White Noise of Poisson Random Measures

We develop a white noise theory for Poisson random measures associated with a pure jump Lévy process. The starting point of this theory is the chaos expansion of Itô. We use this to construct the white noise of a Poisson random measure, which takes values in a certain distribution space. Then we show, how a Skorohod/Itô integral for point processes can be represented by a Bochner integral in terms of white noise of the random measure and a Wick product. Further, based on these concepts we derive a generalized Clark–Haussmann–Ocone theorem with respect to a combination of Gaussian noise and pure jump Lévy noise. We apply this theorem to obtain an explicit formula for partial observation minimal variance portfolios in financial markets, driven by Lévy processes. As an example we compute the closest hedge to a binary option.     

Quantum Lévy Laplacian and associated heat equation

As a non-commutative extension of the Lévy Laplacian for entire functions on a nuclear space, we define the quantum Lévy Laplacian acting on white noise operators. We solve a heat type equation associated with the quantum Lévy Laplacian and study its relation to the classical Lévy heat equation. The solution to the quantum Lévy heat equation is obtained also from a normal-ordered white noise differential equation involving the quadratic quantum white noise.     

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.     

The Segal-Bargmann transform for Lévy white noise functionals associated with non-integrable Lévy processes

By using a method of truncation, we derive the closed form of the Segal-Bargmann transform of Lévy white noise functionals associated with a Lévy process with the Lévy spectrum without the moment condition. Besides, a sufficient and necessary condition to the existence of Lévy stochastic integrals is obtained.     

Itô Formula for Free Stochastic Integrals

The objects under investigation are the stochastic integrals with respect to free Lévy processes. We define such integrals for square-integrable integrands, as well as for a certain general class of bounded integrands. Using the product form of the Itô formula, we prove the full functional Itô formula in this context.     

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.     

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.     

A Review of White Noise Analysis from a Probabilistic Standpoint

The main notions and tools from white noise analysis are set up on the basis of the calculus of Gaussian random variables and the S -transform. A new proof of the formula for the S -transform of Itô integrals is given. Moreover, measurability and the martingale property with respect to the Brownian filtration are characterized in terms of the S-transform. This allows the extension of these notions to random variables and processes, respectively, in the space of Hida distributions.     

Dynamics of transport under random fluxes on the boundary

The impact of boundary noise on the dynamical evolution of the scalar transport equation in shear flows is studied, taking off from earlier studies in shear-flow dispersion in internal waves, a mechanism for horizontal mixing in the ocean. In particular, we model a gravity current evolving under an assumed shear-flow. The transport equation is deterministic, with a noise term at the inlet boundary. This was motivated by observed seasonal fluctuations in some known sources of salty, dense water in the oceans, like the Red Sea overflow, as well as observed thermal and saline anomalies in the thermohaline circulation.The noises used were: Wiener white, Wiener colored, Lévy white, and Lévy colored noise. Lévy processes form a more general class of processes which are generally non-Gaussian in distribution, and may have infinitely many jumps in finite time. They have been used to model pollutant point-sources, the flight time of particles in vortices, and linear and nonlinear anomalous diffusion.The major finding was that white noises (Wiener and Lévy ) and colored Wiener noise all have the effect of impeding the diffusion process, by as much as 33%. However, colored Lévy noise (non-Gaussian, time-correlated) does not have this effect on diffusion. This would suggest that time-correlation is more important in distinguishing noises than the distribution of the process that produced the noise. This also explains why Lévy colored noise showed great sensitivity to the stability parameter α, while Lévy white noise is unaffected by its stability parameter.     

White noise analysis for Lévy processes

We construct a white noise theory for Lévy processes. The starting point of this theory is a chaos expansion for square integrable random variables. We use this approach to Malliavin calculus to prove the following white noise generalization of the Clark-Haussmann-Ocone formula for Lévy processes

     

Adaptive nonparametric estimation for Lévy processes observed at low frequency

This article deals with adaptive nonparametric estimation for Lévy processes observed at low frequency. For general linear functionals of the Lévy measure, we construct kernel estimators, provide upper risk bounds and derive rates of convergence under regularity assumptions.     

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.     

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.     

Lévy white noise measures on infinite-dimensional spaces: Existence and characterization of the measurable support

It is shown that a Lévy white noise measure Λ always exists as a Borel measure on the dual K^′ of the space K of C^∞ functions on R with compact support. Then a characterization theorem that ensures that the measurable support of Λ is contained in S^′ is proved. In the course of the proofs, a representation of the Lévy process as a function on K^′ is obtained and stochastic Lévy integrals are studied.     

Gradient estimate for Ornstein-Uhlenbeck jump processes

By using absolutely continuous lower bounds of the Lévy measure, explicit gradient estimates are derived for the semigroup of the corresponding Lévy process with a linear drift. A derivative formula is presented for the conditional distribution of the process at time t under the condition that the process jumps before t. Finally, by using bounded perturbations of the Lévy measure, the resulting gradient estimates are extended to linear SDEs driven by Lévy-type processes.     

Harnack inequalities for Ornstein-Uhlenbeck processes driven by Lévy processes

By using the existing sharp estimates of the density function for rotationally invariant symmetric α-stable Lévy processes and rotationally invariant symmetric truncated α-stable Lévy processes, we obtain that the Harnack inequalities hold for rotationally invariant symmetric α-stable Lévy processes with α∈(0,2) and Ornstein-Uhlenbeck processes driven by rotationally invariant symmetric α-stable Lévy process, while the logarithmic Harnack inequalities are satisfied for rotationally invariant symmetric truncated α-stable Lévy processes.     

The Lévy-Itô decomposition in free probability

In this paper we prove the free analog of the Lévy-Itô decomposition for Lévy processes. A significant part of the proof consists of introducing free Poisson random measures, proving their existence and developing a theory of integration with respect to such measures. The existence of free Poisson random measures also yields, via the free Lévy-Itô decomposition, an alternative proof of the general existence of free Lévy processes (in law).MaPhySto – The Danish National Research Foundation Network in Mathematical Physics and StochasticsSupported by the Danish Natural Science Research CouncilMathematics Subject Classification (2000): Primary 46L54; Secondary 60G20, 60G57Acknowledgement We are grateful to the referee for many helpful remarks.     

Lévy processes and homogeneous killing transformation

In this paper we introduce homogeneous multiplicative functionals of Lévy processes and investigate their bivariate Revuz measures. We also prove that the sector condition will be inherited by sub-Lévy processes through the killing transformation.This research is supported in part by Zhejiang University Science Foundation.     

On the Besov regularity of periodic Lévy noises

In this paper, we study the Besov regularity of Lévy white noises on the d-dimensional torus. Due to their rough sample paths, the white noises that we consider are defined as generalized stochastic fields. We, initially, obtain regularity results for general Lévy white noises. Then, we focus on two subclasses of noises: compound Poisson and symmetric-α-stable (including Gaussian), for which we make more precise statements. Before measuring regularity, we show that the question is well-posed; we prove that Besov spaces are in the cylindrical σ-field of the space of generalized functions. These results pave the way to the characterization of the n-term wavelet approximation properties of stochastic processes.     

The Lévy Laplacian and stable processes

In this paper we give stochastic processes generated by powers of the Lévy Laplacian acting on a space of generalized white noise distributions using stable processes.