Modern Stochastics: Theory and Applications

We introduce a dynamic cutting approach for the numerical approximation of Lévy-driven stochastic differential equations. The key idea is to remove small jumps according to a time-dependent threshold, so that the retained jumps form a time-inhomogeneous compound Poisson process. We derive ${L^{p}}$-strong convergence rates for the adjusted Euler scheme. Numerical experiments at matched computational cost compare the dynamic cutting scheme with the classical Asmussen–Rosiński truncation and demonstrate consistently smaller strong errors when the time-dependent jump coefficient has a singularity at $t=0$.

Quasi-mixing limits of the killed symmetric Lévy process are studied. It is proved that (intrinsic) ultracontractivity of the underlying process implies the existence of its (uniformly) exponentially quasi-mixing limits. As a by-product, this implication ensures that the process has (uniformly) exponential quasi-ergodicity and (uniformly) exponentially fractional quasi-ergodicity on ${L^{p}}$ ($p\ge 1$). It is noteworthy that precise rates of convergence and precise limiting equalities are provided, which are determined by spectral gaps and eigenfunction ratios of the underlying process. Finally, three examples are provided to demonstrate the theoretical results.