On a fractional jump process with two kinds of jumps and its connection with the Poisson process
Year: 2015
Type: preprint
Abstract: We consider a fractional jump process with jumps of size 1 and 2, whose probabilities satisfy a fractional extension of the difference-differential equations $$ \dfrac{\mathrm{d}p_{k}(t) }{\mathrm{d}t } =\lambda_{2}p_{k-2}(t)+\lambda_{1}p_{k-1}(t)-(\lambda_{1}+\lambda_{2})p_{k}(t), \quad k\geq 0, \;\; t>0. $$ We obtain the probability law of the resulting process in terms of generalized Mittag-Leffler functions. We also discuss two equivalent representations both in terms of a subordinator governed by a suitable fractional Cauchy problem, and of a compound fractional Poisson process. The first occurrence time of a jump of fixed amplitude is proved to have the same distribution as the waiting time of the first event of a classical fractional Poisson process, this extending a well-known result for the Poisson process. We also express the distribution of the first passage time of the fractional jump process in an integral form that involves the joint distribution of the classical fractional Poisson process. Finally, we show that the ratios given by the powers of the jump process over their means converge in probability to 1.
Source: arXiv (Cornell University)
Language: English
Subfield: Modeling and Simulation
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W1916869002