Hyperfinite stochastic integration for Lévy processes with finite-variation jump part

This article links the hyperfinite theory of stochastic integration with respect to certain hyperfinite Lévy processes with the elementary theory of pathwise stochastic integration with respect to pure-jump Lévy processes with finite-variation jump part. Since the hyperfinite Itô integral is also defined pathwise, these results show that hyperfinite stochastic integration provides a pathwise definition of the stochastic integral with respect to Lévy jump-diffusions with finite-variation jump part.As an application, we provide a short and direct nonstandard proof of the generalized Itô formula for stochastic differentials of smooth functions of Lévy jump-diffusions whose jumps are bounded from below in norm.     

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.     

Time regularity of solutions to linear equations with Lévy noise in infinite dimensions

The existence of strong and weak càdlàg versions of a solution to a linear equation in a Hilbert space H$H$, driven by a Lévy process taking values in a Hilbert space U?H$U?H$ is established. The so-called cylindrical càdlàg property is investigated as well. A special emphasis is put on infinite systems of linear equations driven by independent Lévy processes.     

Multivariate fractionally integrated CARMA processes

A multivariate analogue of the fractionally integrated continuous time autoregressive moving average (FICARMA) process defined by Brockwell [Representations of continuous-time ARMA processes, J. Appl. Probab. 41 (A) (2004) 375-382] is introduced. We show that the multivariate FICARMA process has two kernel representations: as an integral over the fractionally integrated CARMA kernel with respect to a Lévy process and as an integral over the original (not fractionally integrated) CARMA kernel with respect to the corresponding fractional Lévy process (FLP). In order to obtain the latter representation we extend FLPs to the multivariate setting. In particular we give a spectral representation of FLPs and consequently, derive a spectral representation for FICARMA processes. Moreover, various probabilistic properties of the multivariate FICARMA process are discussed. As an example we consider multivariate fractionally integrated Ornstein-Uhlenbeck processes.     

Jump-adapted discretization schemes for Lévy-driven SDEs

We present new algorithms for weak approximation of stochastic differential equations driven by pure jump Lévy processes. The method uses adaptive non-uniform discretization based on the times of large jumps of the driving process. To approximate the solution between these times we replace the small jumps with a Brownian motion. Our technique avoids the simulation of the increments of the Lévy process, and in many cases achieves better convergence rates than the traditional Euler scheme with equal time steps. To illustrate the method, we discuss an application to option pricing in the Libor market model with jumps.     

Computation of the invariant measure for a Lévy driven SDE: Rate of convergence

We study the rate of convergence of some recursive procedures based on some “exact” or “approximate” Euler schemes which converge to the invariant measure of an ergodic SDE driven by a Lévy process. The main interest of this work is to compare the rates induced by “exact” and “approximate” Euler schemes. In our main result, we show that replacing the small jumps by a Brownian component in the approximate case preserves the rate induced by the exact Euler scheme for a large class of Lévy processes.     

On the small-time behaviour of Lévy-type processes

We show some Chung-type lim inf$lim\; inf$ law of the iterated logarithm results at zero for a class of (pure-jump) Feller or Lévy-type processes. This class includes all Lévy processes. The norming function is given in terms of the symbol of the infinitesimal generator of the process. In the Lévy case, the symbol coincides with the characteristic exponent.     

Characterization of dependence of multidimensional Lévy processes using Lévy copulas

This paper suggests Lévy copulas in order to characterize the dependence among components of multidimensional Lévy processes. This concept parallels the notion of a copula on the level of Lévy measures. As for random vectors, a version of Sklar's theorem states that the law of a general multivariate Lévy process is obtained by combining arbitrary univariate Lévy processes with an arbitrary Lévy copula. We construct parametric families of Lévy copulas and prove a limit theorem, which indicates how to obtain the Lévy copula of a multivariate Lévy process X from the ordinary copula of the random vector X_t for small t.     

The forest associated with the record process on a Lévy tree

We perform a pruning procedure on a Lévy tree and instead of throwing away the removed sub-tree, we regraft it on a given branch (not related to the Lévy tree). We prove that the tree constructed by regrafting is distributed as the original Lévy tree, generalizing a result of Addario-Berry, Broutin and Holmgren where only Aldous’s tree is considered. As a consequence, we obtain that the “average pruning time” of a leaf is distributed as the height of a leaf picked at random in the Lévy tree.     

A stochastic heat equation with the distributions of Lévy processes as its invariant measures

We consider a linear heat equation on a half line with an additive noise chosen properly in such a manner that its invariant measures are a class of distributions of Lévy processes. Our assumption on the corresponding Lévy measure is, in general, mild except that we need its integrability to show that the distributions of Lévy processes are the only invariant measures of the stochastic heat equation.     

Jump-diffusion processes in random environments

In this paper we investigate jump-diffusion processes in random environments which are given as the weak solutions of SDEs. We formulate conditions ensuring existence and uniqueness in law of solutions. We investigate the Markov property. To prove uniqueness we solve a general martingale problem for càdlàg processes. This result is of independent interest. Application of our results to generalized exponential Lévy model are present in the last section.     

Homogenization of random transport along periodic two-dimensional flows

We present a model for random transport along periodic two-dimensional flows and use the concept of rotation numbers from dynamical systems to prove a functional central limit theorem for this model. The limiting law turns out to be a stable Lévy process.     

Parameter estimation of a bivariate compound Poisson process

In this article, we review the concept of a Lévy copula to describe the dependence structure of a bivariate compound Poisson process. In this first statistical approach we consider a parametric model for the Lévy copula and estimate the parameters of the full dependent model based on a maximum likelihood approach. This approach ensures that the estimated model remains in the class of multivariate compound Poisson processes. A simulation study investigates the small sample behaviour of the MLEs, where we also suggest a new simulation algorithm. Finally, we apply our method to Danish fire insurance data.     

Triangular array limits for continuous time random walks

A continuous time random walk (CTRW) is a random walk subordinated to a renewal process, used in physics to model anomalous diffusion. Transition densities of CTRW scaling limits solve fractional diffusion equations. This paper develops more general limit theorems, based on triangular arrays, for sequences of CTRW processes. The array elements consist of random vectors that incorporate both the random walk jump variable and the waiting time preceding that jump. The CTRW limit process consists of a vector-valued Lévy process whose time parameter is replaced by the hitting time process of a real-valued nondecreasing Lévy process (subordinator). We provide a formula for the distribution of the CTRW limit process and show that their densities solve abstract space–time diffusion equations. Applications to finance are discussed, and a density formula for the hitting time of any strictly increasing subordinator is developed.     

Asymptotic results for time-changed Lévy processes sampled at hitting times

We provide asymptotic results for time-changed Lévy processes sampled at random instants. The sampling times are given by the first hitting times of symmetric barriers, whose distance with respect to the starting point is equal to ε. For a wide class of Lévy processes, we introduce a renormalization depending on ε, under which the Lévy process converges in law to an α-stable process as ε goes to 0. The convergence is extended to moments of hitting times and overshoots. These results can be used to build high frequency statistical procedures. As examples, we construct consistent estimators of the time change and, in the case of the CGMY process, of the Blumenthal-Getoor index. Convergence rates and a central limit theorem for suitable functionals of the increments of the observed process are established under additional assumptions.     

BSDEs driven by time-changed Lévy noises and optimal control

We study backward stochastic differential equations (BSDEs) for time-changed Lévy noises when the time-change is independent of the Lévy process. We prove existence and uniqueness of the solution and we obtain an explicit formula for linear BSDEs and a comparison principle. BSDEs naturally appear in control problems. Here we prove a sufficient maximum principle for a general optimal control problem of a system driven by a time-changed Lévy noise. As an illustration we solve the mean–variance portfolio selection problem.     

Local time–space stochastic calculus for Lévy processes

We develop a stochastic calculus on the plane with respect to the local times of a large class of Lévy processes. We can then extend to these Lévy processes an Itô formula that was established previously for Brownian motion. Our method provides also a multidimensional version of the formula. We show that this formula generates many “Itô formulas” that fit various problems. In the special case of a linear Brownian motion, we recover a recently established Itô formula that involves local times on curves. This formula is already used in financial mathematics.     

Constructing hierarchical Archimedean copulas with Lévy subordinators

A probabilistic interpretation for hierarchical Archimedean copulas based on Lévy subordinators is given. Independent exponential random variables are divided by group-specific Lévy subordinators which are evaluated at a common random time. The resulting random vector has a hierarchical Archimedean survival copula. This approach suggests an efficient sampling algorithm and allows one to easily construct several new parametric families of hierarchical Archimedean copulas.     

A stochastic linear–quadratic problem with Lévy processes and its application to finance

We study a Linear–Quadratic Regulation (LQR) problem with Lévy processes and establish the closeness property of the solution of the multi-dimensional Backward Stochastic Riccati Differential Equation (BSRDE) with Lévy processes. In particular, we consider multi-dimensional and one-dimensional BSRDEs with Teugel’s martingales which are more general processes driven by Lévy processes. We show the existence and uniqueness of solutions to the one-dimensional regular and singular BSRDEs with Lévy processes by means of the closeness property of the BSRDE and obtain the optimal control for the non-homogeneous case. An application of the backward stochastic differential equation approach to a financial (portfolio selection) problem with full and partial observation cases is provided.     

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.