Exponential ergodicity and regularity for equations with Lévy noise

We prove exponential convergence to the invariant measure, in the total variation norm, for solutions of SDEs driven by α-stable noises in finite and in infinite dimensions. Two approaches are used. The first one is based on Liapunov’s function approach by Harris, and the second on Doeblin’s coupling argument in [8]. Irreducibility and uniform strong Feller property play an essential role in both approaches. We concentrate on two classes of Markov processes: solutions of finite dimensional equations, introduced in [27], with Hölder continuous drift and a general, non-degenerate, symmetric α-stable noise, and infinite dimensional parabolic systems, introduced in [29], with Lipschitz drift and cylindrical α-stable noise. We show that if the nonlinearity is bounded, then the processes are exponential mixing. This improves, in particular, an earlier result established in [28], with a different method.     

On solutions to backward stochastic partial differential equations for Lévy processes

In this paper, we prove the existence and uniqueness of the solution for a class of backward stochastic partial differential equations (BSPDEs, for short) driven by the Teugels martingales associated with a Lévy process satisfying some moment conditions and by an independent Brownian motion. An example is given to illustrate the theory.     

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

     

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.     

Existence of mild solutions for stochastic differential equations and semilinear equations with non-Gaussian Lévy noise

Existence and uniqueness of the mild solutions for stochastic differential equations for Hilbert valued stochastic processes are discussed, with the multiplicative noise term given by an integral with respect to a general compensated Poisson random measure. Parts of the results allow for coefficients which can depend on the entire past path of the solution process. In the Markov case Yosida approximations are also discussed, as well as continuous dependence on initial data, and coefficients. The case of coefficients that besides the dependence on the solution process have also an additional random dependence is also included in our treatment. All results are proven for processes with values in separable Hilbert spaces. Differentiable dependence on the initial condition is proven by adapting a method of S. Cerrai.     

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.     

Stochastic hydrodynamic-type evolution equations driven by Lévy noise in 3D unbounded domains—Abstract framework and applications

The existence of martingale solutions of the hydrodynamic-type equations in 3D possibly unbounded domains is proved. The construction of the solution is based on the Faedo–Galerkin approximation. To overcome the difficulty related to the lack of the compactness of Sobolev embeddings in the case of unbounded domain we use certain Fréchet space. Besides, we use compactness and tightness criteria in some nonmetrizable spaces and a version of the Skorohod theorem in non-metric spaces. The general framework is applied to the stochastic Navier–Stokes, magneto-hydrodynamic (MHD) and the Boussinesq equations.     

Stochastic equations of super-Lévy processes with general branching mechanism

In this work, the process of distribution functions of a one-dimensional super-Lévy process with general branching mechanism is characterized as the pathwise unique solution of a stochastic integral equation driven by time–space Gaussian white noises and Poisson random measures. This generalizes the recent work of Xiong (2013), where the result for a super-Brownian motion with binary branching mechanism was obtained.     

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 heat equation with time-independent multiplicative stable Lévy noise

We study the heat equation with a random potential term. The potential is a one-sided stable noise, with positive jumps, which does not depend on time. To avoid singularities, we define the equation in terms of a construction similar to the Skorokhod integral or Wick product. We give a criterion for existence based on the dimension of the space variable, and the parameter p$p$ of the stable noise. Our arguments are different for p<1$p<1$ and p?1$p?1$.     

Lifting Lévy processes to hyperfinite random walks

An internal lifting for an arbitrary measurable Lévy process is constructed. This lifting reflects our intuitive notion of a process which is the infinitesimal sum of its infinitesimal increments, those in turn being independent from and closely related to each other - for short, the process can be regarded as some kind of random walk (where the step size generically will vary). The proof uses the existence of càdlàg modifications of Lévy processes and certain features of hyperfinite adapted probability spaces, commonly known as the “model theory of stochastic processes”.     

Lévy processes and their subordination in matrix Lie groups

Lévy processes in matrix Lie groups are studied. Subordination (random time change) is used to show that quasi-invariance of the Brownian motion in a Lie group induces absolute continuity of the laws of the corresponding pure jump processes. These results are applied to several examples which are discussed in detail.     

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.     

Poincaré inequality for linear SPDE driven by Lévy Noise

In this paper, we prove the Poincaré inequality and the integration by parts formula for the invariant measure of the linear SPDE driven by Lévy Noise. The equation was researched in Dong and Xie [5], which has proved the existence and uniqueness of the weak solution and the ergodicity of the Markov semigroup associated with the solution.     

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.     

The exact packing measure of Lévy trees

We study fine properties of Lévy trees that are random compact metric spaces introduced by Le Gall and Le Jan in 1998 as the genealogy of continuous state branching processes. Lévy trees are the scaling limits of Galton-Watson trees and they generalize the Aldous continuum random tree which corresponds to the Brownian case. In this paper, we prove that Lévy trees always have an exact packing measure: we explicitly compute the packing gauge function and we prove that the corresponding packing measure coincides with the mass measure up to a multiplicative constant.     

The fractional stochastic heat equation on the circle: Time regularity and potential theory

We consider a system of d$d$ linear stochastic heat equations driven by an additive infinite-dimensional fractional Brownian noise on the unit circle S¹$S^1$. We obtain sharp results on the Hölder continuity in time of the paths of the solution u=_{{u(t,x)}t∈R+,x∈S1}$u={\left\{u(t,x)\right\}}_{t\in {\mathbb{R}}_{+},x\in S^1}$. We then establish upper and lower bounds on hitting probabilities of u$u$, in terms of the Hausdorff measure and Newtonian capacity respectively.