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Energy of taut strings accompanying Wiener process

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  • Lifshits, Mikhail
  • Setterqvist, Eric

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 ∫0Th′(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 C∈(0,∞) such that for any q>0rT1/2IW(T,r)⟶LqC,as rT1/2→0, and for any fixed r>0, rT1/2IW(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.

Suggested Citation

  • Lifshits, Mikhail & Setterqvist, Eric, 2015. "Energy of taut strings accompanying Wiener process," Stochastic Processes and their Applications, Elsevier, vol. 125(2), pages 401-427.
    • Handle: RePEc:eee:spapps:v:125:y:2015:i:2:p:401-427
    DOI: 10.1016/j.spa.2014.09.020
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    References listed on IDEAS

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    1. Łochowski, Rafał M. & Miłoś, Piotr, 2013. "On truncated variation, upward truncated variation and downward truncated variation for diffusions," Stochastic Processes and their Applications, Elsevier, vol. 123(2), pages 446-474.
    2. George B. Dantzig, 1971. "A Control Problem of Bellman," Management Science, INFORMS, vol. 17(9), pages 542-546, May.
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    Cited by:

    1. Zakhar Kabluchko & Mikhail Lifshits, 2017. "Least Energy Approximation for Processes with Stationary Increments," Journal of Theoretical Probability, Springer, vol. 30(1), pages 268-296, March.
    2. Schertzer, Emmanuel, 2018. "Renewal structure of the Brownian taut string," Stochastic Processes and their Applications, Elsevier, vol. 128(2), pages 487-504.

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