Subordinator

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

The truncated 0-stable subordinator, renewal theorems, and disordered systems

Abstract: We introduce the subordinator, which we call "truncated 0-stable", whose Levy measure has density 1/x restricted to the interval (0,1). This process emerges naturally in the study of marginally relevant disordered systems, such as pinning and directed polymer models. We show that the truncated 0-stable subordinator admits an explicit marginal density and we study renewal processes in its domain of attraction, for which we prove sharp local renewal theorems. As an application, we derive sharp estimates on the second moment of the partition functions of pinning and directed polymer models.

Strong approximation of time-changed stochastic differential equations involving drifts with random and non-random integrators

Abstract: The rates of strong convergence for various approximation schemes are investigated for a class of stochastic differential equations (SDEs) which involve a random time change given by an inverse subordinator SDEs to be considered are unique in two different aspects: i) they contain two drift terms, one driven by the random time change and the other driven by a regular, non-random time variable; ii) the standard Lipschitz assumption is replaced by that with a time-varying Lipschitz bound The difficulty imposed by the first aspect is overcome via an approach that is significantly different from a well-known method based on the so-called duality principle On the other hand, the second aspect requires the establishment of a criterion for the existence of exponential moments of functions of the random time change

Some Contributions to the Theory of Conditional Gibbs Partitions

Abstract: Conditional Gibbs partitions naturally arise in Bayesian nonparametric analysis of species sampling problems under almost surely discrete priors inducing infinite exchangeable partitions with distribution in Gibbs product form. In this setting interest relies on posterior predictive inference on some characteristics of a population of species, given an initial sample of observations. Here we focus on the subclass of Poisson-Kingman partitions driven by the Stable subordinator, and, relying on the unconditional theory of exchangeable Gibbs partitions, derive some additional results for the posterior partition, the conditional α diversity and a Stirling’s approximation of the Gibbs weights.

Power‐type derivatives for rough volatility with jumps

Abstract: This paper proposes a novel analytical pricing–hedging framework for volatility derivatives which simultaneously takes into account rough volatility and volatility jumps. Directly targeting the instantaneous variance of a risky asset, our model consists of a generalized fractional Ornstein–Uhlenbeck process driven by a Lévy subordinator and an independent sinusoidal-composite Lévy process, and allows the characteristic function of average forward variance to be obtainable in semiclosed form, without having to invoke any geometric-mean approximations. Pricing–hedging formulae are proposed for a general class of power-type derivatives, in the spirit of numerical Fourier transform. A comparative empirical study is conducted on two independent recent data sets on Volatility Index options, before and during the COVID-19 pandemic, to demonstrate that the proposed framework is highly amenable to efficient model calibration under various choices of kernels. The price dynamics of the underlying asset can be readily considered and the possibility of studying rough volatility of volatility is given as well.

On the asymptotic behaviour of increasing positive self-similar Markov processes

Abstract: We are interested by the rate of growth of increasing positive self-similar Markov processes (ipssMp) such that the subordinator associated to it via Lamperti's transformation has infinite mean. We prove that the logarithm of an ipssMp normalized by the logarithm of the time converges weakly, as the time tends to infinity, if and only if the Laplace exponent of the underlying subordinator is regularly varying at zero. Moreover, we prove that the regular variation at zero of the Laplace exponent is essentially nasc for the existence of a function that normalizes the logarithm of an ipssMp. We obtain a law of iterated logarithm for the liminf of the logarithm of an ipssMp and an integral test to study the upper envelope of it. Furthermore, results concerning the rate of growth of the random clock appearing in Lamperti's transformation are obtained.