Arbitrage and approximate arbitrage: the fundamental theorem of asset pricing

We consider an incomplete market model where asset prices are modelled by Ito processes, and derive the first fundamental theorem of asset pricing using standard stochastic calculus techniques. This contrasts with the sophisticated functional analytic theorems required in the comprehensive works of F. Delbaen and W. Schachermayer (1993) No Arbitrage and the Fundamental Theorem of Asset Pricing, pp. 37–38; Math. Finance 4 (1994), pp. 343–348; Math. Ann. 300 (1994), pp. 464–520; Ann. Appl. Probab. 5 (1995), pp. 926–645 and Proc. Sympos. Appl. Math. 57 (1999), pp. 49–58, and the comparative lack of transparency of the associated technical conditions. An additional benefit is that a clear relationship between no arbitrage and the existence of equivalent local martingale measures is also presented.     

A comparison of two no-arbitrage conditions

We give a comparison of two no-arbitrage conditions for the fundamental theorem of asset pricing. The first condition is named as the no free lunch with vanishing risk condition and the second the no good deal condition. We aim to derive a relationship between these two conditions.     

The arbitrage pricing theorem with incomplete preferences

This paper proves existence of equilibrium and the arbitrage pricing theorem for an asset exchange economy, where individuals' preferences may be incomplete or intransitive. This extends existing results to more general preferences. We also prove the arbitrage pricing theorem for a theory of choice under uncertainty by Bewley [Bewley, T. F. (2002), Knightian decision theory: part I, Decisions in Economics and Finance 25, 79–110.]. These preferences model Knightian uncertainty by preferences which may be incomplete but satisfy independence.     

Yan Theorem in L∞ with Applications to Asset Pricing

We prove an L~∞ version of the Yan theorem and deduce from it a necessary condition for theabsence of free lunches in a model of financial markets,in which asset prices are a continuous R~d valued processand only simple investment strategies are admissible.Our proof is based on a new separation theorem for convexsets of finitely additive measures.     

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.     

Normal approximation and fourth moment theorems for monochromatic triangles

Given a graph sequence denote by T₃(G_n) the number of monochromatic triangles in a uniformly random coloring of the vertices of G_n with colors. In this paper we prove a central limit theorem (CLT) for T₃(G_n) with explicit error rates, using a quantitative version of the martingale CLT. We then relate this error term to the well-known fourth-moment phenomenon, which, interestingly, holds only when the number of colors satisfies . We also show that the convergence of the fourth moment is necessary to obtain a Gaussian limit for any , which, together with the above result, implies that the fourth-moment condition characterizes the limiting normal distribution of T₃(G_n), whenever . Finally, to illustrate the promise of our approach, we include an alternative proof of the CLT for the number of monochromatic edges, which provides quantitative rates for the results obtained in [7].     

均值方差偏好和期望损失风险约束下的动态投资组合

本文在均值方差框架下,研究了期望损失风险约束下的连续时间动态投资组合问题。运用鞅理论和凸对偶方法,分别给出了最优财富和最优投资策略的解析式,而且两基金分离定理仍然成立。最后通过数值例子分析了风险约束对最优投资策略的影响。     

Price Operators Analysis in L p -Spaces

An integral type representation and various extension theorems for monotone linear operators in L _p -spaces are considered in relation to market price modelling. As application, a characterization of the existence of a risk-neutral probability measure equivalent to the applied underlying one is provided in terms of the given prices. These results are in the line of the fundamental theorem of asset pricing. Here, in particular, the risk-neutral probability measure considered has the advantage of having its density laying in pre-considered upper and lower bounds.     

The Fundamental Convergence Theorem for <Emphasis Type="Italic">p</Emphasis>(·)-Superharmonic Functions

We study the balayage related to the supersolutions of the variable exponent p(·)-Laplace equation. We prove the fundamental convergence theorem for the balayage and apply it for proving the Kellogg property, boundary regularity results for the balayage, and a removability theorem for p(·)-solutions.     

Three‐space type Hahn‐Banach properties

In set theory without the axiom of choice , three‐space type results for the Hahn‐Banach property are provided. We deduce that for every Hausdorff compact scattered space K , the Banach space C(K ) of real continuous functions on K satisfies the (multiple) continuous Hahn‐Banach property in . We also prove in Rudin's theorem: “Radon measures on Hausdorff compact scattered spaces are discrete”.     

Deficient rational functions and Ahlfors's theory of covering surfaces

We prove a second fundamental theorem in the sense of Nevanlinna's theory of meromorphic functions replacing the constantsa in (r, f, a) by rational functionsR withR(∞)=a. The key argument is Ahlfors's second fundamental theorem from his theory of covering surfaces.     

Right closure of martingale sequences in the sense of theA-integral

In this paper new sufficient (necessary and sufficient for martingales of special form) conditions for the martingale closure from the right in the sense of theA-integral are given. These results follow from the theorem about passing to the limit under theA-integral. The theorem is established using the criterion for transposing iterated limits with respect to the base. It is shown that the sufficient conditions thus obtained are stronger than those previously known. Translated fromMatematicheskie Zametki, Vol.68, No. 1, pp. 98–104, July, 2000.     

On level sets of smooth functions

Based on the established earlier general estimation method of the lengths of level sets of real functions, the paper proves a theorem which is an analog of the second fundamental theorem of the theory of Gamma-lines which, in its turn, is an analog of the second fundamental theorem of R. Nevanlinna.     

A basic relation between invariants of matrices under the action of the special orthogonal group

The first fundamental theorem of invariant theory for the action of the special orthogonal group onm tuples of matrices by simultaneous conjugation is proved in [2]. In this paper, as a first step in the direction of establishing the second fundamental theorem, we study a basic identity betweenSO(n, K) invariants ofm matrices.     

A short proof of Hua's fundamental theorem of the geometry of hermitian matrices

We present a new and simple proof of Hua's fundamental theorem of the geometry of hermitian matrices which characterizes bijective maps preserving adjacency in both directions on the real vector space of all n × n hermitian matrices.     

The fundamental theorem about weak relative [C, H]-Hopf modules

In this paper, we mainly introduce the concept of weak relative [C, H]-Hopf modules and give the fundamental theorem of weak relative right [C, H]-Hopf modules. Published in Russian in Matematicheskie Zametki, 2007, Vol. 82, No. 4, pp. 530–537. The text was submitted by the authors in English.     

Interpolation for weak Orlicz spaces with M Δ condition

An interpolation theorem for weak Orlicz spaces generalized by N-functions satisfying M _Δ condition is given. It is proved to be true for weak Orlicz martingale spaces by weak atomic decomposition of weak Hardy martingale spaces. And applying the interpolation theorem, we obtain some embedding relationships among weak Orlicz martingale spaces. This work was supported by the National Natural Science Foundation of China (Grant No. 10671147)     

A complete representation theorem for G-martingales

In this article, we establish a complete representation theorem for G-martingales. Unlike the existing results in the literature, we provide the existence and uniqueness of the second-order term, which corresponds to the second-order derivative in Markovian case. The main ingredient of the article is a new norm for that second-order term, which is based on an operator introduced by Song.     

Hamilton Cycles,Minimum Degree,and Bipartite Holes

We present a tight extremal threshold for the existence of Hamilton cycles in graphs with large minimum degree and without a large “bipartite hole” (two disjoint sets of vertices with no edges between them). This result extends Dirac's classical theorem, and is related to a theorem of Chvátal and Erd?s. In detail, an ‐bipartite‐hole in a graph G consists of two disjoint sets of vertices S and T with and such that there are no edges between S and T ; and is the maximum integer r such that G contains an ‐bipartite‐hole for every pair of nonnegative integers s and t with . Our central theorem is that a graph G with at least three vertices is Hamiltonian if its minimum degree is at least . From the proof we obtain a polynomial time algorithm that either finds a Hamilton cycle or a large bipartite hole. The theorem also yields a condition for the existence of k edge‐disjoint Hamilton cycles. We see that for dense random graphs , the probability of failing to contain many edge‐disjoint Hamilton cycles is . Finally, we discuss the complexity of calculating and approximating .     

Time Scales in Homogenization of Periodic Flows with Vanishing Molecular Diffusion

We study the long time transport property of conservative systems perturbed by a small white noise. We introduce the dissipation and martingale times and show how they are related to the diffusion time on which a limit theorem is valid. The limit theorem is a probabilistic version of homogenization with vanishing molecular diffusion. Examples of nontrivial time scales are given.