On the dynamics of marcus type stochastic differential equations
In this work metric dynamical systems (MDS) driven by Lévy processes in positive and negative time are constructed. Ergodicity and invariance for such classes of MDS are shown. Further a perfection theorem for càdlàg processes and the conjugacy of solution of Marcus type SDEs driven by Lévy processes and solutions of certain RDEs is proven. This result is applied to verify locally conjugacy of solutions of Marcus type SDEs and solutions of linearised Marcus type SDEs (referring to the results of Hartman–Grobman for deterministic ODEs). Subsequently, stable and unstable manifolds are constructed using the Lyapunov–Perron method. Furthermore, the Lyapunov–Perron method is modified to prove a foliation of the stable manifold. Conclusively, Marcus type stochastic differential delay equations (MSDDEs) are considered. Conditions for existence and uniqueness of solutions are deduced, which implies the semiflow property for solutions of MSDDEs.