Modern Stochastics: Theory and Applications

The existence of density function of the running maximum of a stochastic differential equation (SDE) driven by a Brownian motion and a nontruncated pure-jump process is verified. This is proved by the existence of density function of the running maximum of the Wiener–Poisson functionals resulting from Bismut’s approach to the Malliavin calculus for jump processes.

A new modified Φ-Sobolev inequality for canonical ${L^{2}}$-Lévy processes, which are hybrid cases of the Brownian motion and pure jump-Lévy processes, is developed. Existing results included only a part of the Brownian motion process and pure jump processes. A generalized version of the Φ-Sobolev inequality for the Poisson and Wiener spaces is derived. Furthermore, the theorem can be applied to obtain concentration inequalities for canonical Lévy processes. In contrast to the measure concentration inequalities for the Brownian motion alone or pure jump Lévy processes alone, the measure concentration inequalities for canonical Lévy processes involve Lambert’s W-function. Examples of inequalities are also presented, such as the supremum of Lévy processes in the case of mixed Brownian motion and Poisson processes.

In this paper we present some new limit theorems for power variations of stationary increment Lévy driven moving average processes. Recently, such asymptotic results have been investigated in [Ann. Probab. 45(6B) (2017), 4477–4528, Festschrift for Bernt Øksendal, Stochastics 81(1) (2017), 360–383] under the assumption that the kernel function potentially exhibits a singular behaviour at 0. The aim of this work is to demonstrate how some of the results change when the kernel function has multiple singularity points. Our paper is also related to the article [Stoch. Process. Appl. 125(2) (2014), 653–677] that studied the same mathematical question for the class of Brownian semi-stationary models.