N’Goran, Louis; N’Zi, Modeste Averaging principle for multi-valued stochastic differential equations. (English) Zbl 1020.60053 Random Oper. Stoch. Equ. 9, No. 4, 399-407 (2001). The authors study the limit behaviour as \(\varepsilon\to 0\) of the solution \(\{y_{t}^{\varepsilon}:0\leq t\leq\varepsilon^{-1}\}\) of the multivalued Itô stochastic differential equation \[ dy^{\varepsilon}_t+{\varepsilon}A(y^{\varepsilon}_t) dt\ni {\varepsilon}a(y^{\varepsilon}_t,\xi_t) dt+ {\varepsilon}^{1/2}b(y^{\varepsilon}_t) dW_t,\quad 0\leq t\leq\varepsilon^{-1}, \qquad y^{\varepsilon}_0=y_0, \] where \(A\) is a multi-valued maximal monotone operator and \(\{\xi_t: t\geq 0\}\) is a strictly stationary ergodic process independent of the Brownian motion \(W\). More precisely, it is proved that \(\{y_{t}^{\varepsilon}:0\leq t\leq\varepsilon^{-1}\}\) has the same limit behaviour as the solution of an analogous equation obtained by averaging over the interval \([0,\varepsilon^{-1}],\) the fluctuations in the drift term arising from the process \(\{\xi_t: t\geq 0\}\). The result can be applied to stochastic differential equations with reflecting boundary conditions or (and) with irregular drift. The obtained result coincides in the case \(A=0\) with that one obtained by R. Liptser and J. Stoyanov [Stochastics Stochastics Rep. 32, No. 3/4, 145–163 (1990; Zbl 0729.60047)]. Reviewer: M.P.Moklyachuk (Kyïv) Cited in 5 Documents MSC: 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis) 60J60 Diffusion processes 34A60 Ordinary differential inclusions 34C29 Averaging method for ordinary differential equations 34F05 Ordinary differential equations and systems with randomness 60H99 Stochastic analysis Keywords:stochastic differential equation; limit behaviour; stationary ergodic process; Brownian motion Citations:Zbl 0729.60047 PDFBibTeX XMLCite \textit{L. N'Goran} and \textit{M. N'Zi}, Random Oper. Stoch. Equ. 9, No. 4, 399--407 (2001; Zbl 1020.60053) Full Text: DOI