A stochastic model of the optimal functional extrapolator of a fractal L–Markov process with a quasi-rational spectrum is constructed. When developing the model, methods of spectral and fractal analysis of random processes, the theory of functions of a complex variable, methods for calculating stochastic integrals, and the theory of stochastic differential–difference equations connecting processes with a quasi-rational spectrum with processes with a rational spectrum were used.; as well as an original technique for constructing spectral characteristics of extrapolation, developed by the famous mathematician A. Yaglom. Using the Levinson–McKean theorem, it is established that the random processes studied in this paper are L–Markovian in nature. The fulfillment of the conditions of Mandelbrot's theorem on the shape of the spectral density of fractal random processes, as well as the values of the Hearst exponents and the fractality index, suggest that the random process under study is fractal and, moreover, persistent. It is proved that the optimal extrapolator constructed over the entire past of the process can be represented as the sum of a linear combination of the values of the process itself at three time points in the case of 0 < τ < 1 and at two time points in the case of 1 < τ < 2 and an integral with an exponentially decaying weight function extended to (– ∞; ∞). In the first case, the L – boundary of the L–Markov process under study consists of three points L = {t; t – 2; t + τ – 2}, and in the second case it consists of two points L = {t; t + τ – 2}, where τ is the lead time.

Keywords: extrapolation, L –Markov process, fractality, trend tolerance, spectral characteristic, optimal extrapolator

Explicit formulas for the spectral characteristic and optimal linear filtration operator with a forecast for stochastic L–Markov processes are obtained using methods of spectral analysis of random processes, the theory of functions of a complex variable, and using stochastic differential-difference equations. An interesting example of an optimal filtration operator with a forecast for an L-Markov process with a quasi-rational spectral density generalizing the rational one is constructed for technical applications. It is shown that the forecast filtering operator is the sum of a linear combination of the values of the received signal at some time points and the integral of an exponentially decaying weight function.

Keywords: random process, L-Markov process, prediction filtering, spectral characteristic, filtration operator