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Energy of taut strings accompanying Wiener process
Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, The Institute of Technology. St Petersburg State University, Russia.
Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology.
2015 (English)In: Stochastic Processes and their Applications, ISSN 0304-4149, Vol. 125, no 2, 401-427 p.Article in journal (Refereed) Published
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.

Place, publisher, year, edition, pages
Elsevier , 2015. Vol. 125, no 2, 401-427 p.
Keyword [en]
Gaussian processes; Markovian pursuit; Taut string; Wiener process
National Category
URN: urn:nbn:se:liu:diva-115334DOI: 10.1016/ 000349501200001OAI: diva2:795025

Funding Agencies| [RFBR 13-01-00172]; [SPbSU 6.38.672.2013]

Available from: 2015-03-13 Created: 2015-03-13 Last updated: 2015-03-19

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