On a generalization of the It\^{o}-Wentzel formula for system of generalized It\^{o}'s SDEs and the stochastic first integral

TL;DR

This paper extends the Itô-Wentzel formula to systems of generalized Itô stochastic differential equations with non-centered measures, introducing stochastic first integrals and conditions for their existence.

Contribution

It develops a generalized Itô-Wentzel formula for systems of GSDEs with non-centered measures and introduces the concept of stochastic first integrals for such systems.

Findings

Constructed the generalized Itô-Wentzel formula for GSDEs with non-centered measures.

Defined the stochastic first integral for GSDEs systems.

Established conditions for functions to be first integrals of GSDEs systems.

Abstract

Generalization of the It\^{o}-Wentzel formula for the generalized It\^{o}'s SDEs (It\^{o}'s GSDEs) system with not centered measure is constructed. This construction is based on the basis of the stochastic kernel of integral transformation. The It\^{o}'s GSDEs system for the kernel of the integral invariant is constructed. The concept of a stochastic first integral of the It\^{o}'s GSDEs system with not centered measure is introduced. The conditions for the random function, that it's the first integral of the set It\^{o}'s GSDEs system, are defined.