  In this work, we consider rather general and broad class of Markov chains,
Ito chains, that look like Euler-Maryama discretization of some Stochastic
Differential Equation. The chain we study is a unified framework for
theoretical analysis. It comes with almost arbitrary isotropic and
state-dependent noise instead of normal and state-independent one as in most
related papers. Moreover, in our chain the drift and diffusion coefficient can
be inexact in order to cover wide range of applications as Stochastic Gradient
Langevin Dynamics, sampling, Stochastic Gradient Descent or Stochastic Gradient
Boosting. We prove the bound in $W_{2}$-distance between the laws of our Ito
chain and corresponding differential equation. These results improve or cover
most of the known estimates. And for some particular cases, our analysis is the
first.
