The limit behaviour of integral functional of the solution of stochastic differential equation depending on small parameter (Q2740053)

The convergence in law of the integral functional \((\varepsilon^{k}/t)\int_{0}^{t/\varepsilon^{k}}d(s,\xi(s))ds\), as \(\varepsilon\to 0\), to the functional \((1/t)\int_0^{t}\overline d(\overline\xi(s)) ds\) is proved. Here \(\xi(s)\) is the solution of the stochastic differential equation NEWLINE\[NEWLINEd\xi(t)=\varepsilon^{k_1}f(t,\xi(t)) dt+\varepsilon^{k_2}g(t,\xi(t)) dw(t)+ \varepsilon^{k_3}\int_{R^{d}}q(t,\xi(t),y)\widetilde\nu(dt,dy),NEWLINE\]NEWLINE \(w(t)\) is a \(d\)-dimensional Wiener process, \(\widetilde\nu(dt,dy)\) is the centered Poisson measure independent on \(w(t)\), \(\overline\xi(t)\) is the solution of the averaging stochastic differential equation NEWLINE\[NEWLINE\overline d(x)=\lim_{T\to\infty}(1/T)\int_{A}^{T+A}d(t,x) dt.NEWLINE\]NEWLINE The dependence of the limiting stochastic differential equation on the order of small parameter in every term of the given equation is studied.