Stabilization of Time Delayed Stochastic Jump Discontinuous Systems Driven by Poisson Noise

This section focuses on the properties of a Lévy process, which is a continuous-time stochastic process characterized by several key conditions.

Firstly, the initial value ( X_0 ) is set to 0, reflecting a common starting point in various real-world scenarios. Secondly, the random variable must exhibit independent increments, meaning that changes in the process over non-overlapping time intervals are statistically independent. Additionally, these increments should be stationary, indicating that the statistical properties of the process remain consistent across these independent time intervals.

Moreover, a Lévy process requires stochastically continuous paths. This means that the probability of the difference between random variables at two times, ( S ) and ( T ), being greater than a small positive value ( \varepsilon ) approaches zero as ( T ) converges. In summary, a stochastic process that satisfies the conditions of independent and stationary increments is classified as a Lévy process.

Lévy processes provide a mathematically robust framework for modeling various stochastic disturbances, one notable example being the Poisson process, which is characterized as a pure jump process.