2016/04/15 by Yi Shen, Shen, Yi
Economics, Econometrics and Finance · Mathematics · #60G10 (Secondary) #60G18 (Primary) #60G55 #Complex Systems and Time Series Analysis #FOS: Mathematics #Probability (math.PR) #Random Matrices and Applications #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.1604.04645
openalex publication_date 2016/04/15 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
In this paper we consider the distribution of the location of the path supremum in a fixed interval for self-similar processes with stationary increments. To this end, a point process is constructed and its relation to the distribution of the location of the path supremum is studied. Using this framework, we show that the distribution has a spectral-type representation, in the sense that it is always a mixture of a special group of absolutely continuous distributions, plus point masses on the two boundaries. Bounds on the value and the derivatives of the density function are established. We further discuss self-similar Lévy processes as an example. Most of the results in this paper can be generalized to a group of random locations, including the location of the largest jump, etc.