Stochastic trends based on fuzzy mathematics

Currently, there are a number of ways to determine trends and extremes in stochastic time series, which is not surprising, since time series trends are a fundamental characteristic of the dynamics of the process behind it.
Real stochastic trends are not at all like ideal mathematical ones, because they contain violations. This does not bother the researcher, who initially has an adaptive perception of the fundamental properties of extremeness, continuity, connectedness, trend, etc. He will understand when the violation is insignificant and the trend continues, and when the violation interrupts the trend.
In this paper, we propose a new approach to the recognition of stochastic trends, based on the mathematical construction of regression derivatives for a finite time series. Trends are sought using the derivative from the scenario of classical mathematical analysis.