Projective system approach to the martingale characterization of the absence of arbitrage

The equivalence between the absence of arbitrage and the existence of an equivalent martingale measure fails when an infinite number of trading dates is considered. By enlarging the set of states of nature and the probability measure through a projective system of topological spaces and Radon measures, we characterize the absence of arbitrage when the time set is countable

Measuring arbitrage profits in imperfect markets

In this paper we introduce a measure testing the degree of efficiency in securities markets with bidask spreads. The measure tests relative arbitrage profits when there are transaction costs on the prices and payoffs of the assets. Moreover, we prove that the measure is the minimum of the measures of efficiency in all frictionless markets where the prices and payoffs lie between the bid and the ask prices and payoffs respectively. In particular, we fmd that the model is arbitrage-free if and only if there exist convex combinations of the bid and the ask prices and payoffs such that the corresponding frictionless model is arbitrage-free

Stochastic measures of financial markets efficiency and integration

The notion of integration of different fmancial markets is often related to the absence of crossmarket arbitrage opportunities. Under the appropriated asswnptions and in absence of cross-market arbitrage opportunities, a riskneutral probability measure, shared by both markets, must exist. Some authors have considered this to provide some integration measures when the markets do not share any pricing rule, but always in static (or one period) asset pricing models. The purpose or this paper is to extend the refereed notions to a more general context. This is accomplished by introducing a methodology which may be applied in any intertemporal dynamic asset pricing model and without special asswnptions on the assets prices stochastic process. Then, the integration measures introduced here are stochastic processes testing different relative arbitrage profits and depending on the state of nature and on the date. The measures are introduced in a single fmancial market. When this market is not a global market from different ones, the measures simply test the degree of market efficiency. Transaction costs can be discounted in our model. Therefore, one can measure efficiency and integration in models with frictions. The main results are also interesting form a mathematical pint of view, since some topics of Operational Research are involved. We provide a procedure to solve a vector optimization problem with a non differentiable objective function and prove some properties about its sensitivity

Measuring the degree of fulfillment of the law of one price. Applications to financial markets integration

This paper gives two measures of the degree of fulfillment of the Law of One Price. These measures are characterized by means of saddle point conditions, and are therefore easy to compute in practical situations. Many empirical papers analyze well-Known arbitrage strategies. Our measures present an important advantage over this approach, since we globally focus on the market to find its arbitrage opportunities, without studying special strategies. The developed theory is also applied to markets with frictions, and to study the integration of different financial markets. Our measures are continuous with respect to previous measures in the literature, and seem to be better than them since they compute how much money the agents can win due to the arbitrage opportunities in a financial market, or among different ones

PROJECTIVE SYSTEM APPROACH TO THE MARTINGALE CHARACTERIZATION OF THE ABSENCE OF ARBITRAGE

The equivalence between the absence of arbitrage and the existence of an equivalent martingale measure fails when an infinite number of trading dates is considered. By enlarging the set of states of nature and the probability measure through a projective system of topological spaces and Radon measures, we characterize the absence of arbitrage when the time set is countable.

Sampling-related frames in finite U-invariant subspaces

Recently, a sampling theory for infinite dimensional U-invariant subspaces of a separable Hilbert space H where U denotes a unitary operator on H has been obtained. Thus, uniform average sampling for shift-invariant subspaces of L-2(R) becomes a particular example. As in the general case it is possible to have finite dimensional U-invariant subspaces, the main aim of this paper is to derive a sampling theory for finite dimensional U-invariant subspaces of a separable Hilbert space H. Since the used samples are frame coefficients in a suitable euclidean space C-N, the problem reduces to obtain dual frames with a U-invariance property

Modeling Sampling in Tensor Products of Unitary Invariant Subspaces

The use of unitary invariant subspaces of a Hilbert space H is nowadays a recognized fact in the treatment of sampling problems. Indeed, shift-invariant subspaces of L-2( R) and also periodic extensions of finite signals are remarkable examples where this occurs. As a consequence, the availability of an abstract unitary sampling theory becomes a useful tool to handle these problems. In this paper we derive a sampling theory for tensor products of unitary invariant subspaces. This allows merging the cases of finitely/infinitely generated unitary invariant subspaces formerly studied in the mathematical literature; it also allows introducing the several variables case. As the involved samples are identified as frame coefficients in suitable tensor product spaces, the relevant mathematical technique is that of frame theory, involving both finite/infinite dimensional cases.This work has been supported by the Grant MTM2014-54692-P from the Spanish Ministerio de Economía y Competitividad (MINECO)

The Kramer sampling theorem revisited

The classical Kramer sampling theorem provides a method for obtaining orthogonal sampling formulas. Besides, it has been the cornerstone for a significant mathematical literature on the topic of sampling theorems associated with differential and difference problems. In this work we provide, in an unified way, new and old generalizations of this result corresponding to various different settings; all these generalizations are illustrated with examples. All the different situations along the paper share a basic approach: the functions to be sampled are obtaining by duality in a separable Hilbert space through an -valued kernel K defined on an appropriate domain.This work has been supported by the grant MTM2009–08345 from the Spanish Ministerio de Ciencia e Innovación (MICNN).Publicad

Generalized sampling: from shift-invariant to U-invariant spaces

The aim of this article is to derive a sampling theory in U-invariant subspaces of a separable Hilbert space ℋ where U denotes a unitary operator defined on ℋ. To this end, we use some special dual frames for L2(0, 1), and the fact that any U-invariant subspace with stable generator is the image of L2(0, 1) by means of a bounded invertible operator. The used mathematical technique mimics some previous sampling work for shift-invariant subspaces of L2(ℝ). Thus, sampling frame expansions in U-invariant spaces are obtained. In order to generalize convolution systems and deal with the time-jitter error in this new setting we consider a continuous group of unitary operators which includes the operator U