The diffusion approximation for a class of random processes.(English. Russian original)Zbl 0835.60052

Theory Probab. Math. Stat. 46, 77-87 (1993); translation from Teor. Jmovirn. Mat. Stat. 46, 76-87 (1992).
Summary: Weak convergence of solutions of the stochastic equations $\begin{split} \xi^\varepsilon (t) = x^\varepsilon + \int^t_0 b^\varepsilon \bigl( s, \xi^\varepsilon (s), \eta^\varepsilon (s) \bigr) ds + \int^t_0 \sigma^\varepsilon \bigl( s, \xi^\varepsilon (s), \eta^\varepsilon (s) \bigr) dw^\varepsilon (s) + \\ + \int \int^t_0 c^\varepsilon \bigl( s, \xi^\varepsilon (s), \eta^\varepsilon (s), \theta \bigr) \mu^\varepsilon (d \theta) ds \end{split}$ to the solutions of the diffusion stochastic equation $\xi ( t) = x + \int^t_0 f \bigl( s, \xi (s) \bigr) ds + \int^t_0 g^{1/2} \bigl( s, \xi (s) \bigr) d w(s)$ is considered, without any assumption on the convergence of the coefficients of the equations or the perturbing process $$\eta^\varepsilon (t)$$.

MSC:

 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis) 60F17 Functional limit theorems; invariance principles