On Delayed Geometric Brownian Motion

Averaging procedure; impulse dynamical systems; Markov systems; weak convergence The paper proposes algorithm for time asymptotic analysis of stochastic linear functional differential equations. An approach is based on extension of the defined by deterministic part of this equation resolving semigroup to linear operator semigroup in the space of countable additive symmetric measures. The weak infinitesimal operator of this semigroup helps to find such Lyapunov-Krasovsky type quadratic functional that gives a necessary and sufficient asymptotic stability condition for the equation defined by selected deterministic part of analyzing stochastic equation. And what is more: substituting the solution of the analyzed stochastic equation as an argument of this quadratic functional we have got a stochastic process usable for Ito stochastic differential. This property permits to derive an analogue of Ito formula for the above mentioned stochastic process and to discuss equilibrium asymptotic stochastic stability conditions for initial stochastic functional di_erential equation. As an example we have deduced necessary and sufficient condition for mean square decreasing of stochastic exponent given by Ito type scalar equation with delay.