A Girsanov result for the Pettis integral
A kind of Pettis integral representation for a Banach valued Itô process is given and its drift term is modified using a Girsanov Theorem.
math.PR↗
arXiv subjects
Domenico Candeloro
Publications and source records attributed to Domenico Candeloro.
Relations among Gauge and Pettis integrals for $cwk(X)$-valued multifunctions
The aim of this paper is to study relationships among "gauge integrals" (Henstock, Mc Shane, Birkhoff) and Pettis integral of multifunctions whose values are weakly compact and convex subsets of a general Banach space, not necessarily separable. For this purpose we prove the existence of variationally Henstock integrable selections for variationally Henstock integrable multifunctions. Using this and other known results concerning the existence of selections integrable in the same sense as the corresponding multifunctions, we obtain three decomposition theorems. As applications of such decompositions, we deduce characterizations of Henstock and ${\mathcal H}$ integrable multifunctions, together with an extension of a well-known theorem of Fremlin.
math.FA↗
Some new results on integration for multifunction
It has been proven in previous papers that each Henstock-Kurzweil-Pettis integrable multifunction with weakly compact values can be represented as a sum of one of its selections and a Pettis integrable multifunction. We prove here that if the initial multifunction is also Bochner measurable and has absolutely continuous variational measure of its integral, then it is a sum of a strongly measurable selection and of a variationally Henstock integrable multifunction that is also Birkhoff integrable.
math.FA↗
A vector Girsanov result and its applications to conditional measures via the Birkhoff integrability
Some integration techniques for real-valued functions with respect to vector measures with values in Banach spaces (and viceversa) are investigated in order to establish abstract versions of classical theorems of Probability and Stochastic Processes. In particular the Girsanov Theorem is extended and used with the treated methods.
math.FA↗
Integration of multifunctions with closed convex values in arbitrary Banach spaces
Integral properties of multifunctions with closed convex values are studied. In this more general framework not all the tools and the technique used for weakly compact convex valued multifunctions work. We pay particular attention to the "positive multifunctions". Among them an investigation of multifunctions determined by vector-valued functions is presented. Finally, decomposition results are obtained for scalarly and gauge-defined integrals of multifunctions and a full description of McShane integrability in terms of Henstock and Pettis integrability is given.
math.FA↗
Multi-integrals of finite variation
The aim of this paper is to investigate different types of multi-integrals of finite variation and to obtain decomposition results.
math.FA↗
A Girsanov Result through Birkhoff Integral
A vector-valued version of the Girsanov theorem is presented, for a scalar process with respect to a Banach-valued measure. Previously, a short discussion about the Birkhoff-type integration is outlined, as for example integration by substitution, in order to fix the measure-theoretic tools needed for the main result, Theorem 6, where a martingale equivalent to the underlying vector probability has been obtained in order to represent the modified process as a martingale with the same marginals as the original one.
math.FA↗
Properties of the Riemann-Lebesgue integrability in the non-additive case
We study Riemann-Lebesgue integrability of a vector function relative to an arbitrary non-negative set function. We obtain some classical integral properties. Results regarding the continuity properties of the integral and relationships among Riemann-Lebesgue, Birkhoff simple and Gould integrabilities are also established.
math.FA↗
A multivalued version of the Radon-Nikodym theorem, via the single-valued Gould integral
Some topics concerning the Gould integral are presented here: new results of integrability on finite measure spaces with values in an M-space are given, together with a Radon-Nikodym theorem relative to a Gould-type integral of real functions with respect to a multisubmeasure.
math.FA↗
A special class of fuzzy measures: Choquet integral and applications
Core of an economy and Walras equilibria are considered for a product space $X \times [0,1]$ using the Choquet integral with respect to a fuzzy measure.
math.FA↗
$L^p$ spaces in vector lattices and applications
$L^p$ spaces are investigated for vector lattice-valued functions, with respect to filter convergence. As applications, some classical inequalities are extended to the vector lattice context, and some properties of the Brownian Motion and the Brownian Bridge are studied, to solve some stochastic differential equations.
math.FA↗
Atomicity related to non-additive integrability
In this paper we present some results concerning Gould integrability of vector functions with respect to a monotone measure on finitely purely atomic measure spaces. As an application a Radon-Nikodym theorem in this setting is obtained.
math.FA↗
Henstock multivalued integrability in Banach lattices with respect to pointwise non atomic measures
Henstock-type integrals are considered, for multifunctions taking values in the family of weakly compact and convex subsets of a Banach lattice $X$. The main tool to handle the multivalued case is a Rådström-type embedding theorem established by C. C. A. Labuschagne, A. L. Pinchuck, C. J. van Alten in 2007. In this way the norm and order integrals reduce to that of a single-valued function taking values in an $M$-space, and new proofs are deduced for some decomposition results recently stated in two recent papers by Di Piazza and Musial based on the existence of integrable selections.
math.FA↗
Set-valued Brownian motion
Brownian motions, martingales, and Wiener processes are introduced and studied for set valued functions taking values in the subfamily of compact convex subsets of arbitrary Banach space $X$. The present paper is an application of one the paper of the second author in which an embedding result is obtained which considers also the ordered structure of $ck(X)$ and f-algebras.
math.FA↗
A note on set-valued Henstock--McShane integral in Banach (lattice) space setting
We study Henstock-type integrals for functions defined in a Radon measure space and taking values in a Banach lattice $X$. Both the single-valued case and the multivalued one are considered (in the last case mainly $cwk(X)$-valued mappings are discussed). The main tool to handle the multivalued case is a Rådström-type embedding theorem established in [50]: in this way we reduce the norm-integral to that of a single-valued function taking values in an $M$-space and we easily obtain new proofs for some decomposition results recently stated in [33,36], based on the existence of integrable selections. Also the order-type integral has been studied: for the single-valued case some basic results from [21] have been recalled, enlightning the differences with the norm-type integral, specially in the case of $L$-space-valued functions; as to multivalued mappings, a previous definition ([6]) is restated in an equivalent way, some selection theorems are obtained, a comparison with the Aumann integral is given, and decompositions of the previous type are deduced also in this setting. Finally, some existence results are also obtained, for functions defined in the real interval $[0,1]$.
math.FA↗
An extension of the Birkhoff integrability for multifunctions
A comparison between a set-valued Gould type and simple Birkhoff integrals of $bf(X)$-valued multifunctions with respect to a non-negative set functionis given. Relationships among them and Mc Shane multivalued integrability is given under suitable assumptions.
math.FA↗
Comparison between some norm and order gauge integrals in Banach lattices
Henstock-type integrals are considered, for functions defined in a Radon measure space and taking values in a Banach lattice X considering the norm and the order structure of the space. A number of results are obtained, highlighting the differences between them, specially in the case of L-space-valued functions.
math.FA↗
Order-type Henstock and McShane integrals in Banach lattice setting
We study Henstock-type integrals for functions defined in a compact metric space $T$ endowed with a regular $σ$-additive measure $μ$, and taking values in a Banach lattice $X$. In particular, the space $[0,1]$ with the usual Lebesgue measure is considered.
math.FA↗