No. 27 (286), issue 13Pages 24 - 34

# Investigation of Leontieff Type Equations with White Noise protect by the Methods of Mean Derivatives of Stochastic Processes

Yu.E. Gliklikh
We understand the Leontieff type equation with white noise as the expression of the form \$Ldotxi(t)=Mxi(t)+dot w(t)\$ where L is a degenerate matrix \$n imes n\$, M is a non-degenerate matrix \$n imes n\$, \$xi(t)\$ is a stochastic process we are looking for and \$dot w(t)\$ is the white noise. Since the derivative \$dotxi(t)\$ and the white noise are well-posed only in terms of distributions, the direct investigation of such equations is very complicated. We involve two methods in the investigation. First, we pass to the stochastic differential equation \$Lxi(t)=Mint_0^txi(s)ds+w(t)\$, where w(t) is Wiener process, and then for describing solutions of this equations we apply the so called Nelson mean derivatives that are introduced without using the distributions. By these methods we obtain formulae for solutions of Leotieff type equations with white noise.
Full text
Keywords
mean derivative, current velocity, white nose, Wiener process, Leontieff type equation.
References
1. Shestakov A.L., Sviridyuk G.A. A New Approach to Measuring Dynamically Distorted Signals. Vestnik Yuzhno-Ural'skogo gosudarstvennogo universiteta. Seriya "Matematicheskoe modelirovanie i programmirovanie" - Bulletin of South Ural State University. Seria "Mathematical Modelling, Programming & Computer Software", 2010, no. 16 (192), issue 5, pp. 116-120. (in Russian)
2. Shestakov A.L., Sviridyuk G.A. Optimal Measurement of Dynamically Distorted Signals. Vestnik Yuzhno-Ural'skogo gosudarstvennogo universiteta. Seriya "Matematicheskoe modelirovanie i programmirovanie" - Bulletin of South Ural State University. Seria "Mathematical Modelling, Programming & Computer Software", 2011, no. 17(234), issue 8, pp. 70-75.
3. Nelson E. Derivation of the Schr'odinger Equation from Newtonian Mechanics. Phys. Reviews, 1966, vol. 150, no. 4, pp. 1079-1085.
4. Nelson E. Dynamical Theory of Brownian Motion. Princeton, Princeton University Press, 1967. 142 p.
5. Nelson E. Quantum Fluctuations. Princeton, Princeton University Press, 1985. 147 p.
6. Gliklikh Yu.E. Global and Stochastic Analysis in Problems Mathematical Physics. Moscow, KomKniga, 2005. 416 p. (in Russian)
7. Gliklikh Yu.E. Global and Stochastic Analysis with Applications to Mathematical Physics. London, Springer-Verlag, 2011. 460 p.
8. Partasarati K. An Introduction to Probability Theory and Measure Theory. Moscow, Mir, 1988. 343 p. (in Russian)
9. Gantmakher F.R. The Theory of Matrices. Moscow, Fizmatlit, 1967. 575 p. (in Russian)