1 These notes give a more detailed account of the material discussed in lectures. Exercises are for your own convenience and will not be looked at during the lectures. 1 These lecture notes are based on the first two chapters of Introductory lectures on fluctuations of Lévy processes with applications by A.E.Kyprianou, published by Springer, 2006.

The joint study of functionals of a Brownian motion B and its L evy transform = jBj?L, where L is the local time of B at 0, is motivated by the conjectured ergodicity of the L evy transform. Here, we compute explicitly the covariance of the last zeros before time 1 of B and , which turns out to be strictly positive.

We provide sharp exponential moment bounds for (Stratonovich) iterated stochastic integrals under conditioning by certain small balls, including balls in certain HH older-like norms of exponent greater than 1=3. The proof uses a control of the variation of the L evy area, under conditioning. The results are applied to the computation of the Onsager-Machlup functional of diiusion processes with ...

We study the tail behavior of the distribution of certain subadditive functionals acting on the sample paths of L evy processes. The functionals we consider have, roughly speaking, the following property: only the points of the process that lie above a certain curve contribute to the value of the functional. Our assumptions will make sure that the process ends up eventually below the curve. Our...

Exotic stochastic processes are shown to emerge in the quantum evolution of complex systems Using in uence func tion techniques we consider the dynamics of a system coupled to a chaotic subsystem described through random matrix the ory We nd that the reduced density matrix can display dy namics given by L evy stable laws The classical limit of these dynamics can be related to fractional kinetic...

In our previous publications (IJTAF 2019, Math. Finance 2020), we introduced a general class of SINH-regular processes and demonstrated that efficient numerical methods for the evaluation Wiener-Hopf factors various probability distributions (prices options several types) in L\'evy models can be developed using only few properties characteristic exponent $\psi$. Essentially all popular enjoy th...

In this paper we study a queue with L evy input, without imposing any a priori assumption on the jumps being one-sided. The focus is on computing the transforms of all sorts of quantities related to the transient workload, assuming the workload is in stationarity at time 0. The results are simple expressions that are in terms of the bivariate Laplace exponents of ladder processes. In particular...

Abstract: We propose a feasible method for approximating the marginal distributions and densities of a bounded variation Lévy process using polynomial expansions. We provide a fast recursive formula for approximating the coefficients of the expansions and estimating the order of the approximation error. Our expansions are shown to be the exact counterpart of successive approximations of the Lév...

We introduce the concept of capacitary modulus for a set ⊆Rd, which is a function h that provides simple estimates for the capacity of with respect to an arbitrary kernel f, estimates which depend only on the L inner product (h; f). We show that for a large class of L evy processes, which include the symmetric stable processes and stable subordinators, a capacitary modulus for the range of the ...

? Abstract. We prove a version of the Standardization Theorem and the Discrete Normalization (DN) Theorem in stable Deterministic Residual Structures , which are Abstract Reduction Systems with axiomatized residual relation , and model orthogonal rewrite systems. The latter theorem gives a strategy for construction of reductions L evy-equivalent (or permutation-equivalent) to a given, nite or i...