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
Bibliography: 1. I. I. Gikhman and A. V. Skorokhod, Stochastic Dierential Equations, Springer, Berlin-New York, 1972.
2. I. I. Gikhman and A. V. Skorokhod, Introduction to the Theory of Random Processes, W. B. Saunders Co., PhiladelphiaLondonToronto, 1969.
3. G. L. Kulinich, Limit behavior of the distribution of the solution of a stochastic diusion equation, Ukr. Math. J. 19 (1968), 231-235.
4. G. L. Kulinich, On the limit behavior of the distribution of the solution of a stochastic diusion equation, Theory Probab. Appl. 12 (1967), no. 3, 497-499.
5. G. L. Kulinich, Limit distributions for functionals of integral type of unstable diusion processes, Theory Probab. Math. Statist. 11 (1976), 82-86.
6. G. L. Kulinich and E. P. Kaskun, On the asymptotic behavior of solutions of one-dimensional Ito's stochastic dierential equations with singularity points, Theory Stoch. Process. 4 (20) (1998), no. 1-2, 189-197.
7. G. Kulinich, S. Kushnirenko, Yu. Mishura, Asymptotic behavior of homogeneous additive functionals of the solutions of Ito stochastic dierential equations with nonregular dependence on parameter, Mod. Stoch. Theory Appl. 3 (2016), no. 2, 191-208.
8. G. L. Êulinich, S. V. Êushnirånkî and Yu. S. Mishura, Asymptotic behavior of the martingale type integral functionals for unstable solutions to stochastic dierential equations, Theory Probab. Math. Statist. 90 (2015), 115-126.
9. G. L. Êulinich, S. V. Êushnirånkî and Yu. S. Mishura, Limit behavior of functionals of diusion type processes. Theory Probab. Math. Statist. 92 (2016), 93107.
10. N. V. Krylov, On Ito's stochastic integral equations, Theory Probab. Appl. 14 (1969), no. 2, 330-336.
11. Yu. V. Prokhorov, Convergence of random processes and limit theorems in probability theory,Theory Probab. Appl. 1 (1956), no. 2, 157-214.
12. A. V. Skorokhod, Studies in the Theory of Random Processes, Addison-Wesley Publishing Co., Inc., Reading, Mass., 1965.
13. A. V. Skorokhod and N. P. Slobodenyuk, Limit Theorems for Random Walks, Naukova Dumka, Kiev, 1970. (Russian)
14. A. Yu. Veretennikov, On the strong solutions of stochastic dierential equations, Theory Probab. Appl. 24 (1979), no. 2, 354-366.