On existence of solution in $\mathbb{R}^n$ of~stochastic differential inclusions with current velocities

A. V. Makarova ; A. A. Demchuk ; S. S. Novikova

Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, Tome 21 (2017) no. 1, pp. 42-54.

Résumé

Notion of mean derivatives was introduced by Edward Nelson for the needs of stochastic mechanics (a version of quantum mechanics). Nelson introduced forward and backward mean derivatives while only their half-sum, symmetric mean derivative called current velocity, is a direct analog of ordinary velocity for deterministic processes. Another mean derivative called quadratic, was introduced by Yuri E. Gliklikh and Svetlana V. Azarina. It gives information on the diffusion coefficient of the process and using Nelson's and quadratic mean derivatives together, one can in principle recover the process from its mean derivatives. Since the current velocities are natural analogs of ordinary velocities of deterministic processes, investigation of equations and especially inclusions with current velocities is very much important for applications since there are a lot of models of various physical, economical etc. processes based on such equations and inclusions. 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 $\mathbb{R}^n$. Right-hand sides in both the current velocity part and the quadratic part are set-valued but satisfy some natural conditions.

Détail

Keywords: mean derivatives, current velocities, differential inclusions.

Bibliographie

[1] Gliklikh Yu. E., “Stochastic equations in mean derivatives and their applications. I”, Izv. Ross. Akad. Estestv. Nauk, Mat. Mat. Model. Inform. Upr., 1:4 (1997), 26–52 (In Russian) | Zbl

[2] Azarina S. V., Gliklikh Yu. E., “On existence of solutions to stochastic differential equations with current velocities”, Vestnik YuUrGU. Ser. Mat. Model. Progr., 8:4 (2015), 100–106 | DOI | MR | Zbl

[3] Gliklikh Yu. E., Makarova A. V., “On solvability of stochastic differential inclusions with current velocities”, Applicable Analysis, 91:9 (2012), 1731–1739 | DOI | MR | Zbl

[4] Gliklikh Yu. E., Makarova A. V., “On solution existence theorem for stochastic differential inclusions with current velocities”, Nauchnye vedomosti Belgorodskogo gosudarstvennogo universiteta. Seriia: Matematika. Fizika, 28:17(136) (2012), 5–15 (In Russian)

[5] Makarova A. V., “On solvability of stochastic differential inclusions with current velocities. II”, Global and Stochastic Analysis, 2:1 (2012), 101–112 | MR | Zbl

[6] Gliklikh Yu. E., Makarova A. V., “On stochastic differential inclusions with current velocities”, J. Comp. Eng. Math., 2:3 (2015), 25–33 | DOI | MR | Zbl

[7] Makarova A. V., “On the solvability of differential inclusions with current velocities with a lower semicontinuous right-hand side”, Vestnik Lipetskogo gosudarstvennogo pedagogicheskogo universiteta. Seriia MIFE: matematika, informatsionnye tekhnologii, fizika, estestvoznanie, 2015, no. 1(16), 22–29 (In Russian)

[8] Gliklikh Yu. E., Makarova A. V., “On existence of solutions to stochastic differential inclusions with current velocities. II”, J. Comp. Eng. Math., 3:1 (2016), 48–60 | DOI | MR | Zbl

[9] Gihman I. I., Skorokhod A. V., The Theory of Stochastic Processes I, Classics in Mathematics, 210, Springer, Berlin, 2004, viii+574 pp. | DOI | MR | MR

[10] Nelson E., “Derivation of the Schrödinger equation from Newtonian mechanics”, Phys. Rev., 150:4 (1966), 1079–1085 | DOI

[11] Nelson E., Dynamical theory of Brownian motion, Princeton University Press, Princeton, New Jersey, 1967, 142 pp. | MR

[12] Nelson E., Quantum fluctuations, Princeton Series in Physics, Princeton University Press, Princeton, New Jersey, 1985, viii+146 pp. | MR | Zbl

[13] Gliklikh Yu. E., Global and Stochastic Analysis with Applications to Mathematical Physics, Theoretical and Mathematical Physics, Springer, London, 2011, xxiii+436 pp. | DOI | MR | Zbl

[14] Partasarati K., Vvedenie v teoriiu veroiatnostei i teoriiu mery [ An Introduction to Probability Theory and Measure Theory], Mir, Moscow, 1988, 344 pp. (In Russian) | MR

[15] Borisovich Yu. G., Gel'man B. D., Myshkis A. D., Obukhovskii V. V., Vvedenie v teoriiu mnogoznachnykh otobrazhenii i differentsial'nykh vkliuchenii [Introduction to the theory of multivalued mappings and differential inclusions], Komkniga, Moscow, 2005, 215 pp. (In Russian) | MR

[16] Yosida K., Functional Analysis, Reprint of the 1980 Edition (Grundlehren der mathematischen Wissenschaften, Vol. 123), Classics in Mathematics, Springer-Verlag, Berlin, 1995, xvi+504 pp. | DOI | MR

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

[18] Billingsley P., Convergence of probability measures, Wiley Series in Probability and Statistics, Wiley, Chichester, 1999, x+277 pp. | DOI | MR | Zbl