Construction of a Stochastic Model for a Linear Extrapolator of an L-Markov Fractal Process

Introduction. Problems arising in information theory, automatic control theory, statistical radio engineering and radiophysics, metrology, geology, and other fields often necessitate the study of extrapolation, interpolation, and filtering of random processes and fields. In many cases, when analyzing complex systems characterized by stochastic behavior, the underlying processes are either Markovian or closely related to Markov processes. Today, spectral estimation of random processes and fields is regarded as one of the most promising mathematical methods for investigating such systems. In practice, the spectrum of a random process is typically measured using physical instruments such as spectrum analyzers. Many real-world random processes and fields exhibit invariance under scaling transformations. Consequently, their study reduces to the analysis of stochastic self-similar processes, which can be described using fractal sets. Fractal geometry methods, based on the concept of fractal dimension, have proven to be effective tools for evaluating such complex and heterogeneous processes and fields. The purpose of this work is to confirm the author's hypothesis that a subclass of fractal processes exists within the broader class of L-Markov processes, and to synthesize a stochastic model for the optimal linear extrapolator for this subclass. Basic approaches. This paper presents key aspects of the general theory of stationary random functions, emphasizing the spectral representation of functions whose spectral densities possess specific forms. The physical meaning of spectral representation lies in the ability to isolate spectral components corresponding to different parts of the spectrum using appropriately designed filters. By modeling the input signal as a real stationary random process with a spectral representation, it becomes possible to simulate, forecast, and qualitatively control the filtering process. To construct the optimal linear extrapolator, the study utilizes stochastic methods for analyzing random processes and fields, the theory of analytic and entire functions, the theory of stochastic differential-difference equations with delayed arguments, and methods of fractal analysis for random processes. Conclusions. It was found that within the class of L-Markov processes, for certain values of the parameters characterizing their quasi-rational spectral densities, there exists a subclass of processes with fractal properties. These processes are persistent, exhibiting a high Hurst exponent and, correspondingly, a low fractal dimension. As a result, they possess sufficient memory, enabling reliable and accurate prediction. A model for a linear extrapolator for an L-Markov fractal random process with a quasi-rational spectrum has been synthesized. The findings show that the optimal linear extrapolator for a forward time interval τ depends only on the process values at a finite number of points and does not involve an integral term. This result is particularly important for many technical applications that require the identification, classification, filtering, and recognition of random processes.