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Orthogonal decompositions for generalized stochastic processes with independent values. / Suman Das

Swansea University Author: Suman Das

Abstract

Among all stochastic processes with independent increments, essentially only Brownian motion and Poisson process have a chaotic representation property. In the case of a Levy process, several approaches have been proposed in order to construct an orthogonal decomposition of the corresponding L2-spac...

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Published: 2013
Institution: Swansea University
Degree level: Doctoral
Degree name: Ph.D
URI: https://cronfa.swan.ac.uk/Record/cronfa42660
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spelling 2018-08-02T16:24:30.0241979 v2 42660 2018-08-02 Orthogonal decompositions for generalized stochastic processes with independent values. ff00a4f6d7a587e1dca98978e21bd50f NULL Suman Das Suman Das true true 2018-08-02 Among all stochastic processes with independent increments, essentially only Brownian motion and Poisson process have a chaotic representation property. In the case of a Levy process, several approaches have been proposed in order to construct an orthogonal decomposition of the corresponding L2-space. In this dissertation, we deal with orthogonal (chaotic) decompositions for generalized processes with independent values. We do not suppose stationarity of the process, as a result the Levy measure of the process depends on points of the space. We first construct, in Chapter 3, a unitary isomorphism between a certain symmetric Fock space and the space L2 (D',mu). Here D' is a co-nuclear space of generalized functions (distributions), and mu is a generalized stochastic process with independent values. This isomorphism is constructed by employing the projection spectral theorem for an (uncountable) family of commuting self-adjoint operators. We next derive, in Chapter 4, a counterpart of the Nualart Schoutens decomposition for generalized stochastic process with independent values. Our results here extend those in the papers of Nualart Schoutens and Lytvynov. In Chapter 5, we construct an orthogonal decomposition of the space L2 (D',mu) in terms of orthogonal polynomials on D'. We observe a deep relation between this decomposition and the results of the two previous chapters. Finally, in Chapter 6, we briefly discuss the situation of the generalized stochastic processes of Meixner's type. E-Thesis Mathematics. 31 12 2013 2013-12-31 COLLEGE NANME Mathematics COLLEGE CODE Swansea University Doctoral Ph.D 2018-08-02T16:24:30.0241979 2018-08-02T16:24:30.0241979 Faculty of Science and Engineering School of Mathematics and Computer Science - Mathematics Suman Das NULL 1 0042660-02082018162511.pdf 10805436.pdf 2018-08-02T16:25:11.9430000 Output 2908169 application/pdf E-Thesis true 2018-08-02T16:25:11.9430000 false
title Orthogonal decompositions for generalized stochastic processes with independent values.
spellingShingle Orthogonal decompositions for generalized stochastic processes with independent values.
Suman Das
title_short Orthogonal decompositions for generalized stochastic processes with independent values.
title_full Orthogonal decompositions for generalized stochastic processes with independent values.
title_fullStr Orthogonal decompositions for generalized stochastic processes with independent values.
title_full_unstemmed Orthogonal decompositions for generalized stochastic processes with independent values.
title_sort Orthogonal decompositions for generalized stochastic processes with independent values.
author_id_str_mv ff00a4f6d7a587e1dca98978e21bd50f
author_id_fullname_str_mv ff00a4f6d7a587e1dca98978e21bd50f_***_Suman Das
author Suman Das
author2 Suman Das
format E-Thesis
publishDate 2013
institution Swansea University
college_str Faculty of Science and Engineering
hierarchytype
hierarchy_top_id facultyofscienceandengineering
hierarchy_top_title Faculty of Science and Engineering
hierarchy_parent_id facultyofscienceandengineering
hierarchy_parent_title Faculty of Science and Engineering
department_str School of Mathematics and Computer Science - Mathematics{{{_:::_}}}Faculty of Science and Engineering{{{_:::_}}}School of Mathematics and Computer Science - Mathematics
document_store_str 1
active_str 0
description Among all stochastic processes with independent increments, essentially only Brownian motion and Poisson process have a chaotic representation property. In the case of a Levy process, several approaches have been proposed in order to construct an orthogonal decomposition of the corresponding L2-space. In this dissertation, we deal with orthogonal (chaotic) decompositions for generalized processes with independent values. We do not suppose stationarity of the process, as a result the Levy measure of the process depends on points of the space. We first construct, in Chapter 3, a unitary isomorphism between a certain symmetric Fock space and the space L2 (D',mu). Here D' is a co-nuclear space of generalized functions (distributions), and mu is a generalized stochastic process with independent values. This isomorphism is constructed by employing the projection spectral theorem for an (uncountable) family of commuting self-adjoint operators. We next derive, in Chapter 4, a counterpart of the Nualart Schoutens decomposition for generalized stochastic process with independent values. Our results here extend those in the papers of Nualart Schoutens and Lytvynov. In Chapter 5, we construct an orthogonal decomposition of the space L2 (D',mu) in terms of orthogonal polynomials on D'. We observe a deep relation between this decomposition and the results of the two previous chapters. Finally, in Chapter 6, we briefly discuss the situation of the generalized stochastic processes of Meixner's type.
published_date 2013-12-31T03:53:24Z
_version_ 1763752651598594048
score 11.013148

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