Subadditive Measure Research Articles

Saddle-Point Equilibrium Strategy for Linear Quadratic Uncertain Stochastic Hybrid Differential Games Based on Subadditive Measures

This paper describes a kind of linear quadratic uncertain stochastic hybrid differential game system grounded in the framework of subadditive measures, in which the system dynamics are described by a hybrid differential equation with Wiener–Liu noise and the performance index function is quadratic. Firstly, we introduce the concept of hybrid differential games and establish the Max–Min Lemma for the two-player zero-sum game scenario. Next, we discuss the analysis of saddle-point equilibrium strategies for linear quadratic hybrid differential games, addressing both finite and infinite time horizons. Through the incorporation of a generalized Riccati differential equation (GRDE) and guided by the principles of the Itô–Liu formula, we prove that that solving the GRDE is crucial and serves as both a sufficient and necessary precondition for identifying equilibrium strategies within a finite horizon. In addition, we also acquire the explicit formulations of equilibrium strategies in closed forms, alongside determining the optimal values of the cost function. Through the adoption of a generalized Riccati equation (GRE) and applying a similar approach to that used for the finite horizon case, we establish that the ability to solve the GRE constitutes a sufficient criterion for the emergence of equilibrium strategies in scenarios extending over an infinite horizon. Moreover, we explore the dynamics of a resource extraction problem within a finite horizon and separately delve into an H∞ control problem applicable to an infinite horizon. Finally, we present the conclusions.

Uncertain Stochastic Hybrid Age-Dependent Population Equation Based on Subadditive Measure: Existence, Uniqueness and Exponential Stability

The existing literature lacks a study on age-dependent population equations based on subadditive measures. In this paper, we propose a hybrid age-dependent population dynamic system (referred to as APDS) that incorporates uncertain random perturbations driven by both the well-known Wiener process and the Liu process associated with belief degree, which have similar symmetry in terms of form. Firstly, we redefine the Liu integral in a mean square sense and then extend Liu’s lemma and the Itô-Liu formula. We then utilize the extensions of the Itô-Liu formula, Barkholder-Davis-Gundy (BDG) inequality, the Liu’s lemma, the Gronwall’s lemma and the symmetric nature of calculus itself to establish the uniqueness of a strong solution for the hybrid APDS. Additionally, we prove the existence of the hybrid APDS by combining the proof of uniqueness with some important lemmas. Finally, under appropriate assumptions, we demonstrate the exponential stability of the hybrid system.

Almost Sure Exponential Stability of Uncertain Stochastic Hopfield Neural Networks Based on Subadditive Measures

For this paper, we consider the almost sure exponential stability of uncertain stochastic Hopfield neural networks based on subadditive measures. Firstly, we deduce two corollaries, using the Itô–Liu formula. Then, we introduce the concept of almost sure exponential stability for uncertain stochastic Hopfield neural networks. Next, we investigate the almost sure exponential stability of uncertain stochastic Hopfield neural networks, using the Lyapunov method, Liu inequality, the Liu lemma, and exponential martingale inequality. In addition, we prove two sufficient conditions for almost sure exponential stability. Furthermore, we consider stabilization with linear uncertain stochastic perturbation and present some exceptional examples. Finally, our paper provides our conclusion.

Subadditive measure on projectors of von neumann algebra

In the paper subadditive measure on the lattice of orthogonal projectors of von Neumann algebra is considered. The basic peoperties of the subadditive measure are estabelished and proved.

Convergence in measure theorems of nonlinear integrals of functions integrable to the pth power

One can find a subadditive measure μ and a sequence {fn}n∈N of nonnegative measurable functions converging in μ-measure to such a function f with the property that for any p>1, fnp no longer converges in μ-measure to fp. Therefore, in nonadditive measure theory an additional and intrinsic discussion is needed to prove integral convergence theorems for functions integrable to the pth power.The purpose of the paper is to propose a method for automatically deriving pth power versions of convergence in measure theorems of nonlinear integrals from already established ones. The proposed method is based on the robustness of some properties of integral functional under the operation f↦fp raising a nonnegative function f to the pth power.

Several inequalities for the pan-integral

Several inequalities for the pan-integral are investigated. It is shown that the Chebyshev inequality holds for an arbitrary subadditive measure if and only if the integrands f, g are comonotone. Thus, we provide a new characterization for nonnegative comonotone functions. It is also shown that the Hölder and Minkowski inequalities for the pan-integral hold if the monotone measure μ is subadditive. Since the pan-integral coincides with the concave integral when μ is subadditive, our results can also be applied to the concave integral.

Estimation Of The Mining Damage Risk In The Hypothetical Impact Area Of The Concurrent Processes Of Rock Mass Disorders

Abstract The aim of this work is the estimation of the risk of mining damage occurrence, based on uncertain information regarding the impact of the concurrent processes of deformation and vibration. This problem concerns the experimental and theoretical description of the so-called critical phenomena occurring during the reaction mining area ↔ building object. Post-mining deformations of the rock mass medium and paraseismic vibrations can appear at a considerable distance from the sub-area of the mining operation – hence, the determination of the measures of their impacts is usually somewhat subjective, while the estimation of the mining damage based on deterministic methods is often insufficient. It is difficult to show the correlation between the local maximum of the impact of the velocity vector amplitude and the damage to the building – especially if the measures of interaction are not additive. The parameters of these impacts, as registered by measurements, form finite sets with a highly random character. Formally, it is adequate to the mapping from the probability space to the power set. For the purposes of the present study, the Dempster – Shafer model was used, where space is characterised by subadditive and superadditive measures. Regarding the application layer, the conclusions from the expert evaluations are assumed to be the values of random variables. The model was defined, and the risk of damage occurrence was estimated.

Risk Sharing for Individual Risks

Motivated by reinsurance applications, we consider an optimal risk sharing problem for individual risks. In a framework where the market participants’ preferences are according to the sub-additive and law-invariant risk measures, first we characterize the optimal individual risk sharing allocations. Then, we introduce a Choquet risk measure, associated with a given law-invariant coherent risk measure and, by using that, we characterize the optimal risk sharing allocations for the individual risks.

Approximation of polygonal fuzzy neural networks in sense of Choquet integral norms

Approximation capabilities are important and primary properties of neural networks and fuzzy neural networks (FNNs). Neural networks have been successfully applied in many fields since they can work as approximators in nature. Many scholars research FNNs’ approximation abilities for continuous fuzzy functions. It is concluded that FNNs can work as approximators for continuous fuzzy functions if the fuzzy functions satisfy some specified conditions. However, the problem whether FNNs can work as approximators for discontinuous fuzzy functions is not solved completely until now. In this work, we focus on the approximation of polygonal FNN for discontinuous fuzzy functions in sense of Choquet integral norms. We first introduce the Choquet integral norms in sub-additive fuzzy measure. Then the universal approximation of polygonal FNNs for fuzzy valued functions in sense of Choquet integral norms is analyzed in this paper. It is proved that the polygonal FNNs can work as approximators for fuzzy valued functions in the sense of Choquet integral norms with respect to sub-additive fuzzy measure.

States and Boolean Algebra

We investigate subadditive measures on orthomodular lattices. We show as the main result that the Boolean algebra, the special metric orthomodular lattice and the orthomodular lattice which is unital with respect to subadditive states are equivalent. This result may find an application in the foundation of quantum theories and mathematical logic.

Analog of L p with a subadditive measure

For a subadditive fuzzy measure (not assumed finite), a Minkowski type triangle inequality, with Choquet integrals in place of Lebesgue integrals, is shown to hold. It is immediate that the set of functions for which a certain positive power of the absolute values have finite Choquet integrals is closed under addition, leading to a linear space analogous to the Lebesgue space L p , with a metric related to the integral of that power. Under the additional condition that the subadditive fuzzy measure is inner continuous (Sugeno), the space is shown to be complete. Consequences of the Minkowski type inequality are illustrated in two specific instances.

On Constructing Finite, Finitely Subadditive Outer Measures, and Submodularity

Given a nonempty abstract set , and a covering class , and a finite, finitely subadditive outer measure , we construct an outer measure and investigate conditions for to be submodular. We then consider several other set functions associated with and obtain conditions for equality of these functions on the lattice generated by . Lastly, we describe a construction of a finite, finitely subadditive outer measure given an arbitrary family of subsets, , of and a nonnegative, finite set function defined on .

On the extension of subadditive measures in lattice ordered groups

A lattice ordered group valued subadditive measure is extended from an algebra of subsets of a set to a σ-algebra.

Choquet pricing and equilibrium

Choquet pricing and in particular Wang’s premium principle, have recently been put forward as an alternative to traditional pricing principles in finance and insurance. With Choquet pricing, the price of an insurance contract or financial asset equals the Choquet integral of the corresponding payoff with respect to a concave sub-additive measure. Since the resulting pricing rule is non-linear, existing theories do not provide an answer to the question of whether equilibrium exists. We introduce a general equilibrium model that allows for non-linearity, and show that Choquet pricing is consistent with general equilibrium.

On finitely subadditive outer measures and modularity properties

Let ν be a finite, finitely subadditive outer measure on P(X). Define ρ (E) = ν (X) − ν (E′) for E ⊂ X. The measurable sets Sν and Sρ and the set S = {E ⊂ X/ν (E) = ρ (E)} are investigated in general, and in the presence of regularity or modularity assumptions on ν. This is also done for ν0(E) = inf{ν (M)/E ⊂ M ∈ Sν }. General properties of ν are derived when ν is weakly submodular. Applications and numerous examples are given.

Characterizations of outer measures associated with lattice measures

Letνbe a finite countably subadditive outer measure defined on all subsets of a setX, take a collectionℂof subsets ofXcontainingXand∅, we derive an outer measureρusingνon sets inℂ. By applying this general framework on two special cases in whichν=μ″, one whereμ∈Mσ(𝔏)and the other whereμ∈Mσ(𝔏1),𝔏1⫅𝔏2being lattices on a setX, we obtain new characterizations of the outer measureμ″. These yield useful relationships between various set functions includingμi,μj,μ″, andμ′.

Some remarks concerning finitely Subadditive outer measures with applications

The present paper is intended as a first step toward the establishment of a general theory of finitely subadditive outer measures. First, a general method for constructing a finitely subadditive outer measure and an associated finitely additive measure on any space is presented. This is followed by a discussion of the theory of inner measures, their construction, and the relationship of their properties to those of an associated finitely subadditive outer measure. In particular, the interconnections between the measurable sets determined by both the outer measure and its associated inner measure are examined. Finally, several applications of the general theory are given, with special attention being paid to various lattice related set functions.

Finitely subadditive outer measures, finitely superadditive inner measures and their measurable sets

Consider any set X. A finitely subadditive outer measure on P(X) is defined to be a function v from P(X) to R such that v(ϕ) = 0 and v is increasing and finitely subadditive. A finitely superadditive inner measure on P(X) is defined to be a function p from P(X) to R such that p(ϕ) = 0 and p is increasing and finitely superadditive (for disjoint unions) (It is to be noted that every finitely superadditive inner measure on P(X) is countably superadditive).This paper contributes to the study of finitely subadditive outer measures on P(X) and finitely superadditive inner measures on P(X) and their measurable sets.

Traces and Subadditive Measures on Projections in JBW-Algebras and Von Neumann Algebras

Traces and Subadditive Measures on Projections in JBW-Algebras and Von Neumann Algebras

Traces and subadditive measures on projections in 𝐽𝐵𝑊-algebras and von Neumann algebras

Let P ( M ) P(M) be the projection lattice of an arbitrary JBW-algebra or von Neumann algebra M. It is shown that the tracial states of M correspond by extension precisely to the subadditive probability measures on P ( M ) P(M) . The analogous result for normal semifinite traces is also proved.