Subordinator

About: Subordinator is a research topic. Over the lifetime, 771 publications have been published within this topic receiving 15383 citations.

Time-changed Poisson processes of order $k$

Abstract: In this article, we study the Poisson process of order k (PPoK) time-changed with an independent Levy subordinator and its inverse, which we call respectively, as TCPPoK-I and TCPPoK-II, through various distributional properties, long-range dependence and limit theorems for the PPoK and the TCPPoK-I. Further, we study the governing difference-differential equations of the TCPPoK-I for the case inverse Gaussian subordinator. Similarly, we study the distributional properties, asymptotic moments and the governing difference-differential equation of TCPPoK-II. As an application to ruin theory, we give a governing differential equation of ruin probability in insurance ruin using these processes. Finally, we present some simulated sample paths of both the processes.

Tail asymptotics for exponential functionals of Levy processes: the convolution equivalent case

Abstract: We determine the rate of decrease of the right tail distribution of the exponential functional of a Levy process with a convolution equivalent Levy measure. Our main result establishes that it decreases as the right tail of the image under the exponential function of the Levy measure of the underlying Levy process. The method of proof relies on fluctuation theory of Levy processes and an explicit path-wise representation of the exponential functional as the exponential functional of a bivariate subordinator. Our techniques allow us to establish rather general estimates of the measure of the excursions out from zero for the underlying Levy process reflected in its past infimum, whose area under the exponential of the excursion path exceed a given value.

L\'evy processes with marked jumps I : Limit theorems

Abstract: Consider a sequence (Z_n,Z_n^M) of bivariate L\'evy processes, such that Z_n is a spectrally positive L\'evy process with finite variation, and Z_n^M is the counting process of marks in {0,1} carried by the jumps of Z_n. The study of these processes is justified by their interpretation as contour processes of a sequence of splitting trees with mutations at birth. Indeed, this paper is the first part of a work aiming to establish an invariance principle for the genealogies of such populations enriched with their mutational histories. To this aim, we define a bivariate subordinator that we call the marked ladder height process of (Z_n,Z_n^M), as a generalization of the classical ladder height process to our L\'evy processes with marked jumps. Assuming that the sequence (Z_n) converges towards a L\'evy process Z with infinite variation, we first prove the convergence in distribution, with two possible regimes for the marks, of the marked ladder height process of (Z_n,Z_n^M). Then we prove the joint convergence in law of Z_n with its local time at the supremum and its marked ladder height process.

Nonparametric inference on L\'evy measures of L\'evy-driven Ornstein-Uhlenbeck processes under discrete observations

Abstract: In this paper, we study nonparametric inference for a stationary L\'evy-driven Ornstein-Uhlenbeck (OU) process $X = (X_{t})_{t \geq 0}$ with a compound Poisson subordinator. We propose a new spectral estimator for the L\'evy measure of the L\'evy-driven OU process $X$ under macroscopic observations. We derive multivariate central limit theorems for the estimator over a finite number of design points. We also derive high-dimensional central limit theorems for the estimator in the case that the number of design points increases as the sample size increases. Building upon these asymptotic results, we develop methods to construct confidence bands for the L\'evy measure.