Arbitrage and completeness in financial markets with given <Emphasis Type="Italic">N</Emphasis>-dimensional distributions

Abstract In this paper, we focus on the following problem: given a financial market, modelled by a process , and a family of probability measures on , with N a positive integer and the time space, we search for financially meaningful conditions which are equivalent to the existence and uniqueness of an equivalent (local) martingale measure (EMM) Q such that the price process S has under Q the pre-specified finite-dimensional distributions of order N (N-dds) . We call these two equivalent properties, respectively, N -mixed no free lunch and market N -completeness. They are based on a classification of contingent claims with respect to their path-dependence on S and on the related notion of N-mixed strategy. Finally, we apply this approach to the Black-Scholes model with jumps, by showing a uniqueness result for its equivalent martingale measures set. Mathematics Subject Classification (2000): 60G48, 91B28 Journal of Economic Literature Classification: G12, D52     

On the relationship between absolute prudence and absolute risk aversion

Abstract For a utility function , the functions and are the Arrow-Pratt coefficient of absolute risk aversion (ara) and the coefficient of absolute prudence (ap). They measure respectively an agent’s sensitivity to risk and the strength of the precautionary saving motive under income uncertainty.     

A note on minimum variance

Summary Minimizing is discussed under the unbiasedness condition: and the condition (A):f _i (x) (i=1, ..., p) are linearly independent , and .     

<Emphasis Type="SmallCaps">Notes and Comments:</Emphasis> On the uniqueness of convex-ranged probabilities

Abstract In Marinacci (2000), the following theorem was proved. Theorem 1. (Marinacci (2000) Let P and Q be two finitely additive probabilities on a λ -system Σ . Suppose that P is convex-ranged and that Q is countably additive. If there exists an A ⁺ ∈ Σ with 0<P(A ⁺ )<1 such that whenever B∈ Σ , then P=Q. Mathematics Subject Classification (2000): 28A10, 91B06 Journal of Economic Literature Classification: C60, D81     

A characterization of entropies of degree α

The functional equation has been studied by several authors under various assumptions onf and onk, l. Here we solve this equation iff is measurable andk 3,l 2 are fixed integers. Using the solution we characterize the entropies of degree for all real . Our results generalize the results ofBehara/Nath [1973],Kannappan [1974] andMittal [1976].     

Asymptotically optimal nonparametric one- and two-way analysis of variance tests for the log linear model under censoring

We considerr ×c populations with failure ratesλ _ij(t) satisfying the condition

On a simple characterisation of threshold graphs

In their very interesting paper “Set-packing problems and threshold graphs” [1] V. Chvatal and P. L. Hammer have shown that the constraints $$\begin{gathered} \sum\limits_{j = 1}^n {a_{ij} x_j \leqslant 1(i = 1,2, . . . , m)} \hfill \\ x_j \in (0,1)(j = 1,2, . . . , n) \hfill \\ \end{gathered} $$ are equivalent to the only inequality $$\begin{gathered} \sum\limits_{j - 1}^n {c_j x_j \leqslant d} \hfill \\ x_j \in (0,1)(j = 1,2, . . . , n) \hfill \\ \end{gathered} $$ if and only if the intersection graph associated with the matrix (a _ij ) — see § 1 — is a threshold graph i.e. a graph none of whose induced subgraphs are isomorphic to 2K ₂,P ₄,C ₄: As Chvatal and Hammer have shown [1], threshold graphs can be characterised in many different ways; the main result of this paper is to give a new, very simple characterisation which will enable us to test whether a graph is a threshold by a simple inspection of its incidence matrix.     

Hermite ranks and U-statistics

We focus on the asymptotic behavior of $U$ -statistics of the type $$\begin{aligned} \sum _{1\le i\ne j\le n} h(X_i,X_j)\\ \end{aligned}$$ in the long-range dependence setting, where $(X_i)_{i\ge 1}$ is a stationary mean-zero Gaussian process. Since $(X_i)_{i\ge 1}$ is Gaussian, $h$ can be decomposed in Hermite polynomials. The goal of this paper is to compare the different notions of Hermite rank and to provide conditions for the remainder term in the decomposition to be asymptotically negligeable.     

A Nonclassical Law of the Iterated Logarithm for Functions of Positively Associated Random Variables

Let be a sequence of stationary positively associated random variables and a sequence of positive constants be monotonically approaching infinity and be not asymptotically equivalent to loglog n. Under some suitable conditions, a nonclassical law of the iterated logarithm is investigated, i.e.

Aspetti dinamici di leggi finanziarie scindibili

Vengono studiate le proprietà delle intensità istantanee di interesse di leggi finanziarie scindibili non necessariamente omogeneef(x, s, t). Esse risultano dipendenti dal montante e dal tempo finale secondo il modello . Ciò porta ad ottenere una naturale corrispondenza fra leggi finanziarie scindibili ed equazioni differenziali ordinarie. Si esaminano in dettaglio i casi particolari di leggi uniformi, leggi omogenee e leggi uniformi-omogenee, individuando la forma delle equazioni differenziali ad esse associate. Si estendono infine i risultati a leggi finanziarie del tipo , che dipendono anche dalla variabile istante decisionale .

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Summary We study the properties of the interest rates of the so-called scindibili financial laws (not necessarily homogeneous)f(x, s, t). They explicity depend on the value off andt only, according to the form . This suggests a natural correspondence between such financial laws and ordinary differential equations.The particular cases of uniform laws, homogeneous laws and uniform-homogeneous laws are examined and the structure of the associated differential equations are obtained.The previous results are extended to the financial laws of type which also depend on a decisional time .

     

Berry–Esseen bounds for density estimates under NA assumption

Let {X _j } be a strictly stationary sequence of negatively associated random variables with the marginal probability density function f(x). The recursive kernel estimators of f(x) are defined by

Una nuova caratterizzatione dell'ammissibilità nella teoria delle decisioni

In questo lavoro viene fornita una nuova caratterizzazione dell'ammissibilità attraverso un adeguato uso della nozione di ammissibilità parziale. Questa caratterizzazione consente di affrontare le questioni riguardanti la completezza della classe delle decisioni ammissibiliti sotto condizioni «maneggevoli». Fornisce inoltre un approccio unificante al problema della completezza che consente di derivare, come casi particolari, alcuni risultati già noti nella letteratura sull'argomento.

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In this paper a new characterization of admissibility is given for general decision problems. It is based on an adequate use of the notion of partial admissibility.A general decision problem is usually synthetized by a triplet (, , ) where is the states (or parameters) space, the set of available decisions and is a family of real valued functions defined on and expressing numerically the consequences of choosing when the state is . The set is regarded as a subset of the space of all real valued functions on endowed with the topology of pointwise convergence.As for as admissibility is concerned all the pertinent information about decisions are contained in the corresponding functionsW .This allows to introduce a notion of partial admissibility through the neigh-bourhoods of this topology. Admissibile decisions are then shown to be limits of monotone non increasing sequences of partially admissible decisions.Moreover this topological characterization allows to prove the completeness of classes of admissible decisions under acceptable systems of conditions which contain as special cases, known results in literature.

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Lavoro svolto nell'ambito del Gruppo Nazionale per l'Analisi Funzionale e le sue Applicazioni del C.N.R.     

Stochastic orderings of convolution residuals

In this paper we study convolution residuals, that is, if $X_1,X_2,\ldots ,X_n$ are independent random variables, we study the distributions, and the properties, of the sums $\sum _{i=1}^lX_i-t$ given that $\sum _{i=1}^kX_i>t$ , where $t\in \mathbb R $ , and $1\le k\le l\le n$ . Various stochastic orders, among convolution residuals based on observations from either one or two samples, are derived. As a consequence computable bounds on the survival functions and on the expected values of convolution residuals are obtained. Some applications in reliability theory and queueing theory are described.     

Boundedly complete families which are not complete

Summary Completeness of a family of probability distributions implies its bounded completeness but not conversely. An example of a family which is boundedly complete but not complete was presented by Lehmann and Scheffe [5]. This appears to be the only such example quoted in the statistical literature. The purpose of this note is to provide further examples of this type. It is shown that any given family of power series distributions can be used to construct a class containing infinitely many boundedly complete, but not complete, families. Furthermore, it is shown that the family of continuous distributions , is boundedly complete, but not complete, whereU denotes the uniform distribution on [a, b] and {P _ϑ,ϑ ∈ IR}, is a translation family generated by a distributionP ₀ with mean value zero, which is continuous with respect to the Lebesgue measure.     

Asymptotic normality of linear functions of concomitants of order statistics

Let T( ) be a linear function of concomitants of order statistics, whereT (·) denotes a statistical functional depending on some distribution function (df)F and is an estimator ofF. Under an auxiliary model approach we consider statistics of the form , where denotes a weighted empirical df and a finite population df (t denotes a triangular array). The results can be used to estimate income inequality in finite populations and especially when the survey is based on some design. The paper was written when the author was working at the Statistical Research Unit, Statistics Sweden, Stockholm, Sweden The research was supported by the Joint Committé of the Nordic Social Research Council.