Stability properties of stochastic differential equations driven by Lévy noise.

The main aim of this thesis is to examine stability properties of the solutions to
stochastic differential equations (SDEs) driven by Levy noise.
Using key tools such as Ito's formula for general semimartingales, Kunita's moment
estimates for Levy-type stochastic integrals, and the exponential martingale inequality,
we find conditions under which the solutions to the SDEs under consideration are stable
in probability, almost surely and moment exponentially stable. In addition, stability
properties of stochastic functional differential equations (SFDEs) driven by Levy noise
are examined using Razumikhin type theorems.
In the existing literature the problem of stochastic stabilization and destabilization of
first order non-linear deterministic systems has been investigated when the system is
perturbed with Brownian motion. These results are extended in this thesis to the case
where the deterministic system is perturbed with Levy noise. We mainly focus on
the stabilizing effects of the Levy noise in the system, prove the existence of sample
Lyapunov exponents of the trivial solution of the stochastically perturbed system, and
provide sufficient criteria under which the system is almost surely exponentially stable.
From the results that are established the Levy noise plays a similar role to the Brownian
motion in stabilizing dynamical systems.
We also establish the variation of constants formula for linear SDEs driven by Levy
noise. This is applied to study stochastic stabilization of ordinary functional differential
equation systems perturbed with Levy noise.