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Averaging principle for multi-valued stochastic differential equations. (English) Zbl 1020.60053

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)].

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

Citations:

Zbl 0729.60047
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