Existence and uniqueness theorem of associated solutions of the stochastic differential system with measures
Abstract
The Cauchy problem for stochastic differential system with measures is considered in the paper. Finite-difference with averaging system of stochastic differential equations with correspondence to Cauchy problem is investigated. Stochastic integral equation system with the solution as a limit of the finite-difference with averaging system of stochastic differential equations is studied and also the space of the solutions is described. Moreover, the Cauchy problem associated solutions of the stochastic differential system with measures are defined. Existence and uniqueness theorem of the associated solutions is proved. Besides that, Cauchy problem for higher-order linear stochastic differential equation is considered. Associated solutions of the Cauchy problem are investigated. Also the theorem of the representation of the associated solutions of the higher-order linear stochastic differential equation with help of associated fundamental matrices of the correspondent homogeneous equation system is proved.
References
- Arato M. [Linear stochastic systems with constant coefficients. Statistical approach]. Moscow : Nauka, 1989 (in Russ.).
- Gridnev Y. V., Rusetski A. Y., Rak S. A. [Stochastic simulation of the automatic control systems of unmanned aircraft using optimal Kalman filter]. Dokl. Beloruss. gos. univ. inform. i radioelektron. 2013. No. 8 (78). P. 53–59 (in Russ.).
- Shiryaev A. N. [Essentials of stochastic finance] : in 2 vol. Moscow : Fazis, 1998. Vol. 1 : Facts. Models (in Russ.).
- Gauss C. F. Selected geodetic works. Moscow : Publishing house of geodetic literature, 1957 (in Russ.).
- Vladimirov V. S. [Generalized equations in mathematical physics]. Moscow : Nauka, 1979 (in Russ.).
- Miller B. M., Rubinovich E. J. [Optimization of dynamic systems with a generalized control]. Moscow : Nauka, 2005 (in Russ.).
- Medvedev G. A. [Mathematical models of financial markets] : in 2 parts. Minsk : BSU, 2001. Part 1 (in Russ.).
- Autushka T. S., Lazakovich N. V., Rusetski A. U. The homogeneous linear differential equation of second order in the mnemofunctions algebra. Vestnik BGU. Ser. 1, Fiz. Mat. Inform. 2013. No. 2. P. 74 –79 (in Russ.).
- Autushka T. S. The Cauchy problem for non-homogeneous linear differential equations of higher order with generalized coefficients in algebra of mnemofunctions. Vesti Belarus. dzyarz. pedagagichnaga unіversіteta. Ser. 3, Fiz. Mat. Inform. Biol. Geogr. 2014. No. 2. P. 24 –28 (in Russ.).
- Autushka T. S., Lazakovich N. V., Rusetski A. U. [The Cauchy problem for non-homogeneous linear differential equations of second order with generalized coefficients in algebra of mnemofunctions]. Vesti Nacyjanalʼaj akadjemii navuk Belarusi. Ser. fiz.-mat. navuk. 2013. No. 3. P. 83–92 (in Russ.).
- Tacij R. M., Stasjuk M. F., Mazurenko V. V., et al. [Generalized quasidifferential equation]. Drogobich : Kolo, 2011 (in Ukrainian).
- Rozanov Y. A. [Random processes. Short course]. Moscow : Nauka, 1979 (in Russ.).
The authors who are published in this journal agree to the following:
- The authors retain copyright on the work and provide the journal with the right of first publication of the work on condition of license Creative Commons Attribution-NonCommercial. 4.0 International (CC BY-NC 4.0).
- The authors retain the right to enter into certain contractual agreements relating to the non-exclusive distribution of the published version of the work (e.g. post it on the institutional repository, publication in the book), with the reference to its original publication in this journal.
- The authors have the right to post their work on the Internet (e.g. on the institutional store or personal website) prior to and during the review process, conducted by the journal, as this may lead to a productive discussion and a large number of references to this work. (See The Effect of Open Access.)
