Vector-valued stochastic processes. I. Vector measures and vector-valued stochastic processes with finite variation

1988 ◽  
Vol 1 (2) ◽  
pp. 149-169 ◽  
Author(s):  
Nicolae Dinculeanu
Keyword(s):  
Stochastic Processes ◽  
Finite Variation ◽  
Vector Measures ◽  
Vector Valued

Vector Integration and Stochastic Integration in Banach Spaces

2018 ◽  
Author(s):  
Nicolae Dinculeanu
Keyword(s):  
Banach Spaces ◽  
Positive Measure ◽  
Integration Theory ◽  
Stochastic Integration ◽  
Bochner Integral ◽  
Finite Variation ◽  
Vector Measures ◽  
Vector Integration ◽  
Step Functions ◽  
Integrable Variation

This article deals with vector integration and stochastic integration in Banach spaces. In particular, it considers the theory of integration with respect to vector measures with finite semivariation and its applications. This theory reduces to integration with respect to vector measures with finite variation which, in turn, reduces to the Bochner integral with respect to a positive measure. The article describes the four stages in the development of integration theory. It first provides an overview of the relevant notation for Banach spaces, measurable functions, the integral of step functions, and measurability with respect to a positive measure before discussing the Bochner integral. It then examines integration with respect to measures with finite variation, semivariation of vector measures, integration with respect to a measure with finite semivariation, and stochastic integrals. It also reviews processes with integrable variation or integrable semivariation and concludes with an analysis of martingales.

A Note on Equicontinuous Families of Volumes With an Application to Vector Measures

1974 ◽  
Vol 26 (02) ◽  
pp. 273-280
Author(s):  
Richard Alan Oberle
Keyword(s):  
Abstract Space ◽  
Additive Functions ◽  
Finite Variation ◽  
Real Numbers ◽  
Vector Measures ◽  
The Real ◽  
Positive Integers

Let V denote a ring of subsets of an abstract space X, let R denote the real numbers, and let N denote the positive integers. Denote by a(V, R) (respectively ca(V, R)) the space of real valued, finitely additive (respectively countably additive) functions on the ring V and denote by ab(V, R) the subspace consisting of those members of the space a(V, R) with finite variation on each set in the ring V. Members of the space a(V, R) are referred to as charges and members of the space ab(V, R) are referred to as locally bounded charges. We denote by cab(V, R) the intersection of the spaces ab(V, R) and ca(V, R).

scholarly journals Toward a fundamental theorem of quantal measure theory

2012 ◽  
Vol 22 (5) ◽  
pp. 816-852 ◽  
Author(s):  
RAFAEL D. SORKIN
Keyword(s):  
Stochastic Processes ◽  
Path Integral ◽  
Fundamental Theorem ◽  
Measure Theory ◽  
Finite Lattice ◽  
Extension Problem ◽  
Integral Type ◽  
Causal Sets ◽  
Simple Event ◽  
Vector Valued

In this paper we address the extension problem for quantal measures of path-integral type, concentrating on two cases: sequential growth of causal sets and a particle moving on the finite lattice ℤn. In both cases, the dynamics can be coded into a vector-valued measure μ on Ω, the space of all histories. Initially, μ is just defined on special subsets of Ω called cylinder events, and we would like to extend it to a larger family of subsets (events) in analogy to the way this is done in the classical theory of stochastic processes. Since quantally μ is generally not of bounded variation, a new method is required. We propose a method that defines the measure of an event by means of a sequence of simpler events that in a suitable sense converges to the event whose measure we are seeking to define. To this end, we introduce canonical sequences approximating certain events, and we propose a measure-based criterion for the convergence of such sequences. Applying the method, we encounter a simple event whose measure is zero classically but non-zero quantally.

scholarly journals Vector valued mean-periodic functions on groups

2002 ◽  
Vol 72 (3) ◽  
pp. 363-388 ◽  
Author(s):  
P. Devaraj ◽  
Inder K. Rana
Keyword(s):  
Banach Space ◽  
Spectral Analysis ◽  
Spectral Synthesis ◽  
Complex Banach Space ◽  
Continuous Functions ◽  
Compact Sets ◽  
Periodic Functions ◽  
Finite Variation ◽  
Locally Compact ◽  
Vector Valued

AbstractLet G be a locally compact Hausdorif abelian group and X be a complex Banach space. Let C(G, X) denote the space of all continuous functions f: G → X, with the topology of uniform convergence on compact sets. Let X′ denote the dual of X with the weak* topology. Let Mc(G, X′) denote the space of all X′-valued compactly supported regular measures of finite variation on G. For a function f ∈ C(G, X) and μ ∈ Mc(G, X′), we define the notion of convolution f * μ. A function f ∈ C(G, X) is called mean-periodic if there exists a non-trivial measure μ ∈ Mc(G, X′) such that f * μ = 0. For μ ∈ Mc(G, X′), let M P(μ) = {f ∈ C(G, X): f * μ = 0} and let M P(G, X) = ∪μ M P(μ). In this paper we analyse the following questions: Is M P(G, X) ≠ 0? Is M P(G, X) ≠ C(G, X)? Is M P(G, X) dense in C(G, X)? Is M P(μ) generated by ‘exponential monomials’ in it? We answer these questions for the groups G = ℝ, the real line, and G = T, the circle group. Problems of spectral analysis and spectral synthesis for C(ℝ, X) and C(T, X) are also analysed.

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