On Some Expectation and Derivative Operators Related to Integral Representations of Random Variables with Respect to a PII Process

Given a process with independent increments X (not necessarily a martingale) and a large class of square integrable r.v. H = f(X _T ), f being the Fourier transform of a finite measure μ, we provide a direct expression for Kunita-Watanabe and Föllmer-Schweizer decompositions of H. The representation is expressed by means of two significant maps: the expectation and derivative operators related to the characteristics of X. We also evaluate the expression for the variance optimal error when hedging the claim H with underlying process X. Those questions are motivated by finding the solution of the celebrated problem of global and local quadratic risk minimization in mathematical finance.     

A filtering problem with a small nonlinear term

In this paper, we consider a filtering problem where the signal X _t satisfies a slightly nonlinear stochastic differential equation and we want to obtain estimates of X _t. To this end, we decompose the nonlinearity with two techniques—a deterministic one and a stochastic one—and this leads us to two sequences of estimates which can be computed by solving finite dimensional equations. We want to compare their performances: we solve this problem in most cases if we restrict ourselves to sufficiently small times t and we give conditions which permit to conclude also for larger times     

Robust Approximation in a Filtering Problem with Real State Space and Counting Observations

Let (X _t ,Y _t ) be a pure jump Markov process, where X _t takes values in \bf R and Y _t is a counting process. We compare the filter of this system and a filter of a suitably modified system. We compute an explicit bound for the distance in the so-called bounded Lipschitz metric between the two filters. Finally we show how to use this bound to construct a discrete space approximation of the filter. Accepted 7 December 1999     

Expansion of the global error for numerical schemes solving stochastic differential equations

Given the solution (X_t ) of a Stochastic Differential System, two situat,ions are considered: computat,ion of Ef(X_t ) by a Monte–Carlo method and, in the ergodic case, integration of a function f w.r.t. the invariant probability law of (X_t ) by simulating a simple t,rajectory.

For each case it is proved the expansion of the global approximat,ion error—for a class of discret,isat,ion schemes and of funct,ions f—in powers of the discretisation step size, extending in the fist case a result of Gragg for deterministic O.D.E.

Some nn~nerical examples are shown to illust,rate the applicat,ion of extrapolation methods, justified by the foregoing expansion, in order to improve the approximation accuracy     

On Approximate ℓ 1 Systems in Banach Spaces

Let X be a real Banach space and let (f(n)) be a positive nondecreasing sequence. We consider systems of unit vectors (x_i)^∞_i=1 in X which satisfy ∑_iA±x_i|A|−f(|A|), for all finite A and for all choices of signs. We identify the spaces which contain such systems for bounded (f(n)) and for all unbounded (f(n)). For arbitrary unbounded (f(n)), we give examples of systems for which [x_i] is H.I., and we exhibit systems in all isomorphs of ℓ₁ which are not equivalent to the unit vector basis of ℓ₁. We also prove that certain lacunary Haar systems in L₁ are quasi-greedy basic sequences.     

Some Pólya-Type Irreducibility Criteria for Multivariate Polynomials

We provide irreducibility criteria for multivariate polynomials with coefficients in an arbitrary field that extend a classical result of Pólya for polynomials with integer coefficients. In particular, we provide irreducibility conditions for polynomials of the form f(X)(Y ? f ₁(X))…(Y ? f _n (X)) + g(X), with f, f ₁, ?, f _n , g univariate polynomials over an arbitrary field.     

Entropy,Approximation Quantities and the Asymptotics of the Modulus of Continuity

The paper deals with the approximation of bounded real functions f on a compact metric space (X, d) by so-called controllable step functions in continuation of [Ri/Ste]. These step functions are connected with controllable coverings, that are finite coverings of compact metric spaces by subsets whose sizes fulfil a uniformity condition depending on the entropy numbers ε_n(X) of the space X. We show that a strong form of local finiteness holds for these coverings on compact metric subspaces of IR^m and S^m. This leads to a Bernstein type theorem if the space is of finite convex information. In this case the corresponding approximation numbers ε_n(f) have the same asymptotics its ω(f, ε_n(X)) for f ε C(X). Finally, the results concerning functions f ε M(X) and f ε C(X) are transferred to operators with values in M(X) and C(X), respectively.     

Parametrizing Over ℤ Integral Values of Polynomials Over ℚ

Given a polynomial f ∈ ?[X] such that f(?) ? ?, we investigate whether the set f(?) can be parametrized by a multivariate polynomial with integer coefficients, that is, the existence of g ∈ ?[X ₁,…, X _m ] such that f(?) = g(? ^m ). We offer a necessary and sufficient condition on f for this to be possible. In particular, it turns out that some power of 2 is a common denominator of the coefficients of f, and there exists a rational β with odd numerator and odd prime-power denominator such that f(X) = f(β ?X). Moreover, if f(?) is likewise parametrizable, then this can be done by a polynomial in one or two variables.     

On ideal extensions of partial monounary algebras

In the present paper we introduce the notion of an ideal of a partial monounary algebra. Further, for an ideal (I, f _I ) of a partial monounary algebra (A, f _A ) we define the quotient partial monounary algebra (A, f _A )/(I, f _I ). Let (X, f _X ), (Y, f _Y ) be partial monounary algebras. We describe all partial monounary algebras (P, f _P ) such that (X, f _X ) is an ideal of (P, f _P ) and (P, f _P )/(X, f _X ) is isomorphic to (Y, f _Y ). This work was supported by the Slovak VEGA Grant No. 1/3003/06 and by the Science and Technology Assistance Agency under the contract No. APVT-20-004104.     

The law of the Euler scheme for stochastic differential equations

Summary We study the approximation problem ofE f(X _T ) byE f(X _T ⁿ ), where (X _t ) is the solution of a stochastic differential equation, (X _T ⁿ ) is defined by the Euler discretization scheme with stepT/n, andf is a given function. For smoothf's, Talay and Tubaro have shown that the errorE f(X _T ) –f(X _T ⁿ ) can be expanded in powers of 1/n, which permits to construct Romberg extrapolation precedures to accelerate the convergence rate. Here, we prove that the expansion exists also whenf is only supposed measurable and bounded, under an additional nondegeneracy condition of Hörmander type for the infinitesimal generator of (X _t ): to obtain this result, we use the stochastic variations calculus. In the second part of this work, we will consider the density of the law ofX _T ⁿ and compare it to the density of the law ofX _T .     

Improving the Data Augmentation Algorithm in the Two-Block Setup

The data augmentation (DA) approach to approximate sampling from an intractable probability density f_X is based on the construction of a joint density, f_{X, Y}, whose conditional densities, f_X|Y and f_Y|X, can be straightforwardly sampled. However, many applications of the DA algorithm do not fall in this “single-block” setup. In these applications, X is partitioned into two components, X = (U, V), in such a way that it is easy to sample from f_Y|X, f_{U|V, Y}, and f_{V|U, Y}. We refer to this alternative version of DA, which is effectively a three-variable Gibbs sampler, as “two-block” DA. We develop two methods to improve the performance of the DA algorithm in the two-block setup. These methods are motivated by the Haar PX-DA algorithm, which has been developed in previous literature to improve the performance of the single-block DA algorithm. The Haar PX-DA algorithm, which adds a computationally inexpensive extra step in each iteration of the DA algorithm while preserving the stationary density, has been shown to be optimal among similar techniques. However, as we illustrate, the Haar PX-DA algorithm does not lead to the required stationary density f_X in the two-block setup. Our methods incorporate suitable generalizations and modifications to this approach, and work in the two-block setup. A theoretical comparison of our methods to the two-block DA algorithm, a much harder task than the single-block setup due to nonreversibility and structural complexities, is provided. We successfully apply our methods to applications of the two-block DA algorithm in Bayesian robit regression and Bayesian quantile regression. Supplementary materials for this article are available online.     

Topological groups and convex sets homeomorphic to non-separable Hilbert spaces

Let X be a topological group or a convex set in a linear metric space. We prove that X is homeomorphic to (a manifold modeled on) an infinite-dimensional Hilbert space if and only if X is a completely metrizable absolute (neighborhood) retract with ω-LFAP, the countable locally finite approximation property. The latter means that for any open cover of X there is a sequence of maps (f _n : X → X) _nεgw such that each f _n is -near to the identity map of X and the family {f _n (X)} _n∈ω is locally finite in X. Also we show that a metrizable space X of density dens(X) < is a Hilbert manifold if X has gw-LFAP and each closed subset A ⊂ X of density dens(A) < dens(X) is a Z _∞-set in X.      

Generalized Feynman–Kac Semigroups,Associated Quadratic Forms and Asymptotic Properties

In this paper, we study the Feynman–Kac semigroup T _t f(x)=E _x[f(X _t)exp(N _t)],where X _t is a symmetric Levy process and N _t is a continuous additive functional of zero energy which is not necessarily of bounded variation. We identify the corresponding quadratic form and obtain large time asymptotics of the semigroup. The Dirichlet form theory plays an important role in the whole paper.     

Polarizations of Prym varieties of pairs of coverings

To any pair of coverings f_i:X → X_i, i=1, 2, of smooth projective curves one can associate an abelian subvariety of the Jacobian J_X, the Prym variety P(f₁, f₂) of the pair (f₁, f₂). In some cases we can compute the type of the restriction of the canonical principal polarization of J_X. We obtain 2 families of Prym-Tyurin varieties of exponent 6. Received: 2 September 2004     

Rings of Continuous Functions on Spaces of Finite Rank and the SV Property

Let X be a compact topological space and let C(X) denote the f-ring of all continuous real-valued functions defined on X. A point x in X is said to have rank n if, in C(X), there are n minimal prime ?-ideals contained in the maximal ?-ideal M _x  = {f ? C(X):f(x) = 0}. The space X has finite rank if there is an n ? N such that every point x ? X has rank at most n. We call X an SV space (for survaluation space) if C(X)/P is a valuation domain for each minimal prime ideal P of C(X). Every compact SV space has finite rank. For a bounded continuous function h defined on a cozeroset U of X, we say there is an h-rift at the point z if h cannot be extended continuously to U ∪ {z}. We use sets of points with h-rift to investigate spaces of finite rank and SV spaces. We show that the set of points with h-rift is a subset of the set of points of rank greater than 1 and that whether or not a compact space of finite rank is SV depends on a characteristic of the closure of the set of points with h-rift for each such h. If X has finite rank and the set of points with h-rift is an F-space for each h, then X is an SV space. Moreover, if every x ? X has rank at most 2, then X is an SV space if and only if for each h, the set of points with h-rift is an F-space.     

Finite subgraphs of uncountably chromatic graphs

It is consistent that for every function f:ω → ω there is a graph with size and chromatic number ?₁ in which every n‐chromatic subgraph contains at least f(n) vertices (n ≥ 3). This solves a $ 250 problem of Erd?s. It is consistent that there is a graph X with Chr(X)=|X|=?₁ such that if Y is a graph all whose finite subgraphs occur in X then Chr(Y)≤?₂ (so the Taylor conjecture may fail). It is also consistent that if X is a graph with chromatic number at least ?₂ then for every cardinal λ there exists a graph Y with Chr(Y)≥λ all whose finite subgraphs are induced subgraphs of X. © 2005 Wiley Periodicals, Inc. J Graph Theory 49: 28–38, 2005     

Additively decomposed quasiconvex functions

Letf be a real-valued function defined on the product ofm finite-dimensional open convex setsX ₁, ,X _m .Assume thatf is quasiconvex and is the sum of nonconstant functionsf ₁, ,f _m defined on the respective factor sets. Then everyf _i is continuous; with at most one exception every functionf _i is convex; if the exception arises, all the other functions have a strict convexity property and the nonconvex function has several of the differentiability properties of a convex function.We define the convexity index of a functionf _i appearing as a term in an additive decomposition of a quasiconvex function, and we study the properties of that index. In particular, in the case of two one-dimensional factor sets, we characterize the quasiconvexity of an additively decomposed functionf either in terms of the nonnegativity of the sum of the convexity indices off ₁ andf ₂, or, equivalently, in terms of the separation of the graphs off ₁ andf ₂ by means of a logarithmic function. We investigate the extension of these results to the case ofm factor sets of arbitrary finite dimensions. The introduction discusses applications to economic theory.     

On Itô s formula for multidimensional Brownian motion

Consider a d-dimensional Brownian motion X = (X ¹,…,X ^d ) and a function F which belongs locally to the Sobolev space W ^1,2. We prove an extension of It? s formula where the usual second order terms are replaced by the quadratic covariations [f _k (X), X ^k ] involving the weak first partial derivatives f _k of F. In particular we show that for any locally square-integrable function f the quadratic covariations [f(X), X ^k ] exist as limits in probability for any starting point, except for some polar set. The proof is based on new approximation results for forward and backward stochastic integrals. Received: 16 March 1998 / Revised version: 4 April 1999     

Semi-stable models for rigid-analytic spaces

 Let R be a complete discrete valuation ring with field of fractions K and let X _K be a smooth, quasi-compact rigid-analytic space over S_p K. We show that there exists a finite separable field extension K' of K, a rigid-analytic space X' _K' over S_p K' having a strictly semi-stable formal model over the ring of integers of K', and an étale, surjective morphism f : X' _K' →X _K of rigid-analytic spaces over S_p K. This is different from the alteration result of A.J. de Jong [dJ] who does not obtain that f is étale. To achieve this property we have to work locally on X _K , i.e. our f is not proper and hence not an alteration. Received: 26 October 2001 / Revised version: 14 August 2002 Published online: 14 February 2003