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  • 1.
    Ibragimov, Ildar
    et al.
    Russian Acad Sci, Russia; St Petersburg State Univ, Russia.
    Kabluchko, Zakhar
    Munster Univ, Germany.
    Lifshits, Mikhail
    Linköpings universitet, Matematiska institutionen, Matematisk statistik. Linköpings universitet, Tekniska fakulteten. St Petersburg State Univ, Russia.
    Some extensions of linear approximation and prediction problems for stationary processes2019Inngår i: Stochastic Processes and their Applications, ISSN 0304-4149, E-ISSN 1879-209X, Vol. 129, nr 8, s. 2758-2782Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    Let (B(t))(t is an element of Theta) with Theta = Z or Theta = R be a wide sense stationary process with discrete or continuous time. The classical linear prediction problem consists of finding an element in span{B(s), s amp;lt;= t} providing the best possible mean square approximation to the variable B(tau) with tau amp;gt; t. In this article we investigate this and some other similar problems where, in addition to prediction quality, optimization takes into account other features of the objects we search for. One of the most motivating examples of this kind is an approximation of a stationary process B by a stationary differentiable process X taking into account the kinetic energy that X spends in its approximation efforts. (C) 2018 Elsevier B.V. All rights reserved.

  • 2.
    Lifshits, Mikhail
    et al.
    Linköpings universitet, Matematiska institutionen, Matematisk statistik. Linköpings universitet, Tekniska högskolan. St Petersburg State University, Russia.
    Setterqvist, Eric
    Linköpings universitet, Matematiska institutionen, Matematik och tillämpad matematik. Linköpings universitet, Tekniska högskolan.
    Energy of taut strings accompanying Wiener process2015Inngår i: Stochastic Processes and their Applications, ISSN 0304-4149, E-ISSN 1879-209X, Vol. 125, nr 2, s. 401-427Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    Let W be a Wiener process. For r greater than 0 and T greater than 0 let I-W (T, r)(2) denote the minimal value of the energy integral(T)(0) h(t)(2)dt taken among all absolutely continuous functions h(.) defined on [0, T], starting at zero and satisfying W(t) - r less than= h(t) less than= W(t) + r, 0 less than= t less than= T. The function minimizing energy is a taut string, a classical object well known in Variational Calculus, in Mathematical Statistics, and in a broad range of applications. We show that there exists a constant C E (0, infinity) such that for any q greater than 0 r/T-1/2 I-W (T, r) -greater than(Lq) C, as r/T-1/2 -greater than 0, and for any fixed r greater than 0, r/(TIW)-I-1/2 (T, r)-greater than(a.s.) C, as T -greater than infinity. Although precise value of C remains unknown, we give various theoretical bounds for it, as well as rather precise results of computer simulation. While the taut string clearly depends on entire trajectory of W, we also consider an adaptive version of the problem by giving a construction (called Markovian pursuit) of a random function h(t) based only on the values W(s), s less than= t, and having minimal asymptotic energy. The solution, i.e. an optimal pursuit strategy, turns out to be related with a classical minimization problem for Fisher information on the bounded interval.

  • 3.
    Noble, John
    Institute of Applied Mathematics and Mechanics, University of Warsaw, Poland.
    Time homogeneous diffusions with a given marginal at a deterministic time2013Inngår i: Stochastic Processes and their Applications, ISSN 0304-4149, E-ISSN 1879-209X, Vol. 123, nr 3, s. 675-718Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    In this article, it is proved that for any probability law μ over R with finite first moment and a given deterministic time t>0, there exists a gap diffusion with law μ at the prescribed time t.

    The method starts by constructing a discrete time process X on a finite state space, where Xτ has law μ, for a geometric time τ, independent of the diffusion. This argument is developed, using a fixed point theorem, to give conditions for the existence of a process with prescribed law when stopped at an independent time with negative binomial distribution. Reducing the time mesh gives a continuous time diffusion with prescribed law for τ with Gamma distribution. Keeping E[τ]=t fixed, the parameters of the Gamma distribution are altered, giving the prescribed law for the deterministic time. An approximating sequence establishes the result for arbitrary probability measure over R.

  • 4.
    Privault, Nicolas
    et al.
    School of Physical and Mathematical Sciences, Nanyang Technological University, 21 Nanyang Link, Singapore.
    Yang, Xiangfeng
    Linköpings universitet, Matematiska institutionen, Matematisk statistik. Linköpings universitet, Tekniska fakulteten.
    Zambrini, Jean-Claude
    Group of Mathematical Physics, University of Lisbon, Lisbon, Portugal.
    Large deviations for Bernstein bridges2016Inngår i: Stochastic Processes and their Applications, ISSN 0304-4149, E-ISSN 1879-209X, Vol. 126, nr 5, s. 1285-1305Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    Bernstein processes over a finite time interval are simultaneously forward and backward Markov processes with arbitrarily fixed initial and terminal probability distributions. In this paper, a large deviation principle is proved for a family of Bernstein processes (depending on a small parameter ħ which is called the Planck constant) arising naturally in Euclidean quantum physics. The method consists in nontrivial Girsanov transformations of integral forms, suitable equivalence forms for large deviations and the (local and global) estimates on the parabolic kernel of the Schrödinger operator.

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