Max-stable processes and stationary systems of Levy particles

We study stationary max-stable processes {n(t): t is an element of R} admitting a representation of the form n(t) = max(i is an element of N) (U-i +Y-i(t)), where Sigma(infinity)(i=1) delta U-i is a Poisson point process on R with intensity e(-u)du, and Y1,Y2 are i.i.d. copies of a process {Y (t) : t is an element of R} obtained by running a Levy process for positive t and a dual Levy process for negative t. We give a general construction of such Levy-Brown-Resnick processes, where the restrictions of Y to the positive and negative half-axes are Levy processes with random birth and killing times. We show that these max-stable processes appear as limits of suitably normalized pointwise maxima of the form Mn(t) = max(i=1),...,n xi i (s(n) + t), where xi 1, xi 2... are i.i.d. Levy processes and sn is a sequence such that s(n) similar to clog n with c > 0. Also, we consider maxima of the form max(i=1),...,n Z(i) (t/log n), where Zi, Z2,... are i.i.d. Ornstein-Uhlenbeck processes driven by an alpha-stable noise with skewness parameter beta = -1. After a linear normalization, we again obtain limiting max-stable processes of the above form. This gives a generalization of the results of Brown and Resnick (1977) to the totally skewed a-stable case. (C) 2015 Elsevier B.V. All rights reserved.

An optimal prediction problem in financial modelling

Violetta Bernyk

The subject of the present thesis is an optimal prediction problem concerning the ultimate maximum of a stable Lévy process over a finite interval of time. Such "optimal prediction" problems are of both theoretical and practical interest, in particular they have applications in finance. For instance, suppose that an investor has a long position in one financial asset, whose price is modelled by some stochastic process. The investor's objective is to determine a "best moment" at which to close out the position and to sell the asset at the highest possible price. This optimal decision must be based on continuous observations of the asset price performance and only on the information accumulated to date. Hence, the investor should use a prediction (forecasting) of the future evolution of the price of the financial security. We examine this problem in the case where the asset price is modelled by a Lévy process. Indeed, during the last several years, the application of Lévy processes in the modelling financial asset returns has become one of the active research directions in quantitative finance. Thus, this thesis contains suitable new results concerning Lévy processes. We derive the law of the supremum process associated with a strictly stable Lévy process with no negative jumps which is not a subordinator. We note that the latter problem dates back to 1973. In particular, we show that the probability density function of the supremum process can be expressed using an explicit power series representation or via an integral representation. We also derive the infinitesimal generator of the reflected process associated with a general strictly stable Lévy process. Throughout this thesis, we apply the theory of optimal stopping, the methods of fractional differential calculus, and some results from fluctuation theory. Implementing these theories in the context of Lévy processes requires the development of specific analytical results. In the case where the asset price is modelled by a spectrally positive stable Lévy process, we describe the optimal strategy under certain conditions on the model parameters. The optimal strategy is of the following form: the investor must stop the observation of the price process and sell the asset as soon as the associated reflected process crosses for the first time a particular stopping boundary. We also provide numerical estimates and simulation examples of the results obtained by using this strategy.

EPFL2007

Limit Behaviour of the Voter Model, Exclusion Process and Regenerative Chains

Samuel Schöpfer

Though the following topics seem unlinked, most of the tools used in this thesis are related to random walks and renewal theory. After introducing the voter model, we consider the parabolic Anderson model with the voter model as catalyst. In GÄRTNER, DEN HOLLANDER and MAILLARD [44], the behaviour of the annealed Lyapunov exponents, i.e., the exponential growth rates of the successive moments of the reactant with respect to the catalyst, was investigated. It was shown that these exponents exhibit an interesting dependence on the dimension and on the diffusion constant. In Chapter 3 we address some questions left open in this paper by considering specifically when the Lyapunov exponents are the a priori maximal value. Then, we use exclusion process techniques to show that the evolution of a perturbed threshold voter model is recurrent in the critical case. The key to our approach is to develop the ideas of BRAMSON and MOUNTFORD [9] : we exhibit a Lyapunov-Foster function for the discrete time version of the process. We also make a widespread use of coupling arguments. Finally, using the regenerative scheme of COMETS, FERNÁNDEZ and FERRARI [19], we establish a functional central limit theorem for discrete time stochastic processes with summable memory decay. Furthermore, under stronger assumptions on the memory decay, we identify the limiting variance in terms of the process only. As applications, we define classes of binary autoregressive processes and power-law Ising chains for which the limit theorem is fulfilled.

EPFL2012

A Levy-derived process seen from its supremum and max-stable processes

Sebastian Engelke

We consider a process Z on the real line composed from a Levy process and its exponentially tilted version killed with arbitrary rates and give an expression for the joint law of the supremum (Z) over bar, its time T, and the process Z(T + center dot) - (Z) over bar. This expression is in terms of the laws of the original and the tilted Levy processes conditioned to stay negative and positive respectively. The result is used to derive a new representation of stationary particle systems driven by Levy processes. In particular, this implies that a max-stable process arising from Levy processes admits a mixed moving maxima representation with spectral functions given by the conditioned Levy processes.

Univ Washington, Dept Mathematics2016

An optimal prediction problem in financial modelling

Violetta Bernyk

EPFL2007