Approximations and forecasting quasi-stationary processes with sudden runs

The object of research is heteroskedastic processes in the formation and evaluation of the offset policy of states in international markets, in particular, in the conclusion and execution of offset contracts. The process of concluding and fulfilling the conditions of an offset contract is weakly unsteady, because at its conclusion there can be a wide variety of sudden events, force majeure circumstances that can’t be described and predicted in detail, with acceptable accuracy, in full. It is shown that the term «quasi-stationary process», which has some approximation to the stationary process, is more suitable for describing such processes. The most promising approach to constructing mathematical models of such processes is the use of combined fractal autoregression models and an integrated moving average. During the study, methods of the theory of non-stationary random processes and the theory of runs of random processes were used, it is shown that the combination of a quasi-stationary process with a sequence of random runs is quite satisfactorily modeled by the so-called fractal or self-similar process. As a universal mathematical model of self-similar processes with slowly decreasing dependencies, the model of fractal integrated autoregression and the moving average FARIMA are used. However, this model does not take into account the effect of runs of random processes on the coefficients of the numerator and denominator of the finely rational function, which approximates the process of autoregression and the moving average. Therefore, in the study, the relationship in the parameters of the runs and the coefficients of the approximating function. Mathematical models of discrete time series are considered, which are characterized by quasi-stationary and the presence of sudden runs. It is shown that for approximation of such series, models such as autoregression and moving average are quite suitable, modified to the classes of autoregression models and integrated moving average. And for self-similar (fractal) processes – modified to the classes of models of autoregression and fractal integrated moving average. The relationship between the shear length when calculating autocorrelation coefficients and the total sample length can serve as an acceptable indicator of the correctness of the solution.