Stability properties of stochastic differential equations driven by Lévy noise.

摘要

The main aim of this thesis is to examine stability properties of the solutions toudstochastic differential equations (SDEs) driven by Levy noise.udUsing key tools such as Ito's formula for general semimartingales, Kunita's momentudestimates for Levy-type stochastic integrals, and the exponential martingale inequality,udwe find conditions under which the solutions to the SDEs under consideration are stableudin probability, almost surely and moment exponentially stable. In addition, stabilityudproperties of stochastic functional differential equations (SFDEs) driven by Levy noiseudare examined using Razumikhin type theorems.udIn the existing literature the problem of stochastic stabilization and destabilization ofudfirst order non-linear deterministic systems has been investigated when the system isudperturbed with Brownian motion. These results are extended in this thesis to the caseudwhere the deterministic system is perturbed with Levy noise. We mainly focus onudthe stabilizing effects of the Levy noise in the system, prove the existence of sampleudLyapunov exponents of the trivial solution of the stochastically perturbed system, andudprovide sufficient criteria under which the system is almost surely exponentially stable.udFrom the results that are established the Levy noise plays a similar role to the Brownianudmotion in stabilizing dynamical systems.udWe also establish the variation of constants formula for linear SDEs driven by Levyudnoise. This is applied to study stochastic stabilization of ordinary functional differentialudequation systems perturbed with Levy noise.