Least Energy Approximation for Processes with Stationary Increments

A function is called least energy approximation to a function B on the interval [0, T] with penalty Q if it solves the variational problem integral(T)(0)[f'(t)(2) + Q(f(t) - B(t))] dt SE arrow 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, 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 ), the expectation of energy of optimal approximation per unit of time converges to some limit which we compute explicitly. For Gaussian and L,vy processes, we complete this result with almost sure and convergence. As an example, the asymptotic expression of approximation energy is found for fractional Brownian motion.