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

Makhno, S. Ya.

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

Keywords:

convergence of solutions; solutions of the diffusion stochastic equation; perturbing process