General Probability Space Research Articles

An elementary proof of the dual representation of Expected Shortfall

We provide an elementary proof of the dual representation of Expected Shortfall on the space of integrable random variables over a general probability space. Unlike the results in the extant literature, our proof only exploits basic properties of quantile functions and can thus be easily implemented in any graduate course on risk measures. As a byproduct, we obtain a new proof of the subadditivity of Expected Shortfall.

On Stein’s lemma in hypotheses testing in general non-asymptotic case

The problem of testing two simple hypotheses in a general probability space is considered. For a fixed type-I error probability, the best exponential decay rate of the type-II error probability is investigated. In regular asymptotic cases (i.e., when the length of the observation interval grows without limit) the best decay rate is given by Stein’s exponent. In the paper, for a general probability space, some non-asymptotic lower and upper bounds for the best rate are derived. These bounds represent pure analytic relations without any limiting operations. In some natural cases, these bounds also give the convergence rate for Stein’s exponent. Some illustrating examples are also provided.

Secure List Decoding and its Application to Bit-String Commitment

We propose a new concept of secure list decoding, which is related to bit-string commitment. While the conventional list decoding requires that the list contains the transmitted message, secure list decoding requires the following additional security conditions to work as a modification of bit-string commitment. The first additional security condition is the receiver’s uncertainty for the transmitted message, which is stronger than the impossibility of the correct decoding, even though the transmitted message is contained in the list. The other additional security condition is the impossibility for the sender to estimate another element of the decoded list except for the transmitted message. The first condition is evaluated by the equivocation rate. The asymptotic property is evaluated by three parameters, the rates of the message and list sizes, and the equivocation rate. We derive the capacity region of this problem. We show that the combination of hash function and secure list decoding yields the conventional bit-string commitment. Our results hold even when the input and output systems are general probability spaces including continuous systems. When the input system is a general probability space, we formulate the abilities of the honest sender and the dishonest sender in a different way.

McKean–Vlasov Optimal Control: Limit Theory and Equivalence Between Different Formulations

We study a McKean–Vlasov optimal control problem with common noise in order to establish the corresponding limit theory as well as the equivalence between different formulations, including strong, weak, and relaxed formulations. In contrast to the strong formulation, in which the problem is formulated on a fixed probability space equipped with two Brownian filtrations, the weak formulation is obtained by considering a more general probability space with two filtrations satisfying an (H)-hypothesis type condition from the theory of enlargement of filtrations. When the common noise is uncontrolled, our relaxed formulation is obtained by considering a suitable controlled martingale problem. As for classic optimal control problems, we prove that the set of all relaxed controls is the closure of the set of all strong controls when considered as probability measures on the canonical space. Consequently, we obtain the equivalence of the different formulations of the control problem under additional mild regularity conditions on the reward functions. This is also a crucial technical step to prove the limit theory of the McKean–Vlasov control problem, that is, proving that it consists in the limit of a large population control problem with common noise.

Laws of Large Numbers for Dynamic Coherent Risk Measures

In this paper, we study the asymptotic behavior of dynamic coherent risk measures in general settings regardless of specific representations of the risk measures. In particular, we develop three different types laws of large numbers (LLN) for the average values of portfolios. These LLNs capture the limiting behavior of time-consistent dynamic coherent risk measures under appropriate conditions. Our results apply to general probability spaces with a sequence of financial returns characterized by a set of probability measures. We show that the limit of these averages will generally be multivalued within an identified set. We give examples to illustrate the potential applicability of our results and derive asymptotic results on estimation for the risk of returns of financial assets using a time-consistent dynamic coherent risk measure induced by a class of g-expectations.

Generalized white noise analysis and topological algebras

In this paper we develop a framework to extend the theory of generalized stochastic processes in the Hida white noise space to more general probability spaces which include the grey noise space. To obtain a Wiener-Itô expansion we recast it as a moment problem and calculate the moments explicitly. We further show the importance of a family of topological algebras called strong algebras in this context. Furthermore we show the applicability of our approach to the study of non-Gaussian stochastic processes.

Non-standard limits of graphs and some orbit equivalence invariants

We consider probability measure preserving discrete groupoids, group actions and equivalence relations in the context of general probability spaces. We study for these objects the notions of cost, $\beta$-invariant and some higher-dimensional variants. We also propose various convergence results about $\ell^2$-Betti numbers and rank gradient for sequences of actions, groupoids or equivalence relations under weak finiteness assumptions. In particular we connect the combinatorial cost with the cost of the ultralimit equivalence relations. Finally a relative version of Stuck-Zimmer property is also considered.

Dimensionwise multivariate orthogonal polynomials in general probability spaces

This paper puts forward a new generalized polynomial dimensional decomposition (PDD), referred to as GPDD, encompassing hierarchically organized, measure-consistent multivariate orthogonal polynomials in dependent random variables. In contrast to the existing PDD, which is valid strictly for independent random variables, no tensor-product structure or product-type probability measure is imposed or necessary. Fundamental mathematical properties of GPDD are examined by creating dimensionwise decomposition of polynomial spaces, deriving statistical properties of random orthogonal polynomials, demonstrating completeness of orthogonal polynomials for requisite assumptions, and upholding mean-square convergence to the correct limit. The GPDD approximation proposed should be effective in solving high-dimensional stochastic problems subject to dependent variables with a large class of non-product-type probability measures.

Systems of BSDES with oblique reflection and related optimal switching problems

We consider systems of backward stochastic differential equations with càdlàg upper barrier [Formula: see text] and oblique reflection from below driven by an increasing continuous function [Formula: see text]. Our equations are defined on general probability spaces with a filtration satisfying merely the usual assumptions of right continuity and completeness. We assume that the pair [Formula: see text] satisfies a Mokobodzki-type condition. We prove the existence of a solution for integrable terminal conditions and integrable quasi-monotone generators. Applications to the optimal switching problem are given.

Fefferman\u2013Stein inequalities for the dyadic-like maximal operators

The paper contains a transference theorem which allows to extend a large class of unweighted inequalities for the dyadic maximal operator to their weighted Fefferman–Stein counterparts on general probability spaces.

Simulation algorithms for hierarchical Archimedean copulas beyond the completely monotone case

Abstract Two simulation algorithms for hierarchical Archimedean copulas in the case when intra-group generators are not necessarily completely monotone are presented. Both generalize existing algorithms for the completely monotone case. The underlying stochastic models for both algorithms arise as a particular instance of a more general probability space studied recently in Ressel, P. (2018): A multivariate version of Williamson’s theorem, ℓ 1-symmetric survival functions, and generalized Archimedean copulas. Depend. Model. 6, 356–368. On this probability space the inter-group dependence need not be Archimedean, however, we highlight two particular circumstances that guarantee that a hierarchical Archimedean copula is obtained.

Modeling Stochastic Dominance as Infinite-Dimensional Constraint Systems via the Strassen Theorem

We use the Strassen theorem to solve stochastic optimization problems with stochastic dominance constraints. First, we show that a dominance-constrained problem on general probability spaces can be expressed as an infinite-dimensional optimization problem with a convenient representation of the dominance constraints provided by the Strassen theorem. This result generalizes earlier work which was limited to finite probability spaces. Second, we derive optimality conditions and a duality theory to gain insight into this optimization problem. Finally, we present a computational scheme for constructing finite approximations along with a convergence rate analysis on the approximation quality.

A BKR Operation for Events Occurring for Disjoint Reasons with High Probability

Given events A and B on a product space $S={\prod }_{i = 1}^{n} S_{i}$ , the set $A \Box B$ consists of all vectors x = (x1,…,xn) ∈ S for which there exist disjoint coordinate subsets K and L of {1,…,n} such that given the coordinates xi,i ∈ K one has that x ∈ A regardless of the values of x on the remaining coordinates, and likewise that x ∈ B given the coordinates xj,j ∈ L. For a finite product of discrete spaces endowed with a product measure, the BKR inequality 1 $$ P(A \Box B) \le P(A)P(B) $$ was conjectured by van den Berg and Kesten (J Appl Probab 22:556–569, 1985) and proved by Reimer (Combin Probab Comput 9:27–32, 2000). In Goldstein and Rinott (J Theor Probab 20:275–293, 2007) inequality Eq. 1 was extended to general product probability spaces, replacing $A \Box B$ by the set consisting of those outcomes x for which one can only assure with probability one that x ∈ A and x ∈ B based only on the revealed coordinates in K and L as above. A strengthening of the original BKR inequality Eq. 1 results, due to the fact that . In particular, it may be the case that $A \Box B$ is empty, while is not. We propose the further extension depending on probability thresholds s and t, where is the special case where both s and t take the value one. The outcomes are those for which disjoint sets of coordinates K and L exist such that given the values of x on the revealed set of coordinates K, the probability that A occurs is at least s, and given the coordinates of x in L, the probability of B is at least t. We provide simple examples that illustrate the utility of these extensions.

Ex-post core, fine core and rational expectations equilibrium allocations

This paper investigates the (weak) ex-post core and its relationships to the fine core and the set of rational expectations equilibrium allocations in an oligopolistic economy with asymmetric information, in which the set of agents consists of some large agents and a continuum of small agents and the space of states of nature is a general probability space. We show that under appropriate assumptions, the (weak) ex-post core is not empty and contains the set of rational expectations equilibrium allocations. We provide an example of a pure exchange continuum economy with asymmetric information and infinitely many states of nature, in which the ex-post core does not coincide with the set of rational expectations equilibrium allocations. We also show that when our economic model contains either no large agents or at least two large agents with the same characteristics, the fine core is contained in the ex-post core.

Bounding the integral of powered $i$-th mean curvatures

We get estimates for the integrals of powered i -th mean curvatures, 1\leq i\leq n-1 , of compact and convex hypersurfaces, in terms of the quermaß integrals of the corresponding C^2_+ convex bodies. These bounds will be obtained as consequences of a most general result for functions defined on a general probability space. From this result, similar estimates for the integrals of any convex transformation of the elementary symmetric functions of the radii of curvature of C^2_+ convex bodies will be also proved, both, in terms of the quermaß integrals, and of the roots of their Steiner polynomials. Finally, the radial function is considered, and estimates of the corresponding integrals are obtained in terms of the dual quermaß integrals.

Risk-consistent conditional systemic risk measures

We axiomatically introduce risk-consistent conditional systemic risk measures defined on multidimensional risks. This class consists of those conditional systemic risk measures which can be decomposed into a state-wise conditional aggregation and a univariate conditional risk measure. Our studies extend known results for unconditional risk measures on finite state spaces. We argue in favor of a conditional framework on general probability spaces for assessing systemic risk. Mathematically, the problem reduces to selecting a realization of a random field with suitable properties. Moreover, our approach covers many prominent examples of systemic risk measures from the literature and used in practice.

随机区间收益金融市场的无强占优分析

文章提出了具有随机区间收益的金融市场，在该市场中，证券价格被描述为区间值随机变量，并将研究的金融市场所在的概率空间框架从有限维拓展为一般概率空间，是传统随机市场的推广。在此模型中，提出了强占优策略和线性定价测度的概念，证明了市场无强占优与存在线性定价测度等价的定理，并且研究了随机区间收益市场无强占优与经典市场无占优的关系。该结论既包含经典随机金融市场的结论，又将金融分析推广到随机区间收益金融市场中。 In this article, a new financial market model, in which all securities have random interval pay- offs, is proposed. In the market, the probability space from finite dimensional expands general probability space. Some concepts, such as robust dominant strategy and linear pricing measures, are given and discussed parallel to those in traditional market analysis. With these new concepts, it is discussed that the requirement of no robust dominant strategy is equivalent to the existence of linear pricing measures and the relations between the analysis of the no robust dominant strategy in the market with random interval payoffs and it’s in traditional market. This conclusion includes the traditional random market analysis conclusions, extending to this new financial market.

Surface Measures Generated by Differentiable Measures

We study surface measures on level sets of functions on general probability spaces with measures differentiable along vector fields and suggest a new simple construction. Our construction applies also to level sets of mappings with values in finite-dimensional spaces. The standard surface measures arising for Gaussian measures in the Malliavin calculus can be obtained in this way. A positive answer is given to a question raised by M. Rockner concerning continuity of surface measures with respect to a parameter.

Characterizing common cause closedness of quantum probability theories

We prove new results on common cause closedness of quantum probability spaces, where by a quantum probability space is meant the projection lattice of a non-commutative von Neumann algebra together with a countably additive probability measure on the lattice. Common cause closedness is the feature that for every correlation between a pair of commuting projections there exists in the lattice a third projection commuting with both of the correlated projections and which is a Reichenbachian common cause of the correlation. The main result we prove is that a quantum probability space is common cause closed if and only if it has at most one measure theoretic atom. This result improves earlier ones published in Gyenis and Rédei (2014). The result is discussed from the perspective of status of the Common Cause Principle. Open problems on common cause closedness of general probability spaces (L,ϕ) are formulated, where L is an orthomodular bounded lattice and ϕ is a probability measure on L.

Mean-field backward stochastic differential equations in general probability spaces

In this paper, we deal with a class of mean-field backward stochastic differential equations in continuous time with an arbitrary filtered probability space. We prove the existence and uniqueness of a solution for those equations with strengthened Lipschitz assumption. A comparison theorem is also established.