ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

16. Applications

1991 MS Classification: 60, 93, 39, 73 Rodkina, Alexandra, Voronezh State Academy of Construction and Architecture On stochastic functional-differential equations and its applications Stochastic functional-differential equation with respect to the semimartingale dx_{t} = a(t, x_{t−}^{−∞})dA_{t} + b(t, x_{t−}^{−∞})dM_{t}, t ≥ 0, x_{t} = φ_{t}, t ≤ 0, (1) is considered. Here x_{t−}^{−∞}(s) = x(s), s ∈ (−∞, t], is the trajectory of the process x_{t} up to the moment t, M_{t} is a martingale, A_{t} is an increasing process, φ_{t} is a given initial process. Ito equation and stochastic difference equation are embedded in Eq. (1). As the applications of Eq.

(1) we consider the generalized stochastic approximation procedures, the controlled motion of a rigid body in the presence of random noises and some other technical problems, which can be described by Eq. (1) (see[1]). The method of Lyapunov-Krasovskii functional and some approaches from the martingale theory are used. Computer simulations are presented. Reference: V. B. Kolmanovskii and A. D. Myshkis, Applied Theory of Functional Differential Equations. Kluwer Academic Publishers. Dortrecht-Boston-London. 1992.