Two-parameter Levy processes along decreasing paths

TL;DR

This paper investigates the properties of one-parameter processes derived from two-parameter Lévy processes along decreasing paths, focusing on stationary increments and their characterization.

Contribution

It provides a detailed analysis and characterization of the stationary increment property for processes along decreasing paths in two-parameter Lévy processes.

Findings

Characterization of stationary increments along decreasing paths

Conditions under which the process has stationary increments

Insights into the structure of two-parameter Lévy processes

Abstract

Let {X_{t_1,t_2}: t_1,t_2 >= 0} be a two-parameter L\'evy process on R^d. We study basic properties of the one-parameter process {X_{x(t),y(t)}: t \in T} where x and y are, respectively, nondecreasing and nonincreasing nonnegative continuous functions on the interval T. We focus on and characterize the case where the process has stationary increments.