Book
Lévy Matters IV. Estimation for Discretely Observed Lévy Processes.

In this chapter we discuss different aspects of statistical estimation for Lévybased processes based on lowfrequency observations. In particular, we consider the estimation of the Lévy triplet and the BlumenthalGetoor index in Lévy and timechanged Lévy models. Moreover, a calibration problem in exponential Lévy models based on option data is studied. The common feature of all these statistical problems is that they can be conveniently formulated in the Fourier domain. We introduce a general spectral estimation/calibration approach that can be applied to these and many other statistical problems related to Lévy processes. On the theoretical side, we provide a comprehensive convergence analysis of the proposed algorithms and address each time the question of optimality.
This article gives a brief summary on the main theoretical and practical results for the Scale functions. The article is organized in the following way: the first part describes the main theoretical concepts of Lévy processes, gives the formal definition and analytical properties of the Scale function. The second part describes the most significant practical cases where the Scale functions are applied. Thereafter, the closedform expressions of Scale functions for several classes of spectrally negative Lévy processes are considered. Finally, we concentrate on the very important application of Scale functions, namely, derivation of the distribution of dividends, paid by the insurance company to shareholders. The main contribution of this article is a lemma, describing the asymptotic behaviour of dividends, paid by nidentical companies
We consider Ncomponent synchronization models defined in terms of stochastic particle systems with special interaction. For general (nonsymmetric) Markov models we discuss phenomenon of the long time stochastic synchronization. We study behavior of the system in different limit situations related to appropriate changes of variables and scalings. For N = 2 limit distributions are found explicitly.
In this paper, we introduce a principally new method for modelling the dependence structure between two L{\'e}vy processes. The proposed method is based on some special properties of the timechanged Levy processes and can be viewed as an reasonable alternative to the copula approach.
In this paper, we consider the problem of statistical inference for generalized OrnsteinUhlenbeck processes of the type
\[
X_{t} = e^{\xi_{t}} \left( X_{0} + \int_{0}^{t} e^{\xi_{u}} d u \right),
\]
where \(\xi_s\) is a L{\'e}vy process. Our primal goal is to estimate the characteristics of the L\'evy process \(\xi\) from the lowfrequency observations of the process \(X\). We present a novel approach towards estimating the L{\'e}vy triplet of \(\xi,\) which is based on the Mellin transform technique. It is shown that the resulting estimates attain optimal minimax convergence rates. The suggested algorithms are illustrated by numerical simulations.
Let k be a field of characteristic zero, let G be a connected reductive algebraic group over k and let g be its Lie algebra. Let k(G), respectively, k(g), be the field of k rational functions on G, respectively, g. The conjugation action of G on itself induces the adjoint action of G on g. We investigate the question whether or not the field extensions k(G)/k(G)^G and k(g)/k(g)^G are purely transcendental. We show that the answer is the same for k(G)/k(G)^G and k(g)/k(g)^G, and reduce the problem to the case where G is simple. For simple groups we show that the answer is positive if G is split of type A_n or C_n, and negative for groups of other types, except possibly G_2. A key ingredient in the proof of the negative result is a recent formula for the unramified Brauer group of a homogeneous space with connected stabilizers. As a byproduct of our investigation we give an affirmative answer to a question of Grothendieck about the existence of a rational section of the categorical quotient morphism for the conjugating action of G on itself.
Let G be a connected semisimple algebraic group over an algebraically closed field k. In 1965 Steinberg proved that if G is simply connected, then in G there exists a closed irreducible crosssection of the set of closures of regular conjugacy classes. We prove that in arbitrary G such a crosssection exists if and only if the universal covering isogeny Ĝ → G is bijective; this answers Grothendieck's question cited in the epigraph. In particular, for char k = 0, the converse to Steinberg's theorem holds. The existence of a crosssection in G implies, at least for char k = 0, that the algebra k[G]G of class functions on G is generated by rk G elements. We describe, for arbitrary G, a minimal generating set of k[G]G and that of the representation ring of G and answer two Grothendieck's questions on constructing generating sets of k[G]G. We prove the existence of a rational (i.e., local) section of the quotient morphism for arbitrary G and the existence of a rational crosssection in G (for char k = 0, this has been proved earlier); this answers the other question cited in the epigraph. We also prove that the existence of a rational section is equivalent to the existence of a rational Wequivariant map T   >G/T where T is a maximal torus of G and W the Weyl group.