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Energy of taut strings accompanying Wiener process. / Lifshits, M.; Setterqvist, E.

In: Stochastic Processes and their Applications, Vol. 125, No. 2, 2015, p. 401-427.

Research output: Contribution to journal › Article

Harvard

Lifshits, M & Setterqvist, E 2015, 'Energy of taut strings accompanying Wiener process', Stochastic Processes and their Applications, vol. 125, no. 2, pp. 401-427. https://doi.org/10.1016/j.spa.2014.09.020

APA

Lifshits, M., & Setterqvist, E. (2015). Energy of taut strings accompanying Wiener process. Stochastic Processes and their Applications, 125(2), 401-427. https://doi.org/10.1016/j.spa.2014.09.020

Vancouver

Lifshits M, Setterqvist E. Energy of taut strings accompanying Wiener process. Stochastic Processes and their Applications. 2015;125(2):401-427. https://doi.org/10.1016/j.spa.2014.09.020

Author

Lifshits, M. ; Setterqvist, E. / Energy of taut strings accompanying Wiener process. In: Stochastic Processes and their Applications. 2015 ; Vol. 125, No. 2. pp. 401-427.

BibTeX

@article{7c1c84ab33b24596a9594d1c9df2d9b5,
title = "Energy of taut strings accompanying Wiener process",
abstract = "Let W be a Wiener process. For r>0 and T>0 let IW(T,r)2 denote the minimal value of the energy ℓ0Th′(t)2dt taken among all absolutely continuous functions h(·) defined on [0,T], starting at zero and satisfying W(t)-r≤h(t)≤;W(t)+r,0≤t≤;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 Cε(0,∞) such that for any q>0rT1/2 IW(T,r)→LqC,as rT1/2→0, and for any fixed r>0, rT1/2 IW(T,r) →a.s.C,as T→∞. 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≤t, and having minimal asymptotic energy. The solution, i.e. an optimal pursuit strategy, turns",
author = "M. Lifshits and E. Setterqvist",
year = "2015",
doi = "10.1016/j.spa.2014.09.020",
language = "English",
volume = "125",
pages = "401--427",
journal = "Stochastic Processes and their Applications",
issn = "0304-4149",
publisher = "Elsevier",
number = "2",

}

RIS

TY - JOUR

T1 - Energy of taut strings accompanying Wiener process

AU - Lifshits, M.

AU - Setterqvist, E.

PY - 2015

Y1 - 2015

N2 - Let W be a Wiener process. For r>0 and T>0 let IW(T,r)2 denote the minimal value of the energy ℓ0Th′(t)2dt taken among all absolutely continuous functions h(·) defined on [0,T], starting at zero and satisfying W(t)-r≤h(t)≤;W(t)+r,0≤t≤;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 Cε(0,∞) such that for any q>0rT1/2 IW(T,r)→LqC,as rT1/2→0, and for any fixed r>0, rT1/2 IW(T,r) →a.s.C,as T→∞. 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≤t, and having minimal asymptotic energy. The solution, i.e. an optimal pursuit strategy, turns

AB - Let W be a Wiener process. For r>0 and T>0 let IW(T,r)2 denote the minimal value of the energy ℓ0Th′(t)2dt taken among all absolutely continuous functions h(·) defined on [0,T], starting at zero and satisfying W(t)-r≤h(t)≤;W(t)+r,0≤t≤;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 Cε(0,∞) such that for any q>0rT1/2 IW(T,r)→LqC,as rT1/2→0, and for any fixed r>0, rT1/2 IW(T,r) →a.s.C,as T→∞. 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≤t, and having minimal asymptotic energy. The solution, i.e. an optimal pursuit strategy, turns

U2 - 10.1016/j.spa.2014.09.020

DO - 10.1016/j.spa.2014.09.020

M3 - Article

VL - 125

SP - 401

EP - 427

JO - Stochastic Processes and their Applications

JF - Stochastic Processes and their Applications

SN - 0304-4149

IS - 2

ER -

ID: 3984219