Tempered stable distributions and processes

We investigate the class of tempered stable distributions and their associated processes. Our analysis of tempered stable distributions includes limit distributions, parameter estimation and the study of their densities. Regarding tempered stable processes, we deal with density transformations and compute their p$p$-variation indices. Exponential stock models driven by tempered stable processes are discussed as well.     

On the asymptotic behaviour of Lévy processes,Part I: Subexponential and exponential processes

We study tail probabilities of the suprema of Lévy processes with subexponential or exponential marginal distributions over compact intervals. Several of the processes for which the asymptotics are studied here for the first time have recently become important to model financial time series. Hence our results should be important, for example, in the assessment of financial risk.     

Nonminimal sets,their projections and integral representations of stable processes

New criteria are provided for determining whether an integral representation of a stable process is minimal. These criteria are based on various nonminimal sets and their projections, and have several advantages over and shed light on already available criteria. In particular, they naturally lead from a nonminimal representation to the one which is minimal. Several known examples are considered to illustrate the main results. The general approach is also adapted to show that the so-called mixed moving averages have a minimal integral representation of the mixed moving average type.     

Median,concentration and fluctuations for Lévy processes

We estimate a median of f(_Xt)$f\left(X_t\right)$ where f$f$ is a Lipschitz function, X$X$ is a Lévy process and t$t$ is an arbitrary time. This leads to concentration inequalities for f(_Xt)$f\left(X_t\right)$. In turn, corresponding fluctuation estimates are obtained under assumptions typically satisfied if the process has a regular behavior in small time and a, possibly different, regular behavior in large time.     

On the convergence of LePage series in Skorokhod space

We consider the problem of the convergence of the so-called LePage series in the Skorokhod space D^d=D([0,1],R^d) and provide a simple criterion based on the moments of the increments of the random process involved in the series. This provides a simple sufficient condition for the existence of an α-stable distribution on D^d with given spectral measure.     

Moment properties of multivariate infinitely divisible laws and criteria for multivariate self-decomposability

Ramachandran (1969) [9, Theorem 8] has shown that for any univariate infinitely divisible distribution and any positive real number α, an absolute moment of order α relative to the distribution exists (as a finite number) if and only if this is so for a certain truncated version of the corresponding Lévy measure. A generalized version of this result in the case of multivariate infinitely divisible distributions, involving the concept of g-moments, was given by Sato (1999) [6, Theorem 25.3]. We extend Ramachandran’s theorem to the multivariate case, keeping in mind the immediate requirements under appropriate assumptions of cumulant studies of the distributions referred to; the format of Sato’s theorem just referred to obviously varies from ours and seems to have a different agenda. Also, appealing to a further criterion based on the Lévy measure, we identify in a certain class of multivariate infinitely divisible distributions the distributions that are self-decomposable; this throws new light on structural aspects of certain multivariate distributions such as the multivariate generalized hyperbolic distributions studied by Barndorff-Nielsen (1977) [12] and others. Various points relevant to the study are also addressed through specific examples.     

Weak convergence of censored and reflected stable processes

It is shown that if a sequence of open n$n$-sets _Dk$D_k$ increases to an open n$n$-set D$D$ then reflected stable processes in _Dk$D_k$ converge weakly to the reflected stable process in D$D$ for every starting point x$x$ in D$D$. The same result holds for censored α$\alpha$-stable processes for every x$x$ in D$D$ if D$D$ and _Dk$D_k$ satisfy the uniform Hardy inequality. Using the method in the proof of the above results, we also prove the weak convergence of reflected Brownian motions in unbounded domains.     

Transition density estimates for jump Lévy processes

Upper estimates of densities of convolution semigroups of probability measures are given under explicit assumptions on the corresponding Lévy measure and the Lévy-Khinchin exponent.     

Stochastic integration for Lévy processes with values in Banach spaces

A stochastic integral of Banach space valued deterministic functions with respect to Banach space valued Lévy processes is defined. There are no conditions on the Banach spaces or on the Lévy processes. The integral is defined analogously to the Pettis integral. The integrability of a function is characterized by means of a radonifying property of an integral operator associated with the integrand. The integral is used to prove a Lévy–Itô decomposition for Banach space valued Lévy processes and to study existence and uniqueness of solutions of stochastic Cauchy problems driven by Lévy processes.     

Decomposability for stable processes

We characterize all possible independent symmetric α-stable (SαS) components of an SαS process, 0<α<2. In particular, we focus on stationary SαS processes and their independent stationary SαS components. We also develop a parallel characterization theory for max-stable processes.     

Real harmonizable multifractional stable process and its local properties

A real harmonizable multifractional stable process is defined, its Hölder continuity and localizability are proved. The existence of local time is shown and its regularity is established.     

Martingale representation for Poisson processes with applications to minimal variance hedging

We consider a Poisson process η on a measurable space equipped with a strict partial ordering, assumed to be total almost everywhere with respect to the intensity measure λ of η. We give a Clark-Ocone type formula providing an explicit representation of square integrable martingales (defined with respect to the natural filtration associated with η), which was previously known only in the special case, when λ is the product of Lebesgue measure on R₊ and a σ-finite measure on another space X. Our proof is new and based on only a few basic properties of Poisson processes and stochastic integrals. We also consider the more general case of an independent random measure in the sense of Itô of pure jump type and show that the Clark-Ocone type representation leads to an explicit version of the Kunita-Watanabe decomposition of square integrable martingales. We also find the explicit minimal variance hedge in a quite general financial market driven by an independent random measure.     

Green function estimates for relativistic stable processes in half-space-like open sets

In this paper, we establish sharp two-sided estimates for the Green functions of relativistic stable processes (i.e. Green functions for non-local operators m−(m^2/α−Δ)^α/2) in half-space-like C^1,1 open sets. The estimates are uniform in m∈(0,M] for each fixed M∈(0,∞). When m↓0, our estimates reduce to the sharp Green function estimates for −(−Δ)^α/2 in such kind of open sets that were obtained recently in Chen and Tokle [12]. As a tool for proving our Green function estimates, we show that a boundary Harnack principle for X^m, which is uniform for all m∈(0,∞), holds for a large class of non-smooth open sets.     

An optimal stopping problem for fragmentation processes

In this article we consider a toy example of an optimal stopping problem driven by fragmentation processes. We show that one can work with the concept of stopping lines to formulate the notion of an optimal stopping problem and moreover, to reduce it to a classical optimal stopping problem for a generalized Ornstein–Uhlenbeck process associated with Bertoin’s tagged fragment. We go on to solve the latter using a classical verification technique thanks to the application of aspects of the modern theory of integrated exponential Lévy processes.     

Joint temporal and contemporaneous aggregation of random-coefficient AR(1) processes

We discuss joint temporal and contemporaneous aggregation of N$N$ independent copies of AR(1) process with random-coefficient a∈[0,1)$a\in [0,1)$ when N$N$ and time scale n$n$ increase at different rate. Assuming that a$a$ has a density, regularly varying at a=1$a=1$ with exponent −1<β<1$-1<\beta <1$, different joint limits of normalized aggregated partial sums are shown to exist when N^1/(1+β)/n$N^{1/(1+\beta )}/n$ tends to (i) ∞$\infty$, (ii) 0$0$, (iii) 0<μ<∞$0<\mu <\infty$. The limit process arising under (iii) admits a Poisson integral representation on (0,∞)×C(R)$(0,\infty )\times C\left(\mathbb{R}\right)$ and enjoys ‘intermediate’ properties between fractional Brownian motion limit in (i) and sub-Gaussian limit in (ii).     

A Central Limit Theorem for isotropic flows

We establish that the image of a measure, which satisfies a certain energy condition, moving under a standard isotropic Brownian flow will, when properly scaled, have an asymptotically normal distribution under almost every realization of the flow. We derive the same result for an initial point mass moved by an isotropic Kraichnan flow.