Subordinator

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

Extinction time of a CB-processes with competition in a L\'evy random environment

Abstract: In this paper, we are interested on the extinction time of continuous state branching processes with competition in a L\'evy random environment. In particular we prove, under the so-called Grey's condition together with the assumption that the L\'evy random environment does not drift towards infinity, that for any starting point the process gets extinct in finite time a.s. Moreover if we replace the condition on the L\'evy random environment by a technical integrability condition on the competition mechanism, then the process also gets extinct in finite time a.s. and it comes down from infinity. Then the logistic case in a Brownian random environment is treated. Our arguments are base on a Lamperti-type representation where the driven process turns out to be a perturbed Feller diffusion by an independent spectrally positive L\'evy process. If the independent random perturbation is a subordinator then the process converges to a specified distribution; otherwise, it goes extinct a.s. In the latter case and following a similar approach to Lambert (Ann. Appl. Probab., 15(2):1506-1535, 2005.), we provide the expectation and the Laplace transform of the absorption time, as a functional of the solution to a Riccati differential equation.

Bivariate Bernstein-gamma functions and moments of exponential functionals of subordinators.

Abstract: In this paper, we extend recent work on the functions that we call Bernstein-gamma to the class of bivariate Bernstein-gamma functions. In the more general bivariate setting, we determine Stirling-type asymptotic bounds which generalise, improve upon and streamline those found for the univariate Bernstein-gamma functions. Then, we demonstrate the importance and power of these results through an application to exponential functionals of Levy processes. In more detail, for a subordinator (a non-decreasing Levy process) $(X_s)_{s\geq 0}$, we study its \textit{exponential functional}, $\int_0^t e^{-X_s}ds $, evaluated at a finite, deterministic time $t>0$. Our main result here is an explicit infinite convolution formula for the Mellin transform (complex moments) of the exponential functional up to time $t$ which under very minor restrictions is shown to be equivalent to an infinite series. We believe this work can be regarded as a stepping stone towards a more in-depth study of general exponential functionals of Levy processes on a finite time horizon.

Pure jump increasing processes and the change of variables formula

Abstract: Given an increasing process $(A_t)_{t\geq 0}$, we characterize the right continuous non-decreasing functions $f: \mathbb{R}_+\to \mathbb{R}_+$ that map $A$ to a pure jump process. As an example of application, we show for instance that functions with bounded variations belong to the domain of the extended generator of any subordinator with no drift and infinite Levy measure.

Two Theorems on Hunt's Hypothesis (H) for Markov Processes

Abstract: Hunt's hypothesis (H) and the related Getoor's conjecture is one of the most important problems in the basic theory of Markov processes. In this paper, we investigate the invariance of Hunt's hypothesis (H) for Markov processes under two classes of transformations, which are change of measure and subordination. Our first theorem shows that for two standard processes $(X_t)$ and $(Y_t)$, if $(X_t)$ satisfies (H) and $(Y_t)$ is locally absolutely continuous with respect to $(X_t)$, then $(Y_t)$ satisfies (H). Our second theorem shows that a standard process $(X_t)$ satisfies (H) if and only if $(X_{\tau_t})$ satisfies (H) for some (and hence any) subordinator $(\tau_t)$ which is independent of $(X_t)$ and has a positive drift coefficient. Applications of the two theorems are given.

Term structure of interest rates with stickiness: a subdiffusion approach

Abstract: In this paper, we propose a new class of term structure of interest rate models which is built on the subdiffusion processes. We assume that the spot rate is a function of a time changed diffusion process belonging to a symmetric pricing semigroup for which its spectral representation is known. The time change process is taken to be an inverse Levy subordinator in order to capture the stickiness feature observed in the short-term interest rates. We derive the analytical formulas for both bond and bond option prices based on eigenfunction expansion method. We also numerically implement a specific subdiffusive model by testing the sensitivities of bond and bond option prices with respect to the parameters of time change process.