Least Energy Approximation for Processes with Stationary Increments

TL;DR

This paper investigates the least energy approximation of processes with stationary increments, providing explicit formulas, asymptotic behavior, and convergence results for Gaussian, Lévy, and fractional Brownian motion.

Contribution

It derives explicit solutions and asymptotic analysis for least energy approximations of stationary increment processes, including Gaussian and Lévy types, with convergence results.

Findings

Explicit formula for quadratic penalty case.

Asymptotic energy per unit time converges to a limit.

Almost sure and L^1 convergence for Gaussian and Lévy processes.

Abstract

A function $f=f_T$ is called least energy approximation to a function $B$ on the interval $[0,T]$ with penalty $Q$ if it solves the variational problem $$ \int_0^T \left[ f'(t)^2 + Q(f(t)-B(t)) \right] dt \searrow \min. $$ For quadratic penalty the least energy approximation can be found explicitly. If $B$ is a random process with stationary increments, then on large intervals $f_T$ 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 $T\to \infty$) the expectation of energy of optimal approximation per unit of time converges to some limit which we compute explicitly. For Gaussian and L\'evy processes we complete this result with almost sure and $L^1$ convergence. As an example, the asymptotic expression of approximation energy is found for fractional Brownian motion.