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Existence and uniqueness of solutions for a class of nonlinear stochastic differential equations. (English) Zbl 1279.60071

Summary: Using successive approximation, we prove an existence and uniqueness result for a class of nonlinear stochastic differential equations. Moreover, it is shown that the solution of such equations is a diffusion process and its diffusion coefficients are found.

MSC:

60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
34F05 Ordinary differential equations and systems with randomness
60J60 Diffusion processes
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[1] Kolmanovskii, V. B.; Nosov, V. R., Stability and Periodic Modes of Control System with Aftereffect (1981), Moscow, Russia: Nauka, Moscow, Russia
[2] Kolmanovskii, V.; Koroleva, N.; Maizenberg, T.; Mao, X.; Matasov, A., Neutral stochastic differential delay equations with Markovian switching, Stochastic Analysis and Applications, 21, 4, 819-847 (2003) · Zbl 1025.60028
[3] Mao, X., Stochastic Differential Equations and Their Applications (1997), Chichester, UK: Horwood, Chichester, UK
[4] Liao, X. X.; Mao, X., Exponential stability in mean square of neutral stochastic differential difference equations, Dynamics of Continuous, Discrete and Impulsive Systems, 6, 4, 569-586 (1999) · Zbl 0956.60069
[5] Mao, X., Asymptotic properties of neutral stochastic differential delay equations, Stochastics and Stochastics Reports, 68, 3-4, 273-295 (2000) · Zbl 0964.60066
[6] Samoilenko, A. M.; Mahmudov, N. I.; Stanzhytskii, O. M., Existence, uniqueness, and controllability results for neutral FSDES in Hilbert spaces, Dynamic Systems and Applications, 17, 1, 53-70 (2008) · Zbl 1145.93023
[7] Govindan, T. E., Stability of mild solutions of stochastic evolution equations with variable delay, Stochastic Analysis and Applications, 21, 5, 1059-1077 (2003) · Zbl 1036.60052
[8] Boukfaoui, Y.; Erraoui, M., Remarks on the existence and approximation for semilinear stochastic differential equations in Hilbert spaces, Stochastic Analysis and Applications, 20, 3, 495-518 (2002) · Zbl 1002.60058
[9] Mahmudov, N. I., Existence and uniqueness results for neutral SDEs in Hilbert spaces, Stochastic Analysis and Applications, 24, 1, 79-95 (2006) · Zbl 1110.60063
[10] Kolmanovskiĭ, V. B.; Shaĭkhet, L. E., Control of Systems with Aftereffect, 157 (1996), Providence, RI, USA: American Mathematical Society, Providence, RI, USA
[11] Gikhman, I. I.; Skorokhod, A. V., Stochastic Differential Equations (1968), Kyiv, Ukrania: Naukova Dumka, Kyiv, Ukrania · Zbl 0169.48702
[12] Gikhman, I. I.; Skorokhod, A. V., Introduction in Theory of Random Processes (1977), Kyiv, Ukrania: Naukova Dumka, Kyiv, Ukrania · Zbl 0429.60002
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