Affinely Prime Dynamical Systems(Dedicated to Professor Haim Brezis on the occasion of his 70th birthday)

This paper deals with representations of groups by "affine" automorphisms of compact, convex spaces, with special focus on "irreducible" representations: equivalently"minimal" actions. When the group in question is P SL(2, R), the authors exhibit a oneone correspondence between bounded harmonic functions on the upper half-plane and a certain class of irreducible representations. This analysis shows that, surprisingly, all these representations are equivalent. In fact, it is found that all irreducible affine representations of this group are equivalent. The key to this is a property called "linear Stone-Weierstrass"for group actions on compact spaces. If it holds for the "universal strongly proximal space"of the group(to be defined), then the induced action on the space of probability measures on this space is the unique irreducible affine representation of the group.     

A Categorical Foundation for Bayesian Probability

Building on the work of Lawvere and others, we develop a categorical framework for Bayesian probability. This foundation will then allow for Bayesian representations of uncertainty to be integrated into other categorical modeling applications. The main result uses an existence theorem for regular conditional probabilities by Faden, which holds in more generality than the standard setting of Polish spaces. This more general setting is advantageous, as it allows for non-trivial decision rules (Eilenberg–Moore algebras) on finite (as well as non finite) spaces. In this way, we obtain a common framework for decision theory and Bayesian probability.     

On measure solutions of backward stochastic differential equations

We consider backward stochastic differential equations (BSDEs) with nonlinear generators typically of quadratic growth in the control variable. A measure solution of such a BSDE will be understood as a probability measure under which the generator is seen as vanishing, so that the classical solution can be reconstructed by a combination of the operations of conditioning and using martingale representations. For the case where the terminal condition is bounded and the generator fulfills the usual continuity and boundedness conditions, we show that measure solutions with equivalent measures just reinterpret classical ones. For the case of terminal conditions that have only exponentially bounded moments, we discuss a series of examples which show that in the case of non-uniqueness, classical solutions that fail to be measure solutions can coexist with different measure solutions.     

Affinely Prime Dynamical Systems

This paper deals with representations of groups by "affine" automorphisms of compact,convex spaces,with special focus on "irreducible" representations:equivalently "minimal" actions.When the group in question is PSL(2,R),the authors exhibit a oneone correspondence between bounded harmonic functions on the upper half-plane and a certain class of irreducible representations.This analysis shows that,surprisingly,all these representations are equivalent.In fact,it is found that all irreducible affine representations of this group are equivalent.The key to this is a property called "linear Stone-Weierstrass"for group actions on compact spaces.If it holds for the "universal strongly proximal space"of the group (to be defined),then the induced action on the space of probability measures on this space is the unique irreducible affine representation of the group.     

Quantization dimension of probability measures supported on Cantor-like sets

Let μ be an arbitrary probability measure supported on a Cantor-like set E with bounded distortion. We establish a relationship between the quantization dimension of μ and its mass distribution on cylinder sets under a hereditary condition. As an application, we determine the quantization dimensions of probability measures supported on E which have explicit mass distributions on cylinder sets provided that the hereditary condition is satisfied.     

Small-Ball Probabilities for the Volume of Random Convex Sets

We prove small-deviation estimates for the volume of random convex sets. The focus is on convex hulls and Minkowski sums of line segments generated by independent random points. The random models considered include (Lebesgue) absolutely continuous probability measures with bounded densities and the class of log-concave measures.     

Representability in mixed integer programmiing,I: Characterization results

We define the concept of a representation of a set of either linear constraints in bounded integers, or convex constraints in bounded integers. A regularity condition plays a crucial role in the convex case. Then we characterize the representable sets (Theorem 2.1) and provide several examples of our representations.A consequence of our characterization is that the only representable sets are those from ‘either/or’ constraints. This latter case can be treated by generalizations of techniques from the disjunctive methods of cutting-plane theory (e.g. [2] and [30]).The representations given here are intended for use as part of the constraints of a larger optimization problem, where they often can serve to tighten the (linear or convex) relaxation.The study of representations was initiated by Meyer and in the linear case we continue the development in [35].     

Probability density function of SDEs with unbounded and path-dependent drift coefficient

In this paper, we first prove that the existence of a solution of SDEs under the assumptions that the drift coefficient is of linear growth and path-dependent, and diffusion coefficient is bounded, uniformly elliptic and Hölder continuous. We apply Gaussian upper bound for a probability density function of a solution of SDE without drift coefficient and local Novikov condition, in order to use Maruyama–Girsanov transformation. The aim of this paper is to prove the existence with explicit representations (under linear/super-linear growth condition), Gaussian two-sided bound and Hölder continuity (under sub-linear growth condition) of a probability density function of a solution of SDEs with path-dependent drift coefficient. As an application of explicit representation, we provide the rate of convergence for an Euler–Maruyama (type) approximation, and an unbiased simulation scheme.     

Moreau-Yosida Regularization of State-Dependent Sweeping Processes with Nonregular Sets

The existence and the convergence (up to a subsequence) of the Moreau-Yosida regularization for the state-dependent sweeping process with nonregular (subsmooth and positively alpha-far) sets are established. Then, by a reparametrization technique, the existence of solutions for bounded variation continuous state-dependent sweeping processes with nonregular (subsmooth and positively alpha-far) sets is proved. An application to vector hysteresis is discussed, where it is shown that the Play operator with positively alpha-far sets is well defined for bounded variation continuous inputs.     

An Elementary Approach to the Daniell-Kolmogorov Theorem and Some Related Results

We give a short elementary proof of the Daniell-Kolmogorov existence theorem for probability measures on product spaces, assuming nothing but the existence of Lebesgue measure on the unit interval. Related approaches are used to prove the existence of regular conditional distributions directly on Polish spaces, and to establish the existence of random measures and sets with given finite-dimensional distributions or hitting probabilities, respectively.     

Updating credal networks is approximable in polynomial time

Credal networks relax the precise probability requirement of Bayesian networks, enabling a richer representation of uncertainty in the form of closed convex sets of probability measures. The increase in expressiveness comes at the expense of higher computational costs. In this paper, we present a new variable elimination algorithm for exactly computing posterior inferences in extensively specified credal networks, which is empirically shown to outperform a state-of-the-art algorithm. The algorithm is then turned into a provably good approximation scheme, that is, a procedure that for any input is guaranteed to return a solution not worse than the optimum by a given factor. Remarkably, we show that when the networks have bounded treewidth and bounded number of states per variable the approximation algorithm runs in time polynomial in the input size and in the inverse of the error factor, thus being the first known fully polynomial-time approximation scheme for inference in credal networks.     

Capital requirements with defaultable securities

We study capital requirements for bounded financial positions defined as the minimum amount of capital to invest in a chosen eligible asset targeting a pre-specified acceptability test. We allow for general acceptance sets and general eligible assets, including defaultable bonds. Since the payoff of these assets is not necessarily bounded away from zero, the resulting risk measures cannot be transformed into cash-additive risk measures by a change of numéraire. However, extending the range of eligible assets is important because, as exemplified by the recent financial crisis, assuming the existence of default-free bonds may be unrealistic. We focus on finiteness and continuity properties of these general risk measures. As an application, we discuss capital requirements based on Value-at-Risk and Tail-Value-at-Risk acceptability, the two most important acceptability criteria in practice. Finally, we prove that there is no optimal choice of the eligible asset. Our results and our examples show that a theory of capital requirements allowing for general eligible assets is richer than the standard theory of cash-additive risk measures.     

Dilations for systems of imprimitivity acting on Banach spaces

Motivated by a general dilation theory for operator-valued measures, framings and bounded linear maps on operator algebras, we consider the dilation theory of the above objects with special structures. We show that every operator-valued system of imprimitivity has a dilation to a probability spectral system of imprimitivity acting on a Banach space. This completely generalizes a well-known result which states that every frame representation of a countable group on a Hilbert space is unitarily equivalent to a subrepresentation of the left regular representation of the group. We also prove that isometric group representation induced framings on a Banach space can be dilated to unconditional bases with the same structure for a larger Banach space. This extends several known results on the dilations of frames induced by unitary group representations on Hilbert spaces.     

Generalized measures of noncompactness of sets and operators in Banach spaces

New measures of noncompactness for bounded sets and linear operators, in the setting of abstract measures and generalized limits, are constructed. A quantitative version of a classical criterion for compactness of bounded sets in Banach spaces by R. S. Phillips is provided. Properties of those measures are established and it is shown that they are equivalent to the classical measures of noncompactness. Applications to summable families of Banach spaces, interpolations of operators and some consequences are also given.     

A note on asymptotics of discounted value function and strong 0-discount optimality

We consider a Markov decision process with a Borel state space, bounded rewards, and a bounded transition density satisfying a simultaneous Doeblin-Doob condition. An asymptotics for the discounted value function related to the existence of stationary strong 0-discount optimal policies is extended from the case of finite action sets to the case of compact action sets and continuous in action rewards and transition densities.Supported by NSF grant DMS-9404177     

Absolutely continuous invariant measures for random non-uniformly expanding maps

We prove existence of (at most denumerable many) absolutely continuous invariant probability measures for random one-dimensional dynamical systems with asymptotic expansion. If the rate of expansion (Lyapunov exponents) is bounded away from zero, we obtain finitely many ergodic absolutely continuous invariant probability measures, describing the asymptotics of almost every point. We also prove a similar result for higher-dimensional random non-uniformly expanding dynamical systems. The results are consequences of the construction of such measures for skew-products with essentially arbitrary base dynamics and asymptotic expansion along the fibers. In both cases our method deals with either critical o singular points for the random maps.     

Compact sets of additive measures

In this note one gives compactifications, having a general character, for the family of all probability measures and for convex, bounded sets in linear spaces, considered in weak topologies.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 119, pp. 14–18, 1982.     

On a robust risk measurement approach for capital determination errors minimization

We propose a robust risk measurement approach that minimizes the expectation of overestimation plus underestimation costs. We consider uncertainty by taking the supremum over a collection of probability measures, relating our approach to dual sets in the representation of coherent risk measures. We provide results that guarantee the existence of a solution and explore the properties of minimizer and minimum as risk and deviation measures, respectively. An empirical illustration is carried out to demonstrate the use of our approach in capital determination.     

Constructing Ramsey graphs from Boolean function representations

Explicit construction of Ramsey graphs or graphs with no large clique or independent set, has remained a challenging open problem for a long time. While Erdös’ probabilistic argument shows the existence of graphs on 2n vertices with no clique or independent set of size 2 ⁿ , the best explicit constructions achieve a far weaker bound. There is a connection between Ramsey graph constructions and polynomial representations of Boolean functions due to Grolmusz; a low degree representation for the OR function can be used to explicitly construct Ramsey graphs [17,18]. We generalize the above relation by proposing a new framework. We propose a new definition of OR representations: a pair of polynomials represent the OR function if the union of their zero sets contains all points in {0, 1} ⁿ except the origin. We give a simple construction of a Ramsey graph using such polynomials. Furthermore, we show that all the known algebraic constructions, ones to due to Frankl-Wilson [12], Grolmusz [18] and Alon [2] are captured by this framework; they can all be derived from various OR representations of degree O(√n) based on symmetric polynomials. Thus the barrier to better Ramsey constructions through such algebraic methods appears to be the construction of lower degree representations. Using new algebraic techniques, we show that better bounds cannot be obtained using symmetric polynomials.