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  • 1. Bestill onlineKjøp publikasjonen >>
    Karlsson, John
    Linköpings universitet, Matematiska institutionen, Matematisk statistik. Linköpings universitet, Tekniska högskolan.
    A class of infinite dimensional stochastic processes with unbounded diffusion2013Licentiatavhandling, monografi (Annet vitenskapelig)
    Abstract [en]

    The aim of this work is to provide an introduction into the theory of infinite dimensional stochastic processes. The thesis contains the paper A class of infinite dimensional stochastic processes with unbounded diffusion written at Linköping University during 2012. The aim of that paper is to take results from the finite dimensional theory into the infinite dimensional case. This is done via the means of a coordinate representation. It is shown that for a certain kind of Dirichlet form with unbounded diffusion, we have properties such as closability, quasi-regularity, and existence of local first and second moment of the associated process. The starting chapters of this thesis contain the prerequisite theory for understanding the paper. It is my hope that any reader unfamiliar with the subject will find this thesis useful, as an introduction to the field of infinite dimensional processes.

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    A class of infinite dimensional stochastic processes with unbounded diffusion
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  • 2. Bestill onlineKjøp publikasjonen >>
    Karlsson, John
    Linköpings universitet, Matematiska institutionen, Matematisk statistik. Linköpings universitet, Tekniska fakulteten.
    A class of infinite dimensional stochastic processes with unbounded diffusion and its associated Dirichlet forms2015Doktoravhandling, med artikler (Annet vitenskapelig)
    Abstract [en]

    This thesis consists of two papers which focuses on a particular diffusion type Dirichlet form

     

    where  Here  is the basis in the Cameron-Martin space, H, consisting of the Schauder functions, and ν denotes the Wiener measure.

    In Paper I, we let  vary over the space of wiener trajectories in a way that the diffusion operator A is almost everywhere an unbounded operator on the Cameron–Martin space. In addition we put a weight function  on theWiener measure  and show that under these changes of the reference measure, the Malliavin derivative and divergence are closable operators with certain closable inverses. It is then shown that under certain conditions on , and these changes of reference measure, the Dirichlet form is quasi-regular. This is done first in the classical Wiener space and then the results are transferred to the Wiener space over a Riemannian manifold.

    Paper II focuses on the case when  is a sequence of non-decreasing real numbers. The process X associated to  is then an infinite dimensional Ornstein-Uhlenbeck process. In this case we show that the distributions of a sequence of certain finite dimensional Ornstein-Uhlenbeck processes converge weakly to the distribution of the infinite dimensional Ornstein-Uhlenbeck process. We also investigate the quadratic variation for this process, both in the classical sense and in the recent framework of stochastic calculus via regularization. Since the process is Banach space valued, the tensor quadratic variation is an appropriate tool to establish the Itô formula for the infinite dimensional Ornstein-Uhlenbeck process X. Sufficient conditions are presented for the scalar as well as the tensor quadratic variation to exist.

    Delarbeid
    1. A class of infinite dimensional stochastic processes with unbounded diffusion
    Åpne denne publikasjonen i ny fane eller vindu >>A class of infinite dimensional stochastic processes with unbounded diffusion
    2015 (engelsk)Inngår i: Stochastics: An International Journal of Probablitiy and Stochastic Processes, ISSN 1744-2508, E-ISSN 1744-2516, Vol. 87, nr 3, s. 424-457Artikkel i tidsskrift (Fagfellevurdert) Published
    Abstract [en]

    The paper studies Dirichlet forms on the classical Wiener space and the Wiener space over non-compact complete Riemannian manifolds. The diffusion operator is almost everywhere an unbounded operator on the Cameron-Martin space. In particular, it is shown that under a class of changes of the reference measure, quasi-regularity of the form is preserved. We also show that under these changes of the reference measure, derivative and divergence are closable with certain closable inverses. We first treat the case of the classical Wiener space and then we transfer the results to the Wiener space over a Riemannian manifold.

    sted, utgiver, år, opplag, sider
    Taylor and Francis: STM, Behavioural Science and Public Health Titles, 2015
    Emneord
    Dirichlet form on Wiener space; Dirichlet form on Wiener space over non-compact manifold; closability; weighted Wiener measure; quasi-regularity
    HSV kategori
    Identifikatorer
    urn:nbn:se:liu:diva-118070 (URN)10.1080/17442508.2014.959952 (DOI)000353580300004 ()
    Tilgjengelig fra: 2015-05-20 Laget: 2015-05-20 Sist oppdatert: 2017-12-04
    2. Infinite dimensional Ornstein-Uhlenbeck processes with unbounded diffusion: Approximation, quadratic variation, and Itô formula
    Åpne denne publikasjonen i ny fane eller vindu >>Infinite dimensional Ornstein-Uhlenbeck processes with unbounded diffusion: Approximation, quadratic variation, and Itô formula
    2016 (engelsk)Inngår i: Mathematische Nachrichten, ISSN 0025-584X, E-ISSN 1522-2616, Vol. 289, nr 17-18, s. 2192-2222Artikkel i tidsskrift (Fagfellevurdert) Published
    Abstract [en]

    The paper studies a class of Ornstein-Uhlenbeck processes on the classical Wiener space. These processes are associated with a diffusion type Dirichlet form whose corresponding diffusion operator is unbounded in the Cameron- Martin space. It is shown that the distributions of certain finite dimensional Ornstein-Uhlenbeck processes converge weakly to the distribution of such an infinite dimensional Ornstein-Uhlenbeck process. For the infinite dimensional processes, the ordinary scalar quadratic variation is calculated. Moreover, relative to the stochastic calculus via regularization, the scalar as well as the tensor quadratic variation are derived. A related Itô formula is presented.

    sted, utgiver, år, opplag, sider
    Wiley-VCH Verlagsgesellschaft, 2016
    Emneord
    Infinite dimensional Ornstein-Uhlenbeck process, quadratic variation, Itô formula, weak approximation
    HSV kategori
    Identifikatorer
    urn:nbn:se:liu:diva-122181 (URN)10.1002/mana.201500146 (DOI)000389128100008 ()
    Merknad

    At the time for thesis presentation publication was in status: Manuscript.

    Tilgjengelig fra: 2015-10-23 Laget: 2015-10-23 Sist oppdatert: 2017-12-01bibliografisk kontrollert
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    fulltext
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    omslag
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    presentationsbild
  • 3.
    Karlsson, John
    et al.
    Linköpings universitet, Matematiska institutionen, Matematisk statistik. Linköpings universitet, Tekniska högskolan.
    Löbus, Jörg-Uwe
    Linköpings universitet, Matematiska institutionen, Matematisk statistik. Linköpings universitet, Tekniska högskolan.
    A class of infinite dimensional stochastic processes with unbounded diffusion2015Inngår i: Stochastics: An International Journal of Probablitiy and Stochastic Processes, ISSN 1744-2508, E-ISSN 1744-2516, Vol. 87, nr 3, s. 424-457Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    The paper studies Dirichlet forms on the classical Wiener space and the Wiener space over non-compact complete Riemannian manifolds. The diffusion operator is almost everywhere an unbounded operator on the Cameron-Martin space. In particular, it is shown that under a class of changes of the reference measure, quasi-regularity of the form is preserved. We also show that under these changes of the reference measure, derivative and divergence are closable with certain closable inverses. We first treat the case of the classical Wiener space and then we transfer the results to the Wiener space over a Riemannian manifold.

  • 4.
    Karlsson, John
    et al.
    Linköpings universitet, Matematiska institutionen, Matematisk statistik. Linköpings universitet, Tekniska fakulteten.
    Löbus, Jörg-Uwe
    Linköpings universitet, Matematiska institutionen, Matematisk statistik. Linköpings universitet, Tekniska fakulteten.
    Infinite dimensional Ornstein-Uhlenbeck processes with unbounded diffusion: Approximation, quadratic variation, and Itô formula2016Inngår i: Mathematische Nachrichten, ISSN 0025-584X, E-ISSN 1522-2616, Vol. 289, nr 17-18, s. 2192-2222Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    The paper studies a class of Ornstein-Uhlenbeck processes on the classical Wiener space. These processes are associated with a diffusion type Dirichlet form whose corresponding diffusion operator is unbounded in the Cameron- Martin space. It is shown that the distributions of certain finite dimensional Ornstein-Uhlenbeck processes converge weakly to the distribution of such an infinite dimensional Ornstein-Uhlenbeck process. For the infinite dimensional processes, the ordinary scalar quadratic variation is calculated. Moreover, relative to the stochastic calculus via regularization, the scalar as well as the tensor quadratic variation are derived. A related Itô formula is presented.

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