Energy of taut strings accompanying Wiener process
2015 (English)In: Stochastic Processes and their Applications, ISSN 0304-4149, Vol. 125, no 2, 401-427 p.Article in journal (Refereed) Published
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.
Gaussian processes; Markovian pursuit; Taut string; Wiener process
IdentifiersURN: urn:nbn:se:liu:diva-115334DOI: 10.1016/j.spa.2014.09.020ISI: 000349501200001OAI: oai:DiVA.org:liu-115334DiVA: diva2:795025
Funding Agencies| [RFBR 13-01-00172]; [SPbSU 6.38.672.2013]2015-03-132015-03-132015-03-19