Diffusion approximations of some stochastic difference equations revisited
Let \(\{X_ n^{\epsilon}\}\) be stochastic processes with discrete parameters which are asymptotically expressed in the form \[ X^{\epsilon}_{n+1}-X_ n^{\epsilon}=\epsilon F(X_ n,n+1,\omega)+o(\epsilon^ 2)\quad (n=0,1,2,...),\quad X_ 0^{\epsilon}=X_ 0(\omega)\in R^ d, \] where F(x,n,\(\omega)\) \((E(F(x,n,\omega))=0)\) are certain random fields on a probability space. Then the stochastic process generated by linear interpolation of \(X_ n^{\epsilon}\) converges weakly to a diffusion process. The results are known more or less by \textit{H. J. Kushner} and \textit{Hai Huang} [SIAM J. Anal. Math. 40, 528-541 (1981; Zbl 0458.60074)] and the author [Hiroshima Math. J. 14, 15-34 (1984; Zbl 0571.60074)]. In this paper, the author introduces a kind of potential function which is similar to one introduced by Gordin and improves on the previous results.
scientific article; zbMATH DE number 4067995