Harnack inequalities for stochastic equations driven by Lévy noise

By using coupling argument and regularization approximations of the underlying subordinator, dimension-free Harnack inequalities are established for a class of stochastic equations driven by a Lévy noise containing a subordinate Brownian motion. The Harnack inequalities are new even for linear equations driven by Lévy noise, and the gradient estimate implied by our log-Harnack inequality considerably generalizes some recent results on gradient estimates and coupling properties derived for Lévy processes or linear equations driven by Lévy noise. The main results are also extended to semilinear stochastic equations in Hilbert spaces.     

Large deviations for stochastic PDE with Lévy noise

We prove a large deviation principle result for solutions of abstract stochastic evolution equations perturbed by small Lévy noise. We use general large deviations theorems of Varadhan and Bryc coupled with the techniques of Feng and Kurtz (2006) [15], viscosity solutions of integro-partial differential equations in Hilbert spaces, and deterministic optimal control methods. The Laplace limit is identified as a viscosity solution of a Hamilton-Jacobi-Bellman equation of an associated control problem. We also establish exponential moment estimates for solutions of stochastic evolution equations driven by Lévy noise. General results are applied to stochastic hyperbolic equations perturbed by subordinated Wiener process.     

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.     

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.     

The Jacobi Field of a Lévy Process

We derive an explicit formula for the Jacobi field that is acting in an extended Fock space and corresponds to an ( -valued) Lévy process on a Riemannian manifold. The support of the measure of jumps in the Lévy–Khintchine representation for the Lévy process is supposed to have an infinite number of points. We characterize the gamma, Pascal, and Meixner processes as the only Lévy process whose Jacobi field leaves the set of finite continuous elements of the extended Fock space invariant.     

Parametric estimation of a bivariate stable Lévy process

We propose a parametric model for a bivariate stable Lévy process based on a Lévy copula as a dependence model. We estimate the parameters of the full bivariate model by maximum likelihood estimation. As an observation scheme we assume that we observe all jumps larger than some ε>0 and base our statistical analysis on the resulting compound Poisson process. We derive the Fisher information matrix and prove asymptotic normality of all estimates when the truncation point ε→0. A simulation study investigates the loss of efficiency because of the truncation.     

Asymptotic stability in the pth moment for stochastic differential equations with Lévy noise

This paper is devoted to study a class of stochastic differential equations with Lévy noise. In comparison to the standard Gaussian noise, Lévy noise is more versatile and interesting with a wider range of applications. However, Lévy noise makes the analysis more difficult owing to the discontinuity of its sample paths. In this paper, we attempt to overcome this difficulty. We propose several sufficient conditions under which we investigate the long-time behavior of the solution including the asymptotic stability in the pth moment and almost sure stability. Also, we discuss two types of continuity of the solution: continuous in probability and continuous in the pth moment. Finally, we provide two examples to illustrate the effectiveness of the theoretical results.     

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.     

An optimization approach to weak approximation of stochastic differential equations with jumps

We propose an optimization approach to weak approximation of stochastic differential equations with jumps. A mathematical programming technique is employed to obtain numerically upper and lower bound estimates of the expectation of interest, where the optimization procedure ends up with a polynomial programming. A major advantage of our approach is that we do not need to simulate sample paths of jump processes, for which few practical simulation techniques exist. We provide numerical results of moment estimations for Doléans-Dade stochastic exponential, truncated stable Lévy processes and Ornstein-Uhlenbeck-type processes to illustrate that our method is able to capture very well the distributional characteristics of stochastic differential equations with jumps.     

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 PDIEs and backward doubly stochastic differential equations driven by Lévy processes

In this paper, a new class of backward doubly stochastic differential equations driven by Teugels martingales associated with a Lévy process satisfying some moment condition and an independent Brownian motion is investigated. We obtain the existence and uniqueness of solutions to these equations. A probabilistic interpretation for solutions to a class of stochastic partial differential integral equations is given.     

Stochastic PDIEs with nonlinear Neumann boundary conditions and generalized backward doubly stochastic differential equations driven by Lévy processes

In this paper, a new class of generalized backward doubly stochastic differential equations (GBDSDEs in short) driven by Teugels martingales associated with Lévy process and the integral with respect to an adapted continuous increasing process is investigated. We obtain the existence and uniqueness of solutions to these equations. A probabilistic interpretation for solutions to a class of stochastic partial differential integral equations (PDIEs in short) with a nonlinear Neumann boundary condition is given.     

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.     

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.     

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.     

Large deviation principles for 2-D stochastic Navier-Stokes equations driven by Lévy processes

In this paper, we establish a large deviation principle for the two-dimensional stochastic Navier-Stokes equations driven by Lévy processes, which involves the study of the Lévy noise and the investigation of the effect of the highly nonlinear, unbounded drifts.     

Stochastic Lotka–Volterra systems with Lévy noise

This paper is concerned with stochastic Lotka–Volterra models perturbed by Lévy noise. Firstly, stochastic logistic models with Lévy noise are investigated. Sufficient and necessary conditions for stochastic permanence and extinction are obtained. Then three stochastic Lotka–Volterra models of two interacting species perturbed by Lévy noise (i.e., predator–prey system, competition system and cooperation system) are studied. For each system, sufficient and necessary conditions for persistence in the mean and extinction of each population are established. The results reveal that firstly, both persistence and extinction have close relationships with Lévy noise; Secondly, the interaction rates play very important roles in determining the persistence and extinction of the species.     

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.     

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.     

Sina?ˇ's condition for real valued Lévy processes

We prove that the upward ladder height subordinator H associated to a real valued Lévy process ξ has Laplace exponent φ that varies regularly at ∞ (respectively, at 0) if and only if the underlying Lévy process ξ satisfies Sina?ˇ's condition at 0 (respectively, at ∞). Sina?ˇ's condition for real valued Lévy processes is the continuous time analogue of Sina?ˇ's condition for random walks. We provide several criteria in terms of the characteristics of ξ to determine whether or not it satisfies Sina?ˇ's condition. Some of these criteria are deduced from tail estimates of the Lévy measure of H, here obtained, and which are analogous to the estimates of the tail distribution of the ladder height random variable of a random walk which are due to Veraverbeke and Grübel.