Stochastic Inclusions with Forward Mean Derivatives Having Decomposable Right-Hand Sides
DOI:
https://doi.org/10.14529/mmp190212Keywords:
mean derivatives, decomposable set-valued mappings, differential inclusions.Abstract
In this paper, we prove a theorem on the existence of solutions for stochastic differential inclusions given in terms of the forward mean derivatives and the quadratic mean derivatives. These derivatives present information on the drift and the diffusion coefficient, respectively. The forward mean derivatives were introduced by E. Nelson for the needs of the so-called stochastic mechanics (a version of quantum mechanics), while the quadratic mean derivatives were introduced by Yu.E. Gliklich and S.V. Azarina. In the case of both the forward mean derivatives and the quadratic mean derivatives, we assume that the righthand side is set-valued and lower semi-continuous, but not necessarily convex. Instead of this, we assume that the right-hand side is decomposable. Such inclusions naturally arise in many models of physical processes.References
Nelson E. Derivation of the Schr¨odinger Equation from Newtonian Mechanics. Physical Review, 1966, vol. 150, pp. 1079–1085. DOI: 10.1103/PhysRev.150.1079
Nelson E. Dynamical Theory of Brownian Motion. Princeton, Princeton University Press, 1967.
Nelson E. Quantum Fluctuations. Princeton, Princeton University Press, 1985.
Azarina S.V., Gliklikh Yu.E. Differential Inclusions with Mean Derivatives. Dynamic Systems and Applications, 2007, vol. 16, pp. 49–71.
Azarina S.V., Gliklikh Yu.E. Stochastic Differential Equations and Inclusions with Mean Derivatives Relative to the Past. International Journal of Difference Equations, 2009, vol. 4, no. 1, pp. 27–41.
Parthasarathy K.R. Introduction to Probability and Measure. N.Y., Springer, 1978. DOI: 10.1007/978-1-349-03365-2
Azarina S.V, Gliklikh Yu.E., Obukhovskii A.V. Solvability of Langevin Differential Inclusions with Set-Valued Diffusion Terms on Riemannian Manifolds. Applicable Analysis, 2007, vol. 86,
no. 9, pp. 1105–1116. DOI: 10.1080/00036810701556045
Gliklikh Yu.E. Global and Lochalic Analysis with Applications to Mathematical Physics. London, Springer, 2011.
Deimling K. Multivalued Differential Equations. Berlin, Walter de Gruyter, 1992. DOI: 10.1515/9783110874228
Kamenskii M., Obukhovskii V., Zecca P. Condensing Multivalued Maps and Semilinear Differential Inclusions in Banach Spaces. Berlin, Walter de Gruyter, 2001. DOI: 10.1515/9783110870893
Gihman I.I., Skorohod A.V. Theory of Stochalic Processes. N.Y., Springer, 1979. DOI: 10.1007/978-1-4612-6202-2
Gliklikh Yu.E., Obukhovskii A.V. On a Second Order Differential Inclusion with Random Perturbation of Velocity. Proceedings of Voronezh State University, 2003, no. 1, pp. 93–95.
Gliklikh Yu.E., Makarova A.V. On Stochastic Differential Inclusions with Current Velocities. Journal of Computational and Engineering Mathematics, 2015, vol. 2, no. 3, pp. 25–33.
