On Ito Stochastic Integration with Respect to $p$-Stable Motion: Inner Clock, Integrability of Sample Paths, Double and Multiple Integrals

Abstract

The paper studies in detail the sample paths of Ito-type stochastic integrals with respect to $p$-stable motion $M(t), t \geq 0$. These results, in turn, permit an analysis of the concept of multiple $p$-stable integrals of the form $\int \cdots \int f(t_1,\cdots, t_n)dM(t_1) \cdots dM(t_n)$, and, in particular, a full description of functions of two variables $f(t_1, t_2)$ for which the double stochastic integral $\int \int f(t_1, t_2) dM(t_1) dM(t_2)$ exists.