Delay Differential Equations driven by Levy Processes of Finite Intensity: On the Existence of Invariant Measures for Mackey–Glass Type Equations Including a comprehensive discussion on Stochastic Integration Theory for right-continuous Semimartingales

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Mathematics (MSc)

Bosch, M. van den (4440)

Delay Differential Equations driven by Levy Processes of Finite Intensity: On the Existence of Invariant Measures for Mackey–Glass Type Equations Including a comprehensive discussion on Stochastic Integration Theory for right-continuous Semimartingales

Master thesis | Mathematics (MSc)

We consider a class of stochastic delay differential equations driven by L´evy processes of finite intensity, where jumps have zero expectation and do not exceed a certain height. The class of delay equations is inspired by the original Mackey–Glass equations. We find suitable criteria so that we obtain global strong solutions which are bounded in probability. In here, the system is non-autonomous. Both the drift and noise term are locally Lipschitz and may depend on the past. Subsequently, for autonomous systems, the existence of stationary solutions is shown via the Krylov–Bogoliubov method and, in particular, by proving tightness of the segments of some solution.

right-continuous semimartingale

càdlàg semimartingale

Lévy process

locally Lipschitz functional

existence and uniqueness of solutions

stochastic functional differential equation (SFDE)

bounded in probability