Nonlinear trend exclusion procedure for models defined by stochastic differential and difference equations

We consider a diffusion process and its approximation with a Markov chain whose trends contain a nonlinear unbounded component. The usual parametrix method is inapplicable here since the trend is unbounded. We present a procedure that lets us exclude a nonlinear growing trend and pass to a stochastic differential equation with bounded drift and diffusion coefficients. A similar procedure is also considered for a Markov chain.