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.     

Analysis of generalized Lévy white noise functionals

Employing the Segal-Bargmann transform (S-transform for abbreviation) of regular Lévy white noise functionals, we define and study the generalized Lévy white noise functionals by means of their functional representations acting on test functionals. The main results generalize (Gaussian) white noise analysis initiated by T. Hida to non-Gaussian cases. Thanks to the closed form of the S-transform of Lévy white noise functionals obtained in our previous paper, we are able to define and study the renormalization of products of Lévy white noises, multiplication operator by Lévy white noises, and the differential operators with respect to a Lévy white noise and their adjoint operators. In the courses of our investigation we also obtain a formula for the products of multiple Lévy-Itô stochastic integrals. As applications, we discuss the existence of Hitsuda-Skorokhod integral for Lévy processes, Kubo-Takenaka formula for Lévy processes, and Itô formula for generalized Lévy white noise functionals.     

Stochastic PDEs Driven by Nonlinear Noise and Backward Doubly SDEs

We study a new kind of backward doubly stochastic differential equations, where the nonlinear noise term is given by Itô–Kunita's stochastic integral. This allows us to give a probabilistic interpretation of classical and Sobolev's solutions of semilinear parabolic stochastic partial differential equations driven by a nonlinear space-time noise.     

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.     

A different quantum stochastic calculus for the Poisson process

Summary We show that a gradient operator defined by perturbations of the Poisson process jump times can be used with its adjoint operator instead of the annihilation and creation operators on the Poisson-Charlier chaotic decomposition to represent the Poisson process. The quantum stochastic integration and the Itô formula are developed accordingly, leading to commutation relations which are different from the CCR. An analog of the Weyl representation is defined for a subgroup ofSL(2, ), showing that the exponential and geometric distributions are closely related in this approach.     

Interacting Fock Expansion of Lévy White Noise Functionals

Consider the Lévy white noise space (S ^*,(S ^*),), where S ^* is the Schwartz distributions over R ^d and is a Lévy white noise measure lifted from a 1-dimensional infinitely divisible distribution with finite moments. We give explicit forms and recursion formulas of moment and renormalization kernels for the Lévy white noise measure. By defining inner products (,)_[n] in n-particle spaces, we establish an interacting Fock space _n=0 ⁽ⁿ⁾ and the interacting Fock expansions for Lévy white noise functionals. The usual Fock space (H)= _n=0 can be viewed as a quotient space of the interacting Fock space. As a particular case, we give the interacting Fock expansion for gamma white noise functionals.     

Multiple integration with respect to Poisson and Lévy processes

Summary Necessary and sufficient conditions are given for the existence of a multiple stochastic integral of the form ...fdX ₁...dX_d, where X ₁, ..., X _d are components of a positive or symmetric pure jump type Lévy process in _d. Conditions are also given for a sequence of integrals of this type to converge in probability to zero or infinity, or to be tight. All arguments proceed via reduction to the special case of Poisson integrals.Dedicated to Klaus Krickeberg on the occasion of his 60th birthdaySupported by NSF grant DMS-8703804Supported by NSF grant DMS-8713103     

A nonstandard construction of Lévy Brownian motion

Summary A nonstandard construction of Lévy Brownian motion on ^d is presented, which extends R.M. Anderson's nonstandard representation of Brownian motion. It involves a nonstandard construction of white noise and gives as a classical corollary a new white noise integral representation of Lévy Brownian motion. Moreover, a new invariance principle can be deduced in a similar way as Donsker's invariance principles follows from Anderson's construction.     

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

     

The set-indexed Lévy process: Stationarity,Markov and sample paths properties

We present a satisfactory definition of the important class of Lévy processes indexed by a general collection of sets. We use a new definition for increment stationarity of set-indexed processes to obtain different characterizations of this class. As an example, the set-indexed compound Poisson process is introduced. The set-indexed Lévy process is characterized by infinitely divisible laws and a Lévy–Khintchine representation. Moreover, the following concepts are discussed: projections on flows, Markov properties, and pointwise continuity. Finally the study of sample paths leads to a Lévy–Itô decomposition. As a corollary, the semi-martingale property is proved.     

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.     

Probabilistic and fractal aspects of Lévy trees

We investigate the random continuous trees called Lévy trees, which are obtained as scaling limits of discrete Galton-Watson trees. We give a mathematically precise definition of these random trees as random variables taking values in the set of equivalence classes of compact rooted -trees, which is equipped with the Gromov-Hausdorff distance. To construct Lévy trees, we make use of the coding by the height process which was studied in detail in previous work. We then investigate various probabilistic properties of Lévy trees. In particular we establish a branching property analogous to the well-known property for Galton-Watson trees: Conditionally given the tree below level a, the subtrees originating from that level are distributed as the atoms of a Poisson point measure whose intensity involves a local time measure supported on the vertices at distance a from the root. We study regularity properties of local times in the space variable, and prove that the support of local time is the full level set, except for certain exceptional values of a corresponding to local extinctions. We also compute several fractal dimensions of Lévy trees, including Hausdorff and packing dimensions, in terms of lower and upper indices for the branching mechanism function which characterizes the distribution of the tree. We finally discuss some applications to super-Brownian motion with a general branching mechanism.     

Donsker's Delta Function of Lévy Process

Let X={X(t):tR} be a Lévy process and a non-decreasing, right continuous, bounded function with (–)=0 (((1+u ²)/u ²)d(u) is the Lévy measure). In this paper we define the Donsker delta function (X(t)–a), t>0 and aR, as a generalized Lévy functional under the condition that (0)–(0–)>0. This leads us to define F(X(t)) for any tempered distribution F, and as an application, we derive an Itô formula for F(X(t)) when has jumps at 0 and 1.     

Two-parameter Lévy processes: ItÔ formula,semigroups, and generators

We consider random Lévy fields, i.e., stationary fields continuous in probability and having independent increments. We prove that the trajectories of such fields have at most one jump on every line parallel to the axes. We derive an expression for the ItÔ change of variables for Lévy fields. We also consider semigroups generated by Lévy fields and their generators.Published in Ukrainskii Matematicheskii Zhurnal, Vol.47, No. 7, pp. 952–961, July, 1995.     

Zero-One Law for Symmetric Convolution Semigroups of Measures on Groups

Let ( _t ) _t>0 be a symmetric weakly continuous semigroup of probability measures on a nonabelien complete separable group G and let v be its Lévy measure. The purpose of this paper is to provide a relatively simple proof of the zero-one law for semigroups with the Lévy measure satisfying either v(H ^c) = or v(H ^c) = 0.     

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.     

An action functional for Lévy Brownian motion

Stoll's construction [7] of Lévy Brownian motion l on ^d as a white noise integral is used to obtain an action functional I(x) defined for the surfaces x of l. This provides a Cameron-Martin formula for translation of Lévy measure , and also a large deviation principle for scaled Lévy measures . Proofs follow the lines of [2], where nonstandard techniques were used to give natural proofs of the corresponding results for Wiener measure.The research for this paper was supported partly by a grant from the SERC.     

On a jump-type stochastic fractional partial differential equation with fractional noises

We study a class of stochastic fractional partial differential equations of order α>1$\alpha >1$ driven by a (pure jump) Lévy space–time white noise and a fractional noise. We prove the existence and uniqueness of the global mild solution by the fixed point principle under some suitable assumptions.     

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.     

Some characterizations of unimodal distribution functions

Let F be a distribution function and let Q _F(l)=0 for l<0 and Q _F(l)= sup {F(x+l)–F(x): x} for l0 be its Lévy concentration function. This paper has two purposes: to give a characterization of unimodal distribution functions (Theorem 3.5) and a representation theorem for the class of unimodal distribution functions (Theorem 6.2), both in terms of their Lévy concentration functions.Work supported by the Natural Sciences and Engineering Research Council Canada Grants A-7339 and A-7223, by the Québec Action Concertée Grant ER-1023, and by the Deutsche Forschungsgemeinschaft