Remarks on Taylor Series Expansions and Conditional Expectations for Stratonovich SDEs with Complete V‐Commutativity

Abstract

It may happen that there is not a finite maximum order bound for numerical approximations of stochastic processes X = (X _t : 0 ≤ t ≤ T) satisfying Stratonovich stochastic differential equations (SDEs) with some commutative structure along an appropriate functional V(t, X _t ). This statement can be proven with respect to the concept of mean square convergence under the assumption of “infinite smoothness” of drift a(t, x) and diffusion coefficients b ^j (t, x) and with finite initial second moments. As a result, we obtain an infinite series expansion of the conditional expectation 𝔼[V(t, X _t )|? _t ^N ] on any fixed finite time interval [0, T], provided that the information is collected by discretized σ‐field ? _T ^N  = σ{W _{t 0} , W _{t 1} , …, W _{t N?1} , W _T } at N + 1 given time instants t _i  ∈ [0, T] with t ₀ ≤ t ₁ ≤ ··· ≤ t _N?1 ≤ t _N  = T.     

Exponentiation for unitary structures

Motivated by the observation that both pretopologies and preapproach limits can be characterized as those convergence relations which have a unit for a suitable composition, we introduce the category Algu(T;V) of reflexive and unitary lax algebras, for a symmetric monoidal closed lattice V and a Set-monad T=(T,e,m). For T=U the ultrafilter monad, we characterize exponentiable morphisms in Algu(U;V). Further, we give a sufficient condition for an object to be exponentiable in the category Alg(U;V) of reflexive and transitive lax algebras. This specializes to known and new results for pretopological, preapproach and approach spaces.     

The Number of Out-Pancyclic Vertices in a Strong Tournament

An arc in a tournament T with n ≥ 3 vertices is called pancyclic, if it belongs to a cycle of length l for all 3 ≤ l ≤ n. We call a vertex u of T an out-pancyclic vertex of T, if each out-arc of u is pancyclic in T. Yao et al. (Discrete Appl. Math. 99, 245–249, 2000) proved that every strong tournament contains an out-pancyclic vertex. For strong tournaments with minimum out-degree 1, Yao et al. found an infinite class of strong tournaments, each of which contains exactly one out-pancyclic vertex. In this paper, we prove that every strong tournament with minimum out-degree at least 2 contains three out-pancyclic vertices. Our result is best possible since there is an infinite family of strong tournaments with minimum degree at least 2 and no more than 3 out-pancyclic vertices.     

Trivectors yielding spreads in PG(5, 2)

Given an alternating trilinear form ${T\in {\rm Alt}(\times^{3}V_{6})}$ on V ₆ = V(6, 2) let ${\mathcal{L}_{T}}$ denote the set of those lines ${\langle a, b \rangle}$ in ${{\rm PG}(5,2)=\mathbb{P}V_{6}}$ which are T-singular, satisfying, that is, T(a, b, x) = 0 for all ${x\in {\rm PG}(5, 2).}$ If ${\mathcal{L}_{21}}$ is a Desarguesian line-spread in PG(5, 2) it is shown that ${\mathcal{L}_{T}=\mathcal{L}_{21}}$ for precisely three choices T ₁,T ₂,T ₃ of T, which moreover satisfy T ₁ + T ₂ + T ₃ = 0. For ${T\in\mathcal{T}:=\{T_{1},T_{2},T_{3}\}}$ the ${\mathcal{G}_{T}}$ -orbits of flats in PG(5, 2) are determined, where ${\mathcal{G}_{T}\cong {\rm SL}(3,4).2}$ denotes the stabilizer of T under the action of GL(6, 2). Further, for a representative U of each ${\mathcal{G}_{T}}$ -orbit, the T-associate U ^# is also determined, where by definition $$U^{\#}=\{v\in {\rm PG}(5,2)\, |\, T(u_{1},u_{2},v) = 0\, \,{\rm for\,all }\, \, u_{1},u_{2}\in U\}$$ .     

Nonparametric analysis of doubly truncated data

One of the principal goals of the quasar investigations is to study luminosity evolution. A convenient one-parameter model for luminosity says that the expected log luminosity, T*, increases linearly as θ ₀· log(1  +  Z*), and T*(θ ₀) = T*  −  θ ₀· log(1  +  Z*) is independent of Z*, where Z* is the redshift of a quasar and θ ₀ is the true value of evolution parameter. Due to experimental constraints, the distribution of T* is doubly truncated to an interval (U*, V*) depending on Z*, i.e., a quadruple (T*, Z*, U*, V*) is observable only when U* ≤ T* ≤ V*. Under the one-parameter model, T*(θ ₀) is independent of (U*(θ ₀), V*(θ ₀)), where U*(θ ₀) = U*  −  θ ₀· log(1  +  Z*) and V*(θ ₀) = V*  −  θ ₀· log(1  +  Z*). Under this assumption, the nonparametric maximum likelihood estimate (NPMLE) of the hazard function of T*(θ ₀) (denoted by ĥ) was developed by Efron and Petrosian (J Am Stat Assoc 94:824–834, 1999). In this note, we present an alternative derivation of ĥ. Besides, the NPMLE of distribution function of T*(θ ₀), [^(F)]{\hat F} , will be derived through an inverse-probability-weighted (IPW) approach. Based on Theorem 3.1 of Van der Laan (1996), we prove the consistency and asymptotic normality of the NPMLE [^(F)]{\hat F} under certain condition. For testing the null hypothesis H_q0: T^*(q₀) = T^*-q₀·log(1 + Z^*){H_{\theta_0}: T^{\ast}(\theta_0) = T^{\ast}-\theta_0\cdot \log(1 + Z^{\ast})} is independent of Z*, (Efron and Petrosian in J Am Stat Assoc 94:824–834, 1999). proposed a truncated version of the Kendall’s tau statistic. However, when T* is exponential distributed, the testing procedure is futile. To circumvent this difficulty, a modified testing procedure is proposed. Simulations show that the proposed test works adequately for moderate sample size.     

The metacompactness of spaces with bases of subinfinite rank

Let X be a set. A collection $\text{U}$of subsets of X has subinfinite rank if whenever $\text{V}$ ? $\text{U}$, ∩$\text{V}$≠ø, and $\text{V}$ is infinite, then there are two distinct elements of $\text{V}$, one of which is a subset of the other. Theorem. AT₁space with a base of subinfinite rank is hereditarily metacompact.     

Sum formulas for reductive algebraic groups

Let V be a Weyl module either for a reductive algebraic group G or for the corresponding quantum group Uq. If G is defined over a field of positive characteristic p, respectively if q is a primitive lth root of unity (in an arbitrary field) then V has a Jantzen filtration V=V⁰⊃V¹⊃?⊃Vr=0. The sum of the positive terms in this filtration satisfies a well-known sum formula.If T denotes a tilting module either for G or Uq then we can similarly filter the space HomG(V,T), respectively HomUq(V,T) and there is a sum formula for the positive terms here as well.We give an easy and unified proof of these two (equivalent) sum formulas. Our approach is based on an Euler type identity which we show holds without any restrictions on p or l. In particular, we get rid of previous such restrictions in the tilting module case.     

Hypersurfaces in P n with 1-parameter symmetry groups II

We assume V a hypersurface of degree d in ${P^n({\mathbb C})}$ with isolated singularities and not a cone, admitting a group G of linear symmetries. In earlier work we treated the case when G is semi-simple; here we analyse the unipotent case. Our first main result lists the possible groups G. In each case we discuss the geometry of the action, reduce V to a normal form, find the singular points, study their nature, and calculate the Milnor numbers. The Tjurina number τ(V) ≤ (d ? 1) ^n–2(d ² ? 3d + 3): we call V oversymmetric if this value is attained. We calculate τ in many cases, and characterise the oversymmetric situations. In particular, we list all the cases with dim(G) = 2 which are the oversymmetric cases with d = 3.     

Spanning Connectivity of the Power of a Graph and Hamilton-Connected Index of a Graph

Let G = (V, E) be a connected graph. The hamiltonian index h(G) (Hamilton-connected index hc(G)) of G is the least nonnegative integer k for which the iterated line graph L ^k (G) is hamiltonian (Hamilton-connected). In this paper we show the following. (a) If |V(G)| ≥ k + 1 ≥ 4, then in G ^k , for any pair of distinct vertices {u, v}, there exists k internally disjoint (u, v)-paths that contains all vertices of G; (b) for a tree T,  h(T) ≤ hc(T) ≤ h(T) + 1, and for a unicyclic graph G,  h(G) ≤ hc(G) ≤ max{h(G) + 1, k′ + 1}, where k′ is the length of a longest path with all vertices on the cycle such that the two ends of it are of degree at least 3 and all internal vertices are of degree 2; (c) we also characterize the trees and unicyclic graphs G for which hc(G) = h(G) + 1.     

Linear maps that preserve singular and nonsingular matrices

Let M (n,K) be the algebra of n × n matrices over an algebraically closed field K and T:M (n,K)→M (n,K) a linear transformation with the property that T maps nonsingular (singular) matrices to nonsingular (singular) matrices. Using some elementary facts from commutative algebra we show that T is nonsingular and maps singular matrices to singular matrices (T is nonsingular or T maps all matrices to singular matrices). Using these results we obtain Marcus and Moyl's characterization [T(x) = UXVorU^tXV for fixed U and V] from a result of Dieudonné's. Examples are given to show the hypothesis of algebraic closure in necessary.     

A class of elliptic equations in anisotropic spaces

The equation Δu + V(x)u + b(x)u|u| ^{ρ -1} + h(x) = 0 in ${\mathbb{R}^{n}}$ is studied in anisotropic Lebesgue spaces. We assume ${\frac{n-\theta}{n-2} < \rho < \infty}$ , with n ≥ 3 and 0 ≤ θ < 2, which covers the supercritical range. Our approach relies on estimates of the Riesz potential and allows us to consider a wide class of potentials V, including anisotropic ones. The symmetry and antisymmetry of the solutions are also addressed.     

Tail fitting for truncated and non-truncated Pareto-type distributions

In extreme value analysis, natural upper bounds can appear that truncate the probability tail. At other instances ultimately at the largest data, deviations from a Pareto tail behaviour become apparent. This matter is especially important when extrapolation outside the sample is required. Given that in practice one does not always know whether the distribution is truncated or not, we consider estimators for extreme quantiles both under truncated and non-truncated Pareto-type distributions. We make use of the estimator of the tail index for the truncated Pareto distribution first proposed in Aban et al. (J. Amer. Statist. Assoc. 101(473), 270–277, 2006). We also propose a truncated Pareto QQ-plot and a formal test for truncation in order to help deciding between a truncated and a non-truncated case. In this way we enlarge the possibilities of extreme value modelling using Pareto tails, offering an alternative scenario by adding a truncation point T that is large with respect to the available data. In the mathematical modelling we hence let T→∞ at different speeds compared to the limiting fraction (k/n→0) of data used in the extreme value estimation. This work is motivated using practical examples from different fields, simulation results, and some asymptotic results.     

Almost Injective Transformations of an Infinite Set

Let V be an infinite-dimensional vector space, let n be a cardinal such that ?₀ ≤ n ≤ dim V, and let AM(V, n) denote the semigroup consisting of all linear transformations of V whose nullity is less than n. In recent work, Mendes-Gonçalves and Sullivan studied the ideal structure of AM(V, n). Here, we do the same for a similarly-defined semigroup AM(X, q) of transformations defined on an infinite set X. Although our results are clearly comparable with those already obtained for AM(V, n), we show that the two semigroups are never isomorphic.     

The Ideal Structure of Semigroups of Linear Transformations with Upper Bounds on Their Nullity or Defect

Suppose V is a vector space with dim V = p ≥ q ≥ ?₀, and let T(V) denote the semigroup (under composition) of all linear transformations of V. For α ∈ T (V), let ker α and ran α denote the “kernel” and the “range” of α, and write n(α) = dim ker α and d(α) = codim ran α. In this article, we study the semigroups AM(p, q) = {α ∈ T(V):n(α) < q} and AE(p, q) = {α ∈ T(V):d(α) < q}. First, we determine whether they belong to the class of all semigroups whose sets of bi-ideals and quasi-ideals coincide. Then, for each semigroup, we describe its maximal regular subsemigroup, and we characterise its Green's relations and (two-sided) ideals. As a precursor to further work in this area,, we also determine all the maximal right simple subsemigroups of AM(p, q).     

On certain functional equation related to two-sided centralizers

Let m ≥ 0, n ≥ 0 be fixed integers with m + n ≠ 0 and let R be a prime ring with char(R) = 0 or m + n + 1 ≤ char(R) ≠ 2. Suppose that there exists an additive mapping T : R → R satisfying the relation 2T(x ^m+n+1) = x ^m T(x) x ⁿ  + x ⁿ T(x)x ^m for all ${x\in R}$ . In this case T is a two-sided centralizer.     

Tournois infinis et critiques

Given a tournament T=(V,A), a subset X of V is an interval of T provided that for every a,b∈X and x∈V?X, (a,x)∈A if and only if (b,x)∈A. For example, ?, {x} (x∈V) and V are intervals of T, called trivial intervals. A tournament all the intervals of which are trivial is called indecomposable; otherwise, it is decomposable. An indecomposable tournament T=(V,A) is then said to be critical if for each x∈V, T(V?{x}) is decomposable and if there are x≠y∈V such that T(V?{x,y}) is indecomposable. We introduce the operation of expansion which allows us to describe a process of construction of critical and infinite tournaments. It follows that, for every critical and infinite tournament T=(V,A), there are x≠y∈V such that T and T(V?{x,y}) are isomorphic. To cite this article: I. Boudabbous, C. R. Acad. Sci. Paris, Ser. I 336 (2003).