for each "small market" there exists a probability measure Q n on T n that is equivalent to the original measure P n , such that S n is a local Q"-martingale. So the difference between the two kinds of asymptotic arbitrage is that on one hand by risking nearly nothing we can become very rich but onfy^ on a set of positive probability, on the other hand we can win something, maybe very little, with very high probability, but there...

We formulate the notion of "asymptotic free lunch" which is closely related to the condition "free lunch" of Kreps (1981) and allows us to state and prove a fairly general version of the fundamental theorem of asset pricing in the context of a large financial market as introduced by Kabanov and Kramkov (1994). In a large financial market one considers a sequence ("S"-super-"n") "n"=1 -super-∞ of stochastic stock price processes based on a sequence (Ω-super-"n", "F"-super-"n", ("F" "t" -super-"n") "t" is an element of "I"-super-"n" , P-super-"n") "n"=1 -super-∞ of filtered probability spaces. Under the assumption that for all "n" is an element of N there exists an equivalent sigma-martingale measure for "S"-super-"n", we prove that there exists a "bicontiguous" sequence of equivalent sigma-martingale measures if and only if there is no asymptotic free lunch (Theorem 1.1). Moreover we present an example showing that it is not possible to improve Theorem 1.1 by replacing "no asymptotic free lunch" by some weaker condition such as "no asymptotic free lunch with bounded" or "vanishing risk." Copyright Blackwell Publishers, Inc..

In the context of large financial markets we formulate the notion of no asymptotic free lunch with vanishing risk (NAFLVR), under which we can prove a version of the fundamental theorem of asset pricing (FTAP) in markets with an (even uncountably) infinite number of assets, as it is for instance the case in bond markets. We work in the general setting of admissible portfolio wealth processes as laid down by Y. Kabanov [8] under a substantially relaxed concatenation property and adapt the FTAP proof variant obtained in [1] for the classical small market situation to large financial markets. In the case of countably many assets, our setting includes the large financial market model considered by M. De Donno et al. [2] and its abstract integration theory.The notion of (NAFLVR) turns out to be an economically meaningful "no arbitrage" condition (in particular not involving weak- * -closures), and, (NAFLVR) is equivalent to the existence of a separating measure. Furthermore we show -by means of a counterexample -that the existence of an equivalent separating measure does not lead to an equivalent σ-martingale measure, even in a countable large financial market situation.2000 Mathematics Subject Classification. 60G48, 91B70, 91G99 . Key words and phrases. Fundamental theorem of asset pricing, Large financial markets, Emery topology, (NFLVR) condition, (P-UT) property . The authors gratefully acknowledges the support from ETH-foundation.1 completely analogous in the present setting.Proposition A.3. Let X 1 satisfy (NUPBR) and let (X n ) n≥0 ∈ X 1 be any sequence of semimartingales. Then (X n ) satisfies the (P-UT) property.The following proposition is a reformulation of [18, Proposition 1.10] and corresponds to [1, Proposition 5.2].Proposition A.4. Let (X n ) n≥0 be a sequence of semimartingales with X n 0 = 0, which converges uniformly in probability to X. Assume furthermore the (P-UT) property for this sequence and consider decompositions of form (7.1) for (X n ) and X. Then there exists some C > 0 such that M n,C → M C andX n,C →X C in the Emery topology and B n,C → B C uniformly in probability.

No abstract

Abstract. A general proof of the Dybvig-Ingersoll-Ross Theorem on the monotonicity of long forward rates is presented. Some inconsistencies in the original proof of this theorem are discussed.