Stochastic Processes

This chapter begins with some fundamental ideas concerning random sequences, and related convergence concepts. It discusses the underlying probability model, and develops the idea of infinite dimensional Euclidean space and the associated Borel field, leading on to the Kolmogorov consistency theorem. The chapter concludes with consideration of uniform and limiting properties, including uniform boundedness and uniform integrability.

A new sufficient condition for uniform integrability of exponential local martingales

В статье доказано новое достаточное условие для равномерной интегрируемости экспоненциальных локальных мартингалов. Условие сформулировано в терминах квадратической вариации локального мартингала. Оно обобщает ряд известных достаточных условий подобного рода. In the article it is suggested a new sufficient condition for uniform integrability of continues exponential local martingales. It is shown that the condition is much weaker then a number of known sufficient conditions for uniform integrability of exponential local martingales.

The finiteness of the first meeting time between a Gaussian jump and a Brownian particles in the fluid

In this paper, we present more analysis about the finiteness of the first meeting time between Gaussian jump and Brownian particles in the fluid. Rather than using the uniform integrability conditions which are detailed in El-Hadidy [Mod. Phys. Lett. B 33(22) (2019) 1950256], we show the triangular arrays of the random sequence that represents the probable positions of the first meeting at any time [Formula: see text], to get the stronger moment conditions which are sufficient for the finiteness of this time.