Geodesic
On existence of solutions to stochastic differential inclusions with current velocities II
Journal of computational and engineering mathematics, Tome 3 (2016) no. 1, pp. 48-60.

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Existence of solution theorems are obtained for stochastic differential inclusions given in terms of the so-called current velocities (symmetric mean derivatives, a direct analogs of ordinary velocity of deterministic systems) and quadratic mean derivatives (giving information on the diffusion coefficient) on the flat $n$-dimensional torus. Right-hand sides in both the current velocity part and the quadratic part are set-valued but satisfy some natural conditions, under which they have $\varepsilon$-approximations that point-wise converge to Borel measurable selections of the corresponding set-valued mappings.
Keywords: mean derivatives, current velocities, differential inclusions. 
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Yu. E. Gliklikh; A. V. Makarova. On existence of solutions to stochastic differential inclusions with current velocities II. Journal of computational and engineering mathematics, Tome 3 (2016) no. 1, pp. 48-60. http://geodesic.mathdoc.fr/item/JCEM_2016_3_1_a5/

[1] S. V. Azarina, Yu. E. Gliklikh, “Differential inclusions with mean derivatives”, Dynamic Systems and Applications, 16:1 (2007), 49-71 | MR | Zbl

[2] S. V. Azarina, Yu. E. Gliklikh, “O suschestvovanii reshenii stokhasticheskikh differentsialnykh uravnenii s tekuschimi skorostyami”, Vestnik Yuzhno-Uralskogo gosudarstvennogo universiteta. Seriya: Matematicheskoe modelirovanie i programmirovanie, 8:4 (2015), 100-106 | MR | Zbl

[3] Yu. G. Borisovich, B. D. Gelman, A. D. Myshkis, V. V. Obukhovskii, Vvedenie v teoriyu mnogoznachnykh otobrazhenii i differentsialnykh vklyuchenii, Komkniga, Moskva, 2005

[4] J. Cresson, S. Darses, “Stochastic Embedding of Dynamical Systems”, Journal of Mathematical Physics, 48:7 (2007), 072703-1–072303-54 | DOI | MR

[5] I. I. Gihman, A. V. Skorohod, Theory of Stochastic Processes, v. 1, Springer, New York, 1974 | MR | Zbl

[6] I. I. Gihman, A. V. Skorohod, Theory of Stochastic Processes, v. 3, Springer, New York, 1979 | Zbl

[7] Yu. E. Gliklikh, Global and Stochastic Analysis with Applications to Mathematical Physics, Springer–Verlag, London, 2011 | MR | Zbl

[8] Yu. E. Gliklikh, A. V. Makarova, “On Stochastic Differential Inclusions with Current Velocities”, Journal of Computational and Engineering Mathematics, 2:3 (2015), 25-33 | DOI | MR

[9] A. V. Makarova, “On Solvability of Stochastic Differential Inclusions with Current Velocities II”, Global and Stochastic Analysis, 2:1 (2012), 101-112 | MR | Zbl

[10] E. Nelson, “Derivation of the Schrödinger Equation from Newtonian Mechanics”, Physical Review, 150:4 (1966), 1079-1085 | DOI

[11] E. Nelson, Dynamical Theory of Brownian Motion, Princeton University Press, Princeton, 1967 | MR

[12] E. Nelson, Quantum Fluctuations, Princeton University Press, Princeton, 1985 | MR | Zbl

[13] K. R. Parthasarathy, Introduction to Probability and Measure, Springer, New York, 1978 | MR

[14] D. P. Zhelobenko, Compact Lie Groups and Their Representations, American Mathematical Society, Providence RI, 1973 [] | MR | Zbl

[15] K. Yosida, Functional Analysis, Springer – Verlag, Berlin et. al., 1965 | Zbl