On the minimal integral energy of majorants of the Wiener process

2025/07/23 by Nikitin, S. E.
#60F15 #60G15 #60G17 #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2507.17468

Abstract

We consider the asymptotic behavior (over long time intervals) of the minimal integral energy |h|_T^ψ= ∫₀^T ψ(h^′(t)) dt of majorants of the Wiener process W(⋅) satisfying the constraints h(0) = r , h(t) ≥ W(t) for 0 ≤ t ≤ T . The results significantly generalize previous asymptotic estimates obtained for the case of kinetic energy ψ(u) = u² , revealing that this case, where the minimal energy grows logarithmically, is a critical one, lying between two different asymptotic regimes.

Citations

• Asymptotic distribution of the derivative of the taut string accompanying Wiener process
• Energy saving approximation of Wiener process under unilateral constraints
• Markovian structure in the concave majorant of Brownian motion
• Renewal Structure of the Brownian Taut String

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