# Optimal Solutions for Inclusions of Geometric Brownian Motion Type with Mean Derivatives

Yu.E. Gliklikh, O.O. ZheltikovaThe idea of mean derivatives of stochastic processes was suggested by E. Nelson in 60-th years of XX century. Unlike ordinary derivatives, the mean derivatives are well-posed for a very broad class of stochastic processes and equations with mean derivatives naturally arise in many mathematical models of physics (in particular, E. Nelson introduced the mean derivatives for the needs of Stochastic Mechanics, a version of quantum mechanics). Inclusions with mean derivatives is a natural generalization of those equations in the case of feedback control or in motion in complicated media. The paper is devoted to a brief introduction into the theory of equations and inclusions with mean derivatives and to investigation of a special type of such inclusions called inclusions of geometric Brownian motion type. The existence of optimal solutions maximizing a certain cost criterion, is proved.Full text

- Keywords
- mean derivatives; stochastic differential inclusions; optimal solution.
- References
- 1. Nelson E. Derivation of the Schrodinger Equation from Newtonian Mechanics. Phys. Reviews, 1966, vol. 150, pp. 1079-1085.

2. Nelson E. Dynamical Theory of Brownian Motion. Princeton, Princeton University Press, 1967. 142 p.

3. Nelson E. Quantum Fluctuations. Princeton, Princeton University Press, 1985. 147 p.

4. Gliklikh Yu.E. Global and Stochastic Analysis with Applications to Mathematical Physics. London, Springer-Verlag, 2011. 460 p.

5. Azarina S.V., Gliklikh Yu.E. Differential Inclusions with Mean Derivatives. Dynamic Systems and Applications, 2007, vol. 16, pp. 49-72.

6. Azarina S.V., Gliklikh Yu.E. Inclusions with Mean Derivatives for Porcesses of Geometric Brownian Motion Type and Their Applications [Vklyucheniya s proizvodnymi v srednem dlya protsessov tipa geometricheskogo brounovskogo dvizheniya i ikh prilozheniya]. Seminar po global'nomu i stokhasticheskomu analizu [Seminar on Global and Stochastic Analysis], 2009, issue 4, pp. 3-8.

7. Borisovich Yu.G., Gelman B.D., Myshkis A.D. , Obukhovskii V.V. Vvedenie v teoriyu mnogoznachnykh otobrazheniy i differentsial'nykh vklyucheniy [Introduction to the Theory of Multi-Valued Mappings and Differential Inclusions]. Moscow, KomKniga, 2005. 213 p.

8. Gliklikh Yu.E. Global'nyy i stakhosticheskiy analiz v zadachakh matematicheskoy fiziki [Global and Stochastic Analysis in Problems of Mathematical Physics]. Moscow, KomKniga, 2005. 416 p.

9. Gihman I.I., Skorohod A.V. Theory of Stochastic Processes. Vol. 3. N.Y., Springer-Verlag, 1979. 496 p.

10. Kantorovich L.V., Akilov G.P. Functional analysis. Oxford, Pergamon Press, 1982. 742 p.

11. Parthasarathy K.R. Introduction to Probability and Measure. N.Y., Springer-Verlag, 1978. 343 p.

12. Gliklikh Yu.E., Obukhovskiui A.V. Stochastic Differential Inclusions of Langevin Type on Riemannian Manifolds. Discussiones Mathematicae DICO, 2001, vol. 21, pp. 173-190.

13. Yosida Y. Functional Analysis. Berlin, Springer-Verlag, 1965. 624 p.

14. Billingsley P. Convergence of Probability Measures. N.Y., Wiley, 1969. 351. p.