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Theory of Probability and Mathematical Statistics



WEAK CONVERGENCE OF INTEGRAL FUNCTIONALS DEFINED ON THE SOLUTIONS OF STOCHASTIC DIFFERENTIAL ITO EQUATIONS WITH NON-REGULAR DEPENDENCE ON THE PARAMETER

G. L. Кulіnіch, S. V. Кushnіrеnkо and Yu. S. Mishura

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Abstract: We study the weak convergence as $ T \to \infty $ of functionals $ \int_ {0} ^ {t} g_T (\xi_T (s)) \, dW_T (s) $, $ t \ge 0 $. Here $ \xi_T (t) $ is a strong solution of stochastic differential equation $ d \xi_T (t) = a_T (\xi_T (t)) \, dt + dW_T (t) $, $ T> 0 $ is a parameter, $ a_T (x) $ are real measurable functions, $ x \in \R $, $ \left| a_T (x) \right| \leq C_T \; \hbox{for all} \; x $, $ W_T (t) $ are standard Wiener processes, $ g_T (x) $ are real, measurable, locally bounded, non-random functions. The explicit form of the limiting processes for these functionals is established under non-regular dependence $ g_T (x) $, $ a_T (x) $ on the parameter $T$.

Keywords: Процеси дифузійного типу, гранична поведінка інтегральних функціоналів, нерегулярна залежність від параметра

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