Subordinator

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

A distributional equality for suprema of spectrally positive L\'evy processes

Abstract: Let $Y$ be a spectrally positive Levy process with $E Y_1<0$, $C$ an independent subordinator with finite expectation, and $X=Y+C$. A curious distributional equality proved in Huzak et al., Ann. Appl. Probab. 14 (2004) 1278--1397, states that if $E X_1<0$, then $\sup_{0\le t <\infty}Y_t$ and the supremum of $X$ just before the first time its new supremum is reached by a jump of $C$ have the same distribution. In this paper we give an alternative proof of an extension of this result and offer an explanation why it is true.

Parameter Estimation of Multivariate Lévy Processes

Abstract: In this thesis, we apply Levy copulas to describe the dependence structure of multivariate Levy processes and build some Levy copula-based models. Parameter estimation of the models is the main part of this work. The estimation procedure is based on maximum likelihood principles. For compound Poisson processes (CPP) which have finite Levy measure, we decompose the mass on the axes and outside of the axes. This decomposition for a bivariate CPP generates three independent components and shows either the jumps only in one component, or the bivariate jumps in both components. The likelihood function can be derived based on these independent parts. We also suggest a new simulation algorithm for a bivariate CPP. We apply our method to model Danish fire insurance data and estimate the parameters of the model. The extension of the method for Levy Processes with infinite Levy measure is discussed in the second part. More precisely we take a bivariate stable Levy Process and truncate all the small jumps. We base the statistical analysis on the resulting CPP. The Fisher information matrix is also calculated and the asymptotic normality of the estimators is proved as the number of jumps tends to infinity. In this model this may happen either for the observation period going to infinity, or the truncation point going to 0 for a fixed observation period. A simulation study investigates the loss of efficiency because of the truncation. Finally, a new estimation procedure is introduced in the last chapter. The main idea of this approach, which we call two-step method, is similar to IFM (inference functions for margins) for multivariate distribution functions. First, the parameters of the marginal processes are estimated. Then, given the estimates from the first step, we estimate in a second step only the dependence structure parameters. This method is applied to a bivariate α-stable Clayton subordinator with different or common marginal parameters. For the latter, the Godambe information matrix and asymptotic covariance matrix are analytically calculated. Moreover, the asymptotic normality of the estimators is proved as the time span goes to infinity or the truncation point goes to zero. A simulation study compares the quality of all three estimation methods: the two-step estimates, the MLEs of a full model and the MLEs based on joint jumps only.

On harmonic functions of symmetric Levy processes

Abstract: We consider some classes of Levy processes for which the estimate of Krylov and Safonov (as in [BL02]) fails and thus it is not possible to use the standard iteration technique to obtain a-priori Holder continuity estimates of harmonic functions. Despite the faliure of this method, we obtain some a-priori regularity estimates of harmonic functions for these processes. Moreover, we extend results from [SSV06] and obtain asymptotic behavior of the Green function and the Levy density for a large class of subordinate Brownian motions, where the Laplace exponent of the corresponding subordinator is a slowly varying function.

Numerical Methods for Lévy Processes: Lattice Methods and the Density, the Subordinator and the Time Copula

Abstract: Evidence from the financial markets suggests that empirical returns distributions, both historical and implied, do not arise from diffusion processes. A growing literature models the returns process as a Lévy process, finding a number of explicit formulae for the values of some derivatives in special cases. Practical use of these models has been hindered by a relative paucity of numerical methods that can be used when explicit solutions are not present. This paper investigates lattice methods that can be used when the returns process is Lévy. We relate the transition density function of a Lévy process to its representation as a time-changed Brownian motion and to its time-copula, leading to alternative derivations of the lattice. We apply the lattice to models based on the variance-gamma, NIG and Meixner processes, contrasting the numerical difficulties in each case. We discuss implications for implied pricing, concluding that current methods based, directly or indirectly, on low order branching, are unlikely to be capable of calibrating to market prices. ∗We gratefully acknowledge the help and support of Manfred Gilli and the hospitality of the Department of Econometrics, University of Geneva. We would like to thank Grace Kuan and Stewart Hodges for their comments and advice. The paper has benefited from comments by Lynda McCarthy, Peter Carr, Philip Schönbucher, Steve Heston, Mark Broadie, Chris Rogers, Rupert Brotherton-Ratcliffe and Alessio Sancetta, and from participants at the 8th CAP workshop, New York, and QMF 2002, Sydney..

Ranked increments of a stable subordinator and ranked excursion lengths of Brownian motion

Abstract: The thesis treats three different areas; (i) Ranked increments of stable processes and ranked excursions of Brownian motion, (ii) Sufficient capital levels for banks, and (iii) Trading strategies for reduction of the fluctuations of revenues for power plants.The first part is a theoretcial investigation involved with the calculation of distribution functions concerning special properties of stable processes. The second part is a description of a framework in which the sufficiency of capital levels for banks can be evaluated.The third part is a typical example of how financial mathematics can be used to derive practical methods applicable in risk management of energy derivatives and real options.Altogether, five papers are presented.