Location of the Path Supremum for Self-similar Processes with Stationary Increments

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 L\'{e}vy 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.