Least Energy Approximation for Processes with Stationary Increments

Zakhar Kabluchko () and Mikhail Lifshits ()
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Zakhar Kabluchko: Münster University
Mikhail Lifshits: St. Petersburg State University

Journal of Theoretical Probability, 2017, vol. 30, issue 1, 268-296

Abstract: Abstract A function $$f=f_T$$ f = f T is called least energy approximation to a function B on the interval [0, T] with penalty Q if it solves the variational problem $$\begin{aligned} \int _0^T \left[ f'(t)^2 + Q(f(t)-B(t)) \right] dt \searrow \min . \end{aligned}$$ ∫ 0 T f ′ ( t ) 2 + Q ( f ( t ) - B ( t ) ) d t ↘ min . For quadratic penalty, the least energy approximation can be found explicitly. If B is a random process with stationary increments, then on large intervals, $$f_T$$ f T also is close to a process of the same class, and the relation between the corresponding spectral measures can be found. We show that in a long run (when $$T\rightarrow \infty $$ T → ∞ ), the expectation of energy of optimal approximation per unit of time converges to some limit which we compute explicitly. For Gaussian and Lévy processes, we complete this result with almost sure and $$L^1$$ L 1 convergence. As an example, the asymptotic expression of approximation energy is found for fractional Brownian motion.

Keywords: Least energy approximation; Gaussian process; Lévy process; Fractional Brownian motion; Process with stationary increments; Taut string; Variational calculus; Primary: 60G10; Secondary: 60G15; 49J40; 41A00 (search for similar items in EconPapers)
Date: 2017
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DOI: 10.1007/s10959-015-0642-8

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