Gaussian measure of sections of dilates and translations of convex bodies

In this paper we give a solution for the Gaussian version of the Busemann–Petty problem with additional information about dilates and translations. We also discuss the size of the Gaussian measure of the hyperplane sections of the dilates of the unit cube.     

Second derivatives of convex functions in the sense of A. D. Aleksandrov on infinite-dimensional spaces with measure

We consider convex functions on infinite-dimensional spaces equipped with measures. Our main results give some estimates of the first and second derivatives of a convex function, where second derivatives are considered from two different points of view: as point functions and as measures.     

Sobolev estimates for optimal transport maps on Gaussian spaces

In this work, we will take the standard Gaussian measure as the reference measure and study the variation of optimal transport maps in Sobolev spaces with respect to it; as a by-product, an inequality which gives a precise link between the variation of entropy, Fisher information between source and target measures, with the Sobolev norm of the optimal transport map will be given. As applications, we will construct strong solutions to Monge–Ampère equations in finite dimension, as well as on the Wiener space, when the target measure satisfies the strong log-concavity condition. A result on the regularity on the optimal transport map on the Wiener space will be obtained.     

Reinforcement of an inequality due to Brascamp and Lieb

In this paper, we obtain a reinforcement of an inequality due to Brascamp and Lieb and a reinforcement of Poincaré's inequality for general logarithmical concave measures on Rd. The formula used in the proof is related to theorems concerning the integration of log-concave functions (such as results of Prékopa and of Ball, Barthe and Naor). We also obtain a lower bound for the variance of the same family of measures.     

Probabilistic and average widths of multivariate Sobolev spaces with mixed derivative equipped with the Gaussian measure

We present sharp bounds on the Kolmogorov probabilistic (N,δ)-width and p-average N-width of multivariate Sobolev space with mixed derivative

Full-size image (1K)

, equipped with a Gaussian measure μ in , that is where 1<q<∞,0<p<∞, and ρ>1 is depending only on the eigenvalues of the correlation operator of the measure μ (see (4)).     

Interpolation inequality of logarithmic type in abstract Besov spaces and an application to semilinear evolution equations

An abstract version of Besov spaces is introduced by using the resolvent of nonnegative operators. Interpolation inequalities with respect to abstract Besov spaces and generalized Lorentz spaces are obtained. These inequalities provide a generalization of Sobolev inequalities of logarithmic type. Uniqueness problems to abstract semilinear evolution equations are also discussed (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Localization and tensorization properties of the curvature-dimension condition for metric measure spaces, II

This is an addendum to the paper [K. Bacher, K.T. Sturm, Localization and tensorization properties of the curvature-dimension condition for metric measure spaces, J. Funct. Anal. 259 (2010) 28-56]. We prove the tensorization property for the curvature-dimension condition, add some detailed calculations - including explicit dependence of constants - and comment on assumptions and conjectures concerning the local-to-global statement in Bacher and Sturm (2010) [1] and Villani (2009) [6], respectively.     

Localization and tensorization properties of the curvature-dimension condition for metric measure spaces

This paper is devoted to the analysis of metric measure spaces satisfying locally the curvature-dimension condition CD(K,N) introduced by the second author and also studied by Lott & Villani. We prove that the local version of CD(K,N) is equivalent to a global condition CD^∗(K,N), slightly weaker than the (usual, global) curvature-dimension condition. This so-called reduced curvature-dimension condition CD^∗(K,N) has the local-to-global property. We also prove the tensorization property for CD^∗(K,N). As an application we conclude that the fundamental group π₁(M,x₀) of a metric measure space (M,d,m) is finite whenever it satisfies locally the curvature-dimension condition CD(K,N) with positive K and finite N.     

A new distance measure including the weak preference relation: Application to the multiple criteria aggregation procedure for mixed evaluations

We introduce a new distance measure between two preorders that captures indifference, strict preference, weak preference and incomparability relations. This measure is the first to capture weak preference relations. We illustrate how this distance measure affords decision makers greater modeling power to capture their preferences, or uncertainty and ambiguity around them, by using our proposed distance measure in a multiple criteria aggregation procedure for mixed evaluations.     

The Bismut-Elworthy formula for backward SDE's and applications to nonlinear Kolmogorov equations and control in infinite dimensional spaces

We consider a forward-backward system of stochastic evolution equations in a Hilbert space. Under nondegeneracy assumptions on the diffusion coefficient (that may be nonconstant) we prove an analogue of the well-known Bismut-Elworthy formula. Next, we consider a nonlinear version of the Kolmogorov equation, i.e. a deterministic quasilinear equation associated to the system according to Pardoux, E and Peng, S. (1992). "Backward stochastic differential equations and quasilinear parabolic partial differential equations". In: Rozowskii, B.L., Sowers, R.B. (Eds.), Stochastic Partial Differential Equations and Their Applications , Lecture Notes in Control Inf. Sci., Vol. 176, pp. 200-217. Springer: Berlin. The Bismut-Elworthy formula is applied to prove smoothing effect, i.e. to prove existence and uniqueness of a solution which is differentiable with respect to the space variable, even if the initial datum and (some) coefficients of the equation are not. The results are then applied to the Hamilton-Jacobi-Bellman equation of stochastic optimal control. This way we are able to characterize optimal controls by feedback laws for a class of infinite-dimensional control systems, including in particular the stochastic heat equation with state-dependent diffusion coefficient.     

Approximate solution for some stochastic differential equations involving both Gaussian and Poissonian white noises

By combining the Kramers-Moyal expansion with fractional Brownian motion of order n, in a formal symbolic calculus, one can obtain an approximation for the solution of some stochastic differential equations involving both Gaussian and Poissonian white noises, in terms of rotating Gaussian white noises on the grid defined by the complex roots of the unity. Illustrative examples are outlined.     

On equilibrium problem of abstract economy, generalized quasi-variational inequality, and an optimization problem in locally H-convex spaces

In this paper, the equilibrium existence problem for abstract economies, the existence problem for solution of generalized quasi-variational inequality, and an optimization problem in locally H-convex spaces are researched.     

Stochastic dominance and risk measure: A decision-theoretic foundation for VaR and C-VaR

Is it possible to obtain an objective and quantifiable measure of risk backed up by choices made by some specific groups of rational investors? To answer this question, in this paper we establish some behavior foundations for various types of VaR models, including VaR and conditional-VaR, as measures of downside risk. In this paper, we will establish some logical connections among VaRs, conditional-VaR, stochastic dominance, and utility maximization. Though supported to some extents with unanimous choices by some specific groups of expected or non-expected-utility investors, VaRs as profiles of risk measures at various levels of risk tolerance are not quantifiable – they can only provide partial and incomplete risk assessments for risky prospects.     

On the dual test for SSD efficiency : With an application to momentum investment strategies

This paper analyzes the dual formulation of Post’s [Post, T., 2003. Empirical tests for stochastic dominance efficiency. Journal of Finance 58, 1905–1932] test for second-order stochastic dominance (SSD) efficiency of a given investment portfolio relative to all possible portfolios formed from set of assets. In contrast to the earlier work, we (1) provide a direct proof for the dual that does not rely on expected utility theory, (2) adhere to the original definition of SSD, (3) phrase in terms of a general polyhedral portfolio possibilities set and (4) construct a SSD dominating benchmark portfolio from the optimal solution. To illustrate the dual SSD test, we apply the test to analyze the effect of short-selling restrictions on the profitability of momentum investment strategies.     

A simple formula for an analogue of conditional Wiener integrals and its applications

Let denote the space of real-valued continuous functions on the interval and for a partition of , let be given by .

In this paper, with the conditioning function , we derive a simple formula for conditional expectations of functions defined on which is a probability space and a generalization of Wiener space. As applications of the formula, we evaluate the conditional expectation of functions of the form

for and derive a translation theorem for the conditional expectation of integrable functions defined on the space .

     

A test for the bidirectional stochastic order with an application to quality control theory

A test for the bidirectional stochastic ordering is developed in this paper. The main properties of such a test are investigated. The asymptotic distribution of the statistic of the test is obtained under conditions which allow the construction of critical regions with a specific level of significance. It is also proved that the test is consistent on the whole set of alternatives. An application of such a test to quality control theory is developed.     

A relative compactness criterion in Wiener–Sobolev spaces and application to semi-linear Stochastic PDEs

We prove a relative compactness criterion in Wiener–Sobolev space which represents a natural extension of the compact embedding of Sobolev space H¹ into , at the level of random fields. Then we give a specific statement of this criterion for random fields solutions of semi-linear stochastic partial differential equations with coefficients bounded in an appropriate way. Finally, we employ this result to construct solutions for semi-linear stochastic partial differential equations with distribution as final condition. We also give a probabilistic interpretation of this solution in terms of backward doubly stochastic differential equations formulated in a weak sense.     

涉及单形内点的一个几何不等式及其应用

应用解析方法和几何不等式理论研究了n维欧氏空间En中涉及Ωn维单形Ω'n与其内接单形Ωn以及Ωn中内点之间的几何不等式问题,建立了涉及单形Ωn及其内接单形Ω'n的外接球半径以及Ωn中内点到各侧面距离之间的一个几何不等式,并给出了它的应用.     

Supremum concentration inequality and modulus of continuity for sub-nth chaos processes

This article provides a detailed analysis of the behavior of suprema and moduli of continuity for a large class of random fields which generalize Gaussian processes, sub-Gaussian processes, and random fields that are in the nth chaos of a Wiener process. An upper bound of Dudley type on the tail of the random field's supremum is derived using a generic chaining argument; it implies similar results for the expected supremum, and for the field's modulus of continuity. We also utilize a sharp and convenient condition using iterated Malliavin derivatives, to arrive at similar conclusions for suprema, via a different proof, which does not require full knowledge of the covariance structure.