Analytical Modeling and Estimation of Normal Processes Defined By Stochastic Differential Equations with Unsolved Derivatives

Sinitsyn I

Published on: 2021-02-05


As the environment is full of microorganisms, the human life appeared in devoid of them have never been possible to us. While most of these microorganisms are beneficial to us, unfortunately, some of them have always been as toxic as poison. Though the toxic substances released by several microorganisms have already been documented in the literature but the studies on their effects on human and animals are not as adequate to derive any conclusion due to the delayed toxic effects developing ailments, diseases and even cancer in future courses of time. Therefore, more researches are still required to fill the gap. The present review is an attempt to discuss the various algae producing toxins causing diseases and cancer in human.


Method of analytical modeling (MAM); Multiplicated noise; Pugachev estimators (filters, extrapolators etc); Regression model (deterministic and stochastic); Stochastic differential equations with unsolved derivatives (SDE USD)


Approximate methods of analytical modeling (MAM) of the wide band stochastic processes (StP) in stochastic differential equations with unsolved derivatives (SDE) USD) based on normal approximate method (NAM), orthogonal expansions method and quasimoment methods are developed in [1, 2]. For stochastic integrodifferential equations with unsolved derivatives (SIDE USD) reducible to SDE corresponding Eqs for MAM are given in [3, 4]. In [3,4] problems of mean square (m.s.) synthesis of normal (Gaussian) estimators (filters, extrapolators, etc) where firstly stated and solved for filtering. Results presented in [1-4] are valid only for sooth (in m.s. sense) functions in SDE USD.

Let us generalize [1-3] results for unsmooth functions in SDE USD for normal filtering and extrapolation. Section 2 is dedicated to SDE USD. In Section 3 deterministic and stochastic regression models for SDE USD are discussed. Section 4 and 5 are devoted to normal m.s. filtering and extrapolation. Examples are given in Section 6. In Section 7 applications to SDE USD with multiplicated Gaussian noises are given. Conclusion contains some remarks concerning future generalizations.