Zero-Dimensional Selection Theorem

1998 ◽  
pp. 54-59
Author(s):  
Dušan Repovš ◽  
Pavel Vladimirovič Semenov
Keyword(s):  
Selection Theorem

Stability of stochastic differential equations driven by multifractional Brownian motion

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Oussama El Barrimi ◽  
Youssef Ouknine
Keyword(s):  
Brownian Motion ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Stationary Process ◽  
Strong Stability ◽  
Pathwise Uniqueness ◽  
Uniqueness Property ◽  
Selection Theorem ◽  
Multifractional Brownian Motion ◽  
Stability Results

Abstract Our aim in this paper is to establish some strong stability results for solutions of stochastic differential equations driven by a Riemann–Liouville multifractional Brownian motion. The latter is defined as a Gaussian non-stationary process with a Hurst parameter as a function of time. The results are obtained assuming that the pathwise uniqueness property holds and using Skorokhod’s selection theorem.

A selection theorem for a new class of multivalued maps

1998 ◽  
Vol 53 (5) ◽  
pp. 1089-1090
Author(s):  
S A Drozdovskii ◽  
V V Filippov
Keyword(s):  
Multivalued Maps ◽  
Selection Theorem ◽  
New Class

scholarly journals A Generalization of the Blackwell–Ryll-Nardzewski Measurable Selection Theorem

2021 ◽  
Vol 134 (1) ◽  
pp. 35-41
Author(s):  
K.A. Afonin ◽  
◽  
Keyword(s):  
Metric Spaces ◽  
Transition Probability ◽  
Measurable Space ◽  
Measurable Selection ◽  
Selection Theorem ◽  
Borel Set ◽  
Borel Probability ◽  
Borel Mapping ◽  
The Given ◽  
Selection Of

One of the main forms of the measurable selection theorem is connected with the existence of the graph of a measurable mapping in a given measurable set 𝑆 in the product of two measurable spaces 𝑋 and 𝑌 . Such a graph enables one to pick a point in the section 𝑆𝑥 for each 𝑥 in 𝑋 such that the obtained mapping will be measurable. The indicated selection is called a measurable selection of the multi-valued mapping associating to the point 𝑥 the section 𝑆𝑥 , which is a set in 𝑌 . The classical theorem of Blackwell and Ryll-Nardzewski states that a Borel set 𝑆 in the product of two complete separable metric spaces contains the graph of a Borel mapping (hence admits a Borel selection) provided that there is a transition probability on this product with positive measures for all sections of 𝑆 . The main result of this paper gives a generalization to the case where only one of the two spaces is complete separable and the other one is a general measurable space whose points parameterize a family of Borel probability measures on the first space such that the sections of the given set 𝑆 in the product have positive measures.

Addendum: New Proof of Finite-Dimensional Selection Theorem

1998 ◽  
pp. 126-145
Author(s):  
Dušan Repovš ◽  
Pavel Vladimirovič Semenov
Keyword(s):  
Selection Theorem ◽  
Finite Dimensional

Compact-Valued Selection Theorem

1998 ◽  
pp. 75-83
Author(s):  
Dušan Repovš ◽  
Pavel Vladimirovič Semenov
Keyword(s):  
Selection Theorem

A measurable selection theorem for compact-valued maps

1979 ◽  
Vol 27 (4) ◽  
pp. 341-352 ◽  
Author(s):  
Siegfried Graf
Keyword(s):  
Measurable Selection ◽  
Selection Theorem

An effective selection theorem

1982 ◽  
Vol 47 (2) ◽  
pp. 388-394 ◽  
Author(s):  
Ashok Maitra
Keyword(s):  
Polish Space ◽  
Local Version ◽  
Compact Set ◽  
Selection Theorem ◽  
Analytic Subset ◽  
Effective Version ◽  
Basis Theorem ◽  
Measurable Selector ◽  
Effective Selection ◽  
Borel Measurable

A recent result of J.P. Burgess [1] states:Theorem 0. Let F be a multifunction from an analytic subset T of a Polish space to a Polish space X. If F is Borel measurable, Graph(F) is coanalytic in T × X and F(t) is nonmeager in its closure for each t Є T, then F admits a Borel measurable selector.The above result unifies and significantly extends earlier results of H. Sarbadhikari [8], S.M. Srivastava [9] and G. Debs (unpublished). The reader is referred to [1] for details.The aim of this article is to give an effective version of Theorem 0. We do this by proving a basis theorem for Π11 sets which are nonmeager in their closure and satisfy a local version of the measurability condition in Theorem 0. Our basis theorem generalizes a well-known result of P.G. Hinman [4] and S.K. Thomason [10] (see also [5] and [7, 4F.20]). Our methods are similar to those used by A. Louveau to prove that a , σ-compact set is contained in a , σ-compact set (see [7, 4F.18]).The paper is organized as follows. §2 is devoted to preliminaries. In §3, we prove the basis theorem and deduce as a consequence an effective version of Theorem 0. We show in §4 how our methods can be used to give alternative proofs of some known results.Discussions with R. Barua, B.V. Rao and V.V. Srivatsa are gratefully acknowledged. I am indebted to J.P. Burgess for drawing my attention to an error in an earlier draft of this paper.

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