# Stochastic Leontieff Type Equations and Mean Derivatives of Stochastic Processes

Yu.E. Gliklikh, E.Yu. MashkovWe understand the Leontieff type stochastic differential equations as a special sort of Ito stochastic differential equations, in which the left-hand side contains a degenerate constant linear operator and the right-hand side has a non-degenerate constant linear operator. In the right-hand side there is also a summand with a term depending only on time. Its physical meaning is the incoming signal into the device described by the operators mentioned above. In the papers by A.L. Shestakov and G.A. Sviridyuk the dynamical distortion of signals is described by such equations. Transition to stochastic differential equations arise where it is necessary to take into account the interference (noise). Note that the investigation of solutions of such equations requires the use of derivatives of the incoming signal and the noise of any order. In this paper for differentiation of noise we apply the machinery of the so-called Nelson's mean derivatives of stochastic processes. This allows us to avoid using the machinery of the theory of generalized functions. We present a brief introduction to the theory of mean derivatives, investigate the transformation of the equations to canonical form and find formulae for solutions in terms of Nelson's mean derivatives of Wiener process.Full text

- Keywords
- mean derivative, current velocity, Wiener process, Leontieff type equation.
- References
- 1. Shestakov A.L., Sviridyuk G.A. A New Approach to Measurement of Dynamically Perturbed Signals. Bulletin of the South Ural State University. Series "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. Bulletin of the South Ural State University. Series "Mathematical Modelling, Programming & Computer Software", 2011, no. 17 (234), issue 8, pp. 70-75.

3. Gliklikh Yu.E. Investigation of Leontieff Type Equations with White Noise by the Method of Mean Derivatives of Stochastic Processes. Bulletin of the South Ural State University. Series "Mathematical Modelling, Programming & Computer Software", 2012, no. 27 (286), issue 13, pp. 24-34. (in Russian)

4. Nelson E. Derivation of the Schr'odinger Equation from Newtonian Mechanics. Phys. Reviews, 1966, vol. 150, no. 4, pp. 1079-1085.

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

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

7. Gliklikh Yu.E., Mashkov E.Yu. On Reduction of Leontieff Type Stochastic Equations to Canonical Form. Izmereniya: sostoyanie, perspektivy razvitiya: tez. dokl. mezhdunar. nauch.-prakt. konf., Chelyabinsk 25-27 sentyabrya 2012 g. [Measurements: the State of the Problem and the Prospects of Developmets. Abstracts of Communications of the International Scientific-Practical Conference. Chelyabinsk 25-27 September 2012. Vol. 1]. Chelyabinsk, Publ. Center of the South Ural State University, 2012, pp. 73-75. (in Russian)

8. Shestakov A.L., Sviridyuk G.A. On the Measurement of the 'White Noise'. Bulletin of the South Ural State University. Series "Mathematical Modelling, Programming & Computer Software", 2012, no. 27 (286), issue 13, pp. 99-108.

9. Gliklikh Yu.E. Global and Stochastic Analysis in the Problems of Mathematical Physics. Moscow, KomKniga, 2005. 416 p. (in Russian)

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

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

12. Cresson J., Darses S. Stochastic Embedding of Dynamical Systems. Journal of Mathematical Physics, 2007, vol. 48, pp. 072703-1 - 072303-54. [DOI: 10.1063/1.2736519].

13. Gantmakher F.R. Theory of Matrices. Moscow, Fizmatlit, 1967. 575 p. (in Russian)

14. Gikhman I.I., Skorokhod A.V. Introduction to the Theory of Stochastic Processes. Moscow, Nauka, 1977. 567 p. (in Russian)