Nonlinear stochastic integrals for hyperfinite Lévy processes

I develop a notion of nonlinear stochastic integrals for hyperfinite Lévy processes and use it to find exact formulas for expressions which are intuitively of the form and , where l is a Lévy process. These formulas are then applied to geometric Lévy processes, infinitesimal transformations of hyperfinite Lévy processes, and to minimal martingale measures. Some of the central concepts and results are closely related to those found in S. Cohen’s work on stochastic calculus for processes with jumps on manifolds, and the paper may be regarded as a reworking of his ideas in a different setting and with totally different techniques.     

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 new family of time-space harmonic polynomials with respect to Lévy processes

By means of a symbolic method, a new family of time-space harmonic polynomials with respect to Lévy processes is given. The coefficients of these polynomials involve a formal expression of Lévy processes by which many identities are stated. We show that this family includes classical families of polynomials such as Hermite polynomials. Poisson–Charlier polynomials result to be a linear combinations of these new polynomials, when they have the property to be time-space harmonic with respect to the compensated Poisson process. The more general class of Lévy–Sheffer polynomials is recovered as a linear combination of these new polynomials, when they are time-space harmonic with respect to Lévy processes of very general form. We show the role played by cumulants of Lévy processes, so that connections with boolean and free cumulants are also stated.     

Ergodicity of stochastic Boussinesq equations driven by Lévy processes

We consider a class of stochastic Boussinesq equations driven by Lévy processes and establish the uniqueness of its invariant measure. The proof is based on the progressive stopping time technique.     

Symmetric stochastic integrals with respect to p-adic Brownian motion†

In this paper, we consider complex-valued Brownian motion with p-adic time index and the associated abstract Wiener space. We define symmetric stochastic integrals with respect to p-adic Brownian motion. We also provide a sufficient condition for the existence of symmetric stochastic integrals and present a relation to the adjoint of the Malliavin derivatives.     

Simulation of Student–Lévy processes using series representations

Lévy processes have become very popular in many applications in finance, physics and beyond. The Student–Lévy process is one interesting special case where increments are heavy-tailed and, for 1-increments, Student t distributed. Although theoretically available, there is a lack of path simulation techniques in the literature due to its complicated form. In this paper we address this issue using series representations with the inverse Lévy measure method and the rejection method and prove upper bounds for the mean squared approximation error. In the numerical section we discuss a numerical inversion scheme to find the inverse Lévy measure efficiently. We extend the existing numerical inverse Lévy measure method to incorporate explosive Lévy tail measures. Monte Carlo studies verify the error bounds and the effectiveness of the simulation routine. As a side result we obtain series representations of the so called inverse gamma subordinator which are used to generate paths in this model.     

Unique strong solutions of Lévy processes driven stochastic differential equations with discontinuous coefficients

We study the strong solutions for a class of one-dimensional stochastic differential equations driven by a Brownian motion and a pure jump Lévy process. Under fairly general conditions on the coefficients, we prove the pathwise uniqueness by showing the weak uniqueness and applying a local time technique.     

Symmetric rearrangements around infinity with applications to Lévy processes

We prove a new rearrangement inequality for multiple integrals, which partly generalizes a result of Friedberg and Luttinger (Arch Ration Mech 61:35–44, 1976) and can be interpreted as involving symmetric rearrangements of domains around $\infty $ . As applications, we prove two comparison results for general Lévy processes and their symmetric rearrangements. The first application concerns the survival probability of a point particle in a Poisson field of moving traps following independent Lévy motions. We show that the survival probability can only increase if the point particle does not move, and the traps and the Lévy motions are symmetrically rearranged. This essentially generalizes an isoperimetric inequality of Peres and Sousi (Geom Funct Anal 22(4):1000–1014, 2012) for the Wiener sausage. In the second application, we show that the $q$ -capacity of a Borel measurable set for a Lévy process can only decrease if the set and the Lévy process are symmetrically rearranged. This result generalizes an inequality obtained by Watanabe (Z Wahrsch Verw Gebiete 63:487–499, 1983) for symmetric Lévy processes.     

Abrupt Lévy processes

Among Lévy processes with unbounded variation, we distinguish the abrupt ones, which are characterised by infinitely sharp extrema. Stable processes with parameter α>1 and creeping Lévy processes are abrupt. We give a characterisation of abrupt processes and study their Dini derivatives at all points of their trajectories.     

Decomposition results for stochastic storage processes and queues with alternating Lévy inputs

In this paper we generalize known workload decomposition results for Lévy queues with secondary jump inputs and queues with server vacations or service interruptions. Special cases are polling systems with either compound Poisson or more general Lévy inputs. Our main tools are new martingale results, which have been derived in a companion paper.     

Optimal variational principle for backward stochastic control systems associated with Lévy processes

The paper is concerned with optimal control of backward stochastic differential equation (BSDE) driven by Teugel’s martingales and an independent multi-dimensional Brownian motion,where Teugel’s martin- gales are a family of pairwise strongly orthonormal martingales associated with Lévy processes (see e.g.,Nualart and Schoutens’ paper in 2000).We derive the necessary and sufficient conditions for the existence of the op- timal control by means of convex variation methods and duality techniques.As an application,the optimal control problem of linear backward stochastic differential equation with a quadratic cost criteria (or backward linear-quadratic problem,or BLQ problem for short) is discussed and characterized by a stochastic Hamilton system.     

Structural properties of reflected Lévy processes

This paper considers a number of structural properties of reflected Lévy processes, where both one-sided reflection (at 0) and two-sided reflection (at both 0 and K>0) are examined. With V _t being the position of the reflected process at time t, we focus on the analysis of $\zeta(t):=\mathbb{E}V_{t}$ and $\xi(t):=\mathbb{V}\mathrm{ar}V_{t}$ . We prove that for the one- and two-sided reflection, ζ(t) is increasing and concave, whereas for the one-sided reflection, ξ(t) is increasing. In most proofs we first establish the claim for the discrete-time counterpart (that is, a reflected random walk), and then use a limiting argument. A key step in our proofs for the two-sided reflection is a new representation of the position of the reflected process in terms of the driving Lévy process.     

Harmonic analysis of additive Lévy processes

Let X ₁, . . . ,X _N denote N independent d-dimensional Lévy processes, and consider the N-parameter random field $$\mathfrak{X}(t) := X_1(t_1)+\cdots+ X_N(t_N).$$ First we demonstrate that for all nonrandom Borel sets ${F\subseteq{{\bf R}^d}}$ , the Minkowski sum ${\mathfrak{X}({{\bf R}^{N}_{+}})\oplus F}$ , of the range ${\mathfrak{X}({{\bf R}^{N}_{+}})}$ of ${\mathfrak{X}}$ with F, can have positive d-dimensional Lebesgue measure if and only if a certain capacity of F is positive. This improves our earlier joint effort with Yuquan Zhong by removing a certain condition of symmetry in Khoshnevisan et al. (Ann Probab 31(2):1097–1141, 2003). Moreover, we show that under mild regularity conditions, our necessary and sufficient condition can be recast in terms of one-potential densities. This rests on developing results in classical (non-probabilistic) harmonic analysis that might be of independent interest. As was shown in Khoshnevisan et al. (Ann Probab 31(2):1097–1141, 2003), the potential theory of the type studied here has a large number of consequences in the theory of Lévy processes. Presently, we highlight a few new consequences.     

The multifractal nature of Lévy processes

We show that the sample paths of most Lévy processes are multifractal functions and we determine their spectrum of singularities. Received: 21 February 1997 / Revised version: 27 July 1998     

Remark on pathwise uniqueness of stochastic differential equations driven by Lévy processes

Consider real-valued processes determined by stochastic differential equations driven by Lévy processes. The jump parts of the driving Lévy process are not always α-stable ones, nor symmetric ones. In the present article, we shall study the pathwise uniqueness of the solutions to the stochastic differential equations under the conditions on the coefficients that the diffusion and the jump terms are Hölder continuous, while the drift one is monotonic. Our approach is based on Gronwall’s inequality.     

Timing portfolio strategies with exponential Lévy processes

Computational Management Science - This paper analyses the impact of parametric timing portfolio strategies on the U.S. stock market. In particular, we assume that the log-returns follow a given...     

The ergodicity of stochastic partial differential equations with Lévy jump

In this article, the authors prove the uniqueness in law of a class of stochastic equations in infinite dimension, then we apply it to establish the existence and uniqueness of invariant measure of the generalized stochastic partial differential equation perturbed by Lévy process.