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  • 1.
    Lifshits, Mikhail
    et al.
    Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, The Institute of Technology. St Petersburg State University, Russia.
    Setterqvist, Eric
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology.
    Energy of taut strings accompanying Wiener process2015In: Stochastic Processes and their Applications, ISSN 0304-4149, E-ISSN 1879-209X, Vol. 125, no 2, p. 401-427Article in journal (Refereed)
    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.

  • 2.
    Noble, John
    Institute of Applied Mathematics and Mechanics, University of Warsaw, Poland.
    Time homogeneous diffusions with a given marginal at a deterministic time2013In: Stochastic Processes and their Applications, ISSN 0304-4149, E-ISSN 1879-209X, Vol. 123, no 3, p. 675-718Article in journal (Refereed)
    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.

  • 3.
    Privault, Nicolas
    et al.
    School of Physical and Mathematical Sciences, Nanyang Technological University, 21 Nanyang Link, Singapore.
    Yang, Xiangfeng
    Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, Faculty of Science & Engineering.
    Zambrini, Jean-Claude
    Group of Mathematical Physics, University of Lisbon, Lisbon, Portugal.
    Large deviations for Bernstein bridges2016In: Stochastic Processes and their Applications, ISSN 0304-4149, E-ISSN 1879-209X, Vol. 126, no 5, p. 1285-1305Article in journal (Refereed)
    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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