A new look at some basic concepts in arbitrage pricing theory

The notion of No Free Lunch with Vanishing Risk (or NFLVR in short) w.r.t. admissible strategies depends on the choice of numeraire. Yan introduced the notion of allowable strategy and showed that condition of NFLVR w.r.t. allowable strategies is independent of the choice of numeraire and is equivalent to the existence of an equivalent martingale measure for the deflated price process. In this paper we establish a version of the Kramkov's optional decomposition theorem in the setting of equivalent martingale measures. Based on this theorem, we have a new look at some basic concepts in arbitrage pricing theory: superhedging, fair price, attainable contingent claims, complete markets and etc.     

On the correlation measure of a family of commuting Hermitian operators with applications to particle densities of the quasi-free representations of the CAR and CCR

Let X be a locally compact, second countable Hausdorff topological space. We consider a family of commuting Hermitian operators a(Δ) indexed by all measurable, relatively compact sets Δ in X (a quantum stochastic process over X). For such a family, we introduce the notion of a correlation measure. We prove that, if the family of operators possesses a correlation measure which satisfies some condition of growth, then there exists a point process over X having the same correlation measure. Furthermore, the operators a(Δ) can be realized as multiplication operators in the L²-space with respect to this point process. In the proof, we utilize the notion of ?-positive definiteness, proposed in [Y.G. Kondratiev, T. Kuna, Harmonic analysis on the configuration space I. General theory, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 5 (2002) 201-233]. In particular, our result extends the criterion of existence of a point process from that paper to the case of the topological space X, which is a standard underlying space in the theory of point processes. As applications, we discuss particle densities of the quasi-free representations of the CAR and CCR, which lead to fermion, boson, fermion-like, and boson-like (e.g. para-fermions and para-bosons of order 2) point processes. In particular, we prove that any fermion point process corresponding to a Hermitian kernel may be derived in this way.     

Variational Formulation of a General Equilibrium Model with Incomplete Financial Markets and Numeraire Assets: Existence

We present a general equilibrium model with incomplete financial markets and numeraire assets. We assume that there are 2 periods of time, say today and tomorrow. In period 0, households exchange goods and assets and then consumption takes place; in period 1, one of S possible states of nature occurs. In each of them, assets pay their returns, which are measured in units of a given physical good, i.e., the numeraire commodity; households exchange goods; finally, consumption takes place. We define a consumption, portfolio holding, commodity and asset price vector as an equilibrium vector associated with a given economy, if at those prices and economies households maximize, and market clears. While the existence proof by Geneakoplos and Polemarchakis (Essays in honor of K.J. Arrow, vol 3, Cambridge University Press, Cambridge, pp 65–95, 1986) uses a fixed point argument, we provide an independent existence result in terms of variational inequalities. That approach allows us to get the desired existence result under some different and more general or realistic assumptions than those usually made in the literature.     

Asymptotic connectivity of infinite graphs

We introduce the notion of the asymptotic connectivity of a graph by generalizing to infinite graphs average connectivity as defined by Beineke, Oellermann, and Pippert. Combinatorial and geometric properties of asymptotic connectivity are then explored. In particular, we compute the asymptotic connectivity of a number of planar graphs in order to determine the extent to which this measure correlates with the large-scale geometry of the graph.     

Monotone and cash-invariant convex functions and hulls

This paper provides some useful results for convex risk measures. In fact, we consider convex functions on a locally convex vector space E which are monotone with respect to the preference relation implied by some convex cone and invariant with respect to some numeraire (‘cash’). As a main result, for any function f, we find the greatest closed convex monotone and cash-invariant function majorized by f. We then apply our results to some well-known risk measures and problems arising in connection with insurance regulation.     

On linear complexity of binary lattices,II

The linear complexity is an important and frequently used measure of unpredictably and pseudorandomness of binary sequences. In Part I of this paper, we extended this notion to two dimensions: we defined and studied the linear complexity of binary and bit lattices. In this paper, first we will estimate the linear complexity of a truly random bit (M,N)-lattice. Next we will extend the notion of k-error linear complexity to bit lattices. Finally, we will present another alternative definition of linear complexity of bit lattices.     

Differential Algebras with Banach-Algebra Coefficients II: The Operator Cross-Ratio Tau-Function and the Schwarzian Derivative

Several features of an analytic (infinite-dimensional) Grassmannian of (commensurable) subspaces of a Hilbert space were developed in the context of integrable PDEs (KP hierarchy). We extended some of those features when polarized separable Hilbert spaces are generalized to a class of polarized Hilbert modules and then consider the classical Baker and τ-functions as operator-valued. Following from Part I we produce a pre-determinant structure for a class of τ-functions defined in the setting of the similarity class of projections of a certain Banach *-algebra. This structure is explicitly derived from the transition map of a corresponding principal bundle. The determinant of this map leads to an operator τ-function. We extend to this setting the operator cross-ratio which had previously been used to produce the scalar-valued τ-function, as well as the associated notion of a Schwarzian derivative along curves inside the space of similarity classes of a given projection. We link directly this cross-ratio with Fay’s trisecant identity for the τ-function. By restriction to the image of the Krichever map, we use the Schwarzian to introduce the notion of an operator-valued projective structure on a compact Riemann surface: this allows a deformation inside the Grassmannian (as it varies its complex structure). Lastly, we use our identification of the Jacobian of the Riemann surface in terms of extensions of the Burchnall–Chaundy C*-algebra (Part I) provides a link to the study of the KP hierarchy.     

How much randomness is needed for statistics?

In algorithmic randomness, when one wants to define a randomness notion with respect to some non-computable measure λ, a choice needs to be made. One approach is to allow randomness tests to access the measure λ as an oracle (which we call the “classical approach”). The other approach is the opposite one, where the randomness tests are completely effective and do not have access to the information contained in λ (we call this approach “Hippocratic”). While the Hippocratic approach is in general much more restrictive, there are cases where the two coincide. The first author showed in 2010 that in the particular case where the notion of randomness considered is Martin-Löf randomness and the measure λ is a Bernoulli measure, classical randomness and Hippocratic randomness coincide. In this paper, we prove that this result no longer holds for other notions of randomness, namely computable randomness and stochasticity.     

Introducing validity in fuzzy probability for judicial decision-making

Since the Age of Enlightenment, most philosophers have associated reasoning with the rules of probability and logic. This association has been enhanced over the years and now incorporates the theory of fuzzy logic as a complement to the probability theory, leading to the concept of fuzzy probability. Our insight, here, is integrating the concept of validity into the notion of fuzzy probability within an extended fuzzy logic (FLe) framework keeping with the notion of collective intelligence. In this regard, we propose a novel framework of possibility–probability–validity distribution (PPVD). The proposed distribution is applied to a real world setting of actual judicial cases to examine the role of validity measures in automated judicial decision-making within a fuzzy probabilistic framework. We compute valid fuzzy probability of conviction and acquittal based on different factors. This determines a possible overall hypothesis for the decision of a case, which is valid only to a degree. Validity is computed by aggregating validities of all the involved factors that are obtained from a factor vocabulary based on the empirical data. We then map the combined validity based on the Jaccard similarity measure into linguistic forms, so that a human can understand the results. Then PPVDs that are obtained based on the relevant factors in the given case yield the final valid fuzzy probabilities for conviction and acquittal. Finally, the judge has to make a decision; we therefore provide a numerical measure. Our approach supports the proposed hypothesis within the three-dimensional contexts of probability, possibility, and validity to improve the ability to solve problems with incomplete, unreliable, or ambiguous information to deliver a more reliable decision.     

Dirac structures and paracomplex manifolds

We introduce the notion of a generalized paracomplex structure. This is a natural notion which unifies several geometric structures such as symplectic forms, paracomplex structures, and Poisson structures. We show that generalized paracomplex structures are in one-to-one correspondence with pairs of transversal Dirac structures on a smooth manifold. To cite this article: A. Wade, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Program equilibrium—a program reasoning approach

The concept of program equilibrium, introduced by Howard (Theory and Decision 24(3):203–213, 1988) and further formalised by Tennenholtz (Game Econ Behav 49:363–373, 2004), represents one of the most ingenious and potentially far-reaching applications of ideas from computer science in game theory to date. The basic idea is that a player in a game selects a strategy by entering a program, whose behaviour may be conditioned on the programs submitted by other players. Thus, for example, in the prisoner’s dilemma, a player can enter a program that says “If his program is the same as mine, then I cooperate, otherwise I defect”. It can easily be shown that if such programs are permitted, then rational cooperation is possible even in the one-shot prisoner’s dilemma. In the original proposal of Tennenholtz, comparison between programs was limited to syntactic comparison of program texts. While this approach has some considerable advantages (not the least being computational and semantic simplicity), it also has some important limitations. In this paper, we investigate an approach to program equilibrium in which richer conditions are allowed, based on model checking—one of the most successful approaches to reasoning about programs. We introduce a decision-tree model of strategies, which may be conditioned on strategies of others. We then formulate and investigate a notion of “outcome” for our setting, and investigate the complexity of reasoning about outcomes. We focus on coherent outcomes: outcomes in which every decision by every player is justified by the conditions in his program. We identify a condition under which there exist a unique coherent outcome. We also compare our notion of (coherent) outcome with that of (supported) semantics known from logic programming. We illustrate our approach with many examples.     

Uniform Distribution,Discrepancy, and Reproducing Kernel Hilbert Spaces

In this paper we define a notion of uniform distribution and discrepancy of sequences in an abstract set E through reproducing kernel Hilbert spaces of functions on E. In the case of the finite-dimensional unit cube these discrepancies are very closely related to the worst case error obtained for numerical integration of functions in a reproducing kernel Hilbert space. In the compact case we show that the discrepancy tends to zero if and only if the sequence is uniformly distributed in our sense. Next we prove an existence theorem for such uniformly distributed sequences and investigate the relation to the classical notion of uniform distribution. Some examples conclude this paper.     

Ambiguity functions, Wigner distributions and Cohen's class for LCA groups

In this paper we construct a general class of time-frequency representations for LCA groups which parallel Cohen's class for the real line. For this, we generalize the notion of ambiguity function and Wigner distribution to the setting of general LCA groups in such a way that the Plancherel transform of the ambiguity function coincides with the Wigner distribution. Furthermore, properties of the general ambiguity function and Wigner distribution are studied. In detail we characterize those groups whose ambiguity functions and Wigner distributions vanish at infinity or are square-integrable. Finally, we explicitly construct Cohen's class for the group of p-adic numbers, p prime.     

On sign changes of tempered distributions having a spectral gap at the origin

It is known that if a real finite Borel measure has a spectral gap at the origin then either it must have many sign changes or it is zero identically. Assume the Fourier transform of a real temperate distribution agrees in a neighborhood of the origin with the sum of an analytic function and a lacunary trigonometric series. We conjecture that either the distribution must have many sign changes or the Fourier transform agrees with the sum on the whole line. The Note contains some results related to the conjecture. In particular, our results imply that a real temperate measure having spectral gap at the origin must have many oscillations with large amplitudes. To cite this article: I. Ostrovskii, A. Ulanovskii, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Boolean Valued and Stone Algebra Valued Measure Theories

In conventional generalization of the main results of classical measure theory to Stone algebra valued measures, the values that measures and functions can take are Booleanized, while the classical notion of a σ-field is retained. The main purpose of this paper is to show by abundace of illustrations that if we agree to Booleanize the notion of a σ-field as well, then all the glorious legacy of classical measure theory is preserved completely. Mathematics Subject Classification: 03C90, 28B15.     

Gleichmäßige Verteilung von Punkten in gewissen metrischen Räumen,speziell auf der Kugel

To measure the uniformity of such a distribution, a new notion of discrepancy (D _LV) is studied, which combines two interesting features: 1) The infinite sequences, which are uniformly distributed (in the usual sense), are characterized byD _LV→0 under rather general conditions on the space. 2) Forn points on the sphere,n∈{4, 6, 12},D _LV attains its minimum for the vertices of a regular polyhedron.     

New results on q-positivity

In this paper we discuss symmetrically self-dual spaces, which are simply real vector spaces with a symmetric bilinear form. Certain subsets of the space will be called q-positive, where q is the quadratic form induced by the original bilinear form. The notion of q-positivity generalizes the classical notion of the monotonicity of a subset of a product of a Banach space and its dual. Maximal q-positivity then generalizes maximal monotonicity. We discuss concepts generalizing the representations of monotone sets by convex functions, as well as the number of maximally q -positive extensions of a q-positive set. We also discuss symmetrically self-dual Banach spaces, in which we add a Banach space structure, giving new characterizations of maximal q-positivity. The paper finishes with two new examples.     

分数布朗运动环境下的期权定价与测度变换

研究分数B-S市场中的期权定价问题.通过选取不同的资产作为计价单位及相应的测度变换,将经典模型中的计价单位变换方法推广到分数布朗运动市场环境,既丰富了分数期权定价的拟鞅方法,也得到了分数期权定价公式的新的推导方法.     

Stream-based inconsistency measurement

Inconsistency measures have been proposed to assess the severity of inconsistencies in knowledge bases of classical logic in a quantitative way. In general, computing the value of inconsistency is a computationally hard task as it is based on the satisfiability problem which is itself NP-complete. In this work, we address the problem of measuring inconsistency in knowledge bases that are accessed in a stream of propositional formulæ. That is, the formulæ of a knowledge base cannot be accessed directly but only once through processing of the stream. This work is a first step towards practicable inconsistency measurement for applications such as Linked Open Data, where huge amounts of information is distributed across the web and a direct assessment of the quality or inconsistency of this information is infeasible due to its size. Here we discuss the problem of stream-based inconsistency measurement on classical logic, in order to make use of existing measures for classical logic. However, it turns out that inconsistency measures defined on the notion of minimal inconsistent subsets are usually not apt to be used in the streaming scenario. In order to address this issue, we adapt measures defined on paraconsistent logics and also present a novel inconsistency measure based on the notion of a hitting set. We conduct an extensive empirical analysis on the behavior of these different inconsistency measures in the streaming scenario, in terms of runtime, accuracy, and scalability. We conclude that for two of these measures, the stream-based variant of the new inconsistency measure and the stream-based variant of the contension inconsistency measure, large-scale inconsistency measurement in streaming scenarios is feasible.     

Asymptotic coupling and a general form of Harris�� theorem with applications to stochastic delay equations

There are many Markov chains on infinite dimensional spaces whose one-step transition kernels are mutually singular when starting from different initial conditions. We give results which prove unique ergodicity under minimal assumptions on one hand and the existence of a spectral gap under conditions reminiscent of Harris?? theorem. The first uses the existence of couplings which draw the solutions together as time goes to infinity. Such ??asymptotic couplings?? were central to (Mattingly and Sinai in Comm Math Phys 219(3):523?C565, 2001; Mattingly in Comm Math Phys 230(3):461?C462, 2002; Hairer in Prob Theory Relat Field 124:345?C380, 2002; Bakhtin and Mattingly in Commun Contemp Math 7:553?C582, 2005) on which this work builds. As in Bakhtin and Mattingly (2005) the emphasis here is on stochastic differential delay equations. Harris?? celebrated theorem states that if a Markov chain admits a Lyapunov function whose level sets are ??small?? (in the sense that transition probabilities are uniformly bounded from below), then it admits a unique invariant measure and transition probabilities converge towards it at exponential speed. This convergence takes place in a total variation norm, weighted by the Lyapunov function. A second aim of this article is to replace the notion of a ??small set?? by the much weaker notion of a ??d-small set,?? which takes the topology of the underlying space into account via a distance-like function d. With this notion at hand, we prove an analogue to Harris?? theorem, where the convergence takes place in a Wasserstein-like distance weighted again by the Lyapunov function. This abstract result is then applied to the framework of stochastic delay equations. In this framework, the usual theory of Harris chains does not apply, since there are natural examples for which there exist no small sets (except for sets consisting of only one point). This gives a solution to the long-standing open problem of finding natural conditions under which a stochastic delay equation admits at most one invariant measure and transition probabilities converge to it.