Subordinator

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

Convergence of nonhomogeneous random walks generated by compound Cox processes to generalized variance-gamma Lévy processes

Abstract: Functional limit theorems on the convergence of nonhomogeneous random walks generated by compound Cox processes to Levy processes with generalized one-dimensional variance-gamma distributions, in particular, to subordinate Wiener processes with subordinator being a Levy–Weibull process, are proved.

Space-fractional versions of the negative binomial and Polya-type processes

Abstract: In this paper, we introduce a space fractional negative binomial (SFNB) process by subordinating the space fractional Poisson process to a gamma subordinator. Its one-dimensional distributions are derived in terms of generalized Wright functions and their governing equations are obtained. It is a Levy process and the corresponding Levy measure is given. Extensions to the case of distributed order SFNB process, where the fractional index follows a two-point distribution, is analyzed in detail. The connections of the SFNB process to a space fractional Polya-type process is also pointed out. Moreover, we define and study a multivariate version of the SFNB obtained by subordinating a $d$-dimensional space-fractional Poisson process by a common independent gamma subordinator. Some applications of the SFNB process to the studies of population's growth and epidemiology are pointed out. Finally, we discuss an algorithm for the simulation of the SFNB process.

Option Surface Econometrics with Applications

Abstract: At each maturity a discrete return distribution is inferred from option prices. Option pricing models imply a comparable theoretical distribution. As both the transformed data and the option pricing model deliver points on a simplex, the data is statistically modeled by a Dirichlet distribution with expected values given by the option pricing model. The resulting setup allows for maximum likelihood estimation of option pricing model parameters with standard errors enabling the test of hypotheses. Hypothesis testing is illustrated by testing for risk neutral return distributions being consistent with Brownian motion with drift time changed by a subordinator. Models mixing processes of independent increments with processes related to solution of Ornstein Uhlenbeck (OU) equations are tested for the presence of the OU component. OU equations are a form of perpetual motion processes continuously responding to their past changes. The tests support the rejection of Brownian subordination and the presence of a perpetual motion component.

Path properties of levy processes

Abstract: This thesis can be split into two main components, the �first of which looks at the fractal dimension, specifically, box-counting dimension, of sets related to subordinators (non-decreasing L�evy processes). It was recently shown in [111] that lim�!0 U(�)N(t; �) = t almost surely, where N(t; �) is the minimal number of boxes of size at most � needed to cover a subordinator's range up to time t, and U(�) is the subordinator's renewal function. The main result in this section is a central limit theorem (CLT) for N(t; �), complementing and refining work in [111]. Box-counting dimension is defined in terms of N(t; �), but for subordinators we prove that it can also be defined using a new process obtained by shortening the original subordinator's jumps of size greater than �. This new process can be manipulated with remarkable ease in comparison to N(t; �), and allows better understanding of the box-counting dimension of a subordinator's range in terms of its L�evy measure, improving upon [111, Corollary 1]. We prove corresponding CLT and almost sure convergence results for the new process. The second main component of this thesis studies Markov processes conditioned so that their local time must grow slower than a prescribed function. Building upon recent work on Brownian motion with constrained local time in [8] and [78], we study whether or not the conditioned process is transient or recurrent, working with a broad class of Markov processes. In order to understand the local time, it is equivalent to study the inverse local time, which is itself a subordinator. The problem at hand is effectively equivalent to determining the distribution of a subordinator (the inverse local time) conditioned to remain above a given function. In conditioning a subordinator to remain above a curve of the form g(t); t � 0, the process is restricted to a time-dependent region, in contrast to previous works in which a process is conditioned to remain in a fixed region (e.g. cones in [43] and [60]). This means that we study boundary crossing probabilities for a family of curves, and must obtain uniform asymptotics for such a family. The main result in this section is a necessary and su�fficient condition for transience or recurrence of the conditioned Markov process. We will explicitly determine the distribution of the inverse local time for the conditioned process, and in the transient case, we explicitly determine the law of the conditioned Markov process. In the recurrent case, we characterise the entropic repulsion envelope via necessary and suffi�cient conditions.