On multiparameter IDT processes

Details On multiparameter IDT processes On multiparameter IDT processes

2012-03-05

arxiv.org/abs/1203.0897v1

Mathematics

Probability

Scientific paper

We give a new approach for multi-parameter IDT processes and the main example of this kind of processes is the L\'{e}vy's $\mathbb{R}^{M}$-parameter Brownian motion studied by P. L\'{e}vy $\cite{Levy}$, N. N. Chentsov $\cite{Chentsov}$ and H. P. Mckean Jr. $\cite{Mckean}$. This new approach turn out to be very interesting since it provide several extension of most properties of the one-parameter IDT processes invested by R. Mansuy $\cite{Mansuy}$, K. Es-sebaiy and Y. Ouknine $\cite{Ouknine}$, and in our previous work $\cite{Hakassou}$. For instance, we prove that $\mathbb{R}_{+}^{N}$-parameter L\'{e}vy processes introduced by O. E. Barndorff-Nielsen et al. $\cite{Barndorff}$, are IDT in our sense and we prove the inheritance of multi-parameter IDT property under time change when base processes are $\mathbb{R}_{+}^{N}$-parameter L\'{e}vy processes. In this paper too, we are going to study intensively the multi-parameter IDT in the sense of K. Es-sebaiy and Y. Ouknine $\cite{Ouknine}$, and then we shall do a comparison with our approach.

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