Covariance Semigroup for Delayed Geometric Brownian Motion
Proceedings of the 12th International Conference on Applied Mathematics (APLIMAT 2013) 2013
Jevgeņijs Carkovs, Kārlis Šadurskis

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 In- selected deterministic part of analyzed stochastic equation. Furthermore, we have got a stochastic process, usable for Ito stochastic differential, by substituting the solution of the analyzed stochastic equation as an argument of this quadratic functional. 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 differential equation. As an example, we have deduced necessary and sufficient condition for mean square decrease of stochastic exponent given by Ito type scalar equation with delay.


Atslēgas vārdi
Functional differential equations, Second Lyapunov method. Mean square stability, Delayed stochastic exponent

Carkovs, J., Šadurskis, K. Covariance Semigroup for Delayed Geometric Brownian Motion. No: Proceedings of the 12th International Conference on Applied Mathematics (APLIMAT 2013), Slovākija, Bratislava, 5.-7. februāris, 2013. Bratislava: Slovak University of Technology, 2013, P13-1.-P13-7.lpp. ISBN 978-80-227-3865-1.

Publikācijas valoda
English (en)
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