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

TL;DR

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

Contribution

It constructs a generalized Itô-Wentzell formula for systems with non-centered measures and defines stochastic first integrals within this framework.

Findings

Derived a generalized Itô-Wentzell formula for non-centered measure systems

Formed an Itô's GSDE system with solutions as kernels of integral invariants

Introduced the concept and conditions for stochastic first integrals in this context

Abstract

Generalization of the It\^{o}-Wentzell formula for the generalized It\^{o}'s SDE (It\^{o}'s GSDE) system with a non-centered measure is constructed on the basis of the stochastic kernel of integral transformation. The It\^{o}'s GSDE system for the kernel the solution of which is the kernel of the integral invariant, is formed. This invariant is connected with the solution of the It\^{o}'s GSDE system non-centered measure. The concept of a stochastic first integral of the It\^{o}'s GSDE system with non-centered measure is introduced and conditions that when being performed the random function is the first integral of the set It\^{o}'s GSDE system are defined.