Well-posedness for the stochastic 2D primitive equations with Lévy noise

The two-dimensional primitive equations with Lévy noise are studied in this paper. We proved the existence and uniqueness of the solutions in a fixed probability space which based on a priori estimates, weak convergence method and monotonicity arguments.     

Almost automorphic solutions for stochastic differential equations driven by Lévy noise with exponential dichotomy

In this article, we study almost automorphic solutions for semilinear stochastic differential equations driven by Lévy noise. We establish the existence and uniqueness of bounded solutions by using the Banach fixed point theorem, the exponential dichotomy property and stochastic analysis techniques. Furthermore, this unique bounded solution is almost automorphic in distribution under slightly stronger conditions. We also give two examples to illustrate our results.     

Strong solutions to stochastic equations with a Lévy noise and a non-constant diffusion coefficient

The goal of this study is to prove an existence and uniqueness theorem for the solution of a stochastic differential equation with Lévy noise in the case where the drift coefficient can be discontinuous. Additionally, the differentiability of the solution with respect to the initial condition is proved.     

Periodic averaging method for impulsive stochastic differential equations with Lévy noise

The purpose of this paper is to present a periodic averaging method for impulsive stochastic differential equations with Lévy noise under non-Lipschitz condition. It is shown that the solutions of impulsive stochastic differential equations with Lévy noise converge to the solutions of the corresponding averaged stochastic differential equations without impulses     

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.     

Doubly-weighted pseudo almost automorphic solutions for nonlinear stochastic differential equations driven by Lévy noise

This paper introduces the definitions of Poisson doubly-weighted pseudo almost automorphy and doubly-weighted pseudo almost automorphy (DWPAA) in distribution. Based on some suitable assumptions, we establish some basic theory for these definitions, and investigate the existence, uniqueness and exponential stability of the DWPAA solution in distribution for a class of nonlinear stochastic differential equations driven by Lévy noise. Finally, an example is further given to illustrate the effectiveness of our results.     

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.     

Synchronization for stochastic hybrid coupled controlled systems with Lévy noise

In this paper, synchronization for stochastic hybrid-delayed coupled systems with Lévy noise on a network (SHDCLN) is investigated via aperiodically intermittent control. Here time delays, Markovian switching and Lévy noise are considered on a network simultaneously for the first time. After that, by means of Lyapunov method, graph theory, and some techniques of inequality, some sufficient conditions are derived to guarantee the synchronization for SHDCLN. In addition, the designed range of aperiodically intermittent controller parameters is shown. Meanwhile, the coupling strength and the perturbed intensity of noise have a great impact on the intensity of control. Then, we investigate synchronization for stochastic hybrid delayed Chua's circuits with Lévy noise on a network as a practical application of our theoretical results. Finally, a numerical example is given to illustrate the effectiveness of the theoretical results.     

Existence and uniqueness of path wise solutions for stochastic integral equations driven by Lévy noise on separable Banach spaces

The stochastic integrals of M- type 2 Banach valued random functions w.r.t. compensated Poisson random measures introduced in (Rüdiger, B., 2004, In: Stoch. Stoch. Rep., 76, 213–242.) are discussed for general random functions. These are used to solve stochastic integral equations driven by non Gaussian Lévy noise on such spaces. Existence and uniqueness of the path wise solutions are proven under local Lipshitz conditions for the drift and noise coefficients on M-type 2 as well as general separable Banach spaces. The continuous dependence of the solution on the initial data as well as on the drift and noise coefficients are shown. The Markov properties for the solutions are analyzed.     

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.     

Small time and semiclassical asymptotics for stochastic heat equations driven by Lévy noise

The paper is devoted to the study of stochastic heat equations driven by Lévy noise. Applying the WKB method, we obtain multiplicative small time and semiclassical asymptotics for the Green functions and for solutions of the Cauchy problem for the heat equation under some natural additional assumptions on their coefficients. The first step in this construction consists in solving the corresponding stochastic Hamilton-Jacobi equations which constitute the "classical part" of the semiclassical approximation. In its turn, the corresponding Hamilton-Jacobi equations can be solved via solutions of the corresponding Hamiltonian systems, which gives rise to the method of stochastic characteristics. The relevant theory of stochastic Hamiltonian systems and stochastic Hamilton-Jacobi equations was developed in our previous papers. Here we put the final rung on the ladder: stochastic Hamiltonian systems, stochastic Hamilton-Jacobi equations, stochastic heat equations.     

Asymptotically almost automorphic solutions to stochastic differential equations driven by a Lévy process

In this paper, a new concept of Poisson asymptotically almost automorphy for stochastic processes is introduced. And then, some fundamental properties including composition theorems for the space of such processes are proved. Subsequently, this concept is applied to investigate the existence and uniqueness of asymptotically almost automorphic solutions in distribution to some linear and semilinear stochastic differential equations driven by a Lévy process under some suitable conditions. Finally, an example is given to illustrate the main results.     

Smooth solution of a nonlocal Fokker–Planck equation associated with stochastic systems with Lévy noise

It is shown that the solution of a nonlocal Fokker–Planck equation is smooth with respect to both time and space variable whenever the divergence of the smooth drift has a lower bound.     

On the pathwise uniqueness of solutions of stochastic differential equations driven by symmetric stable Lévy processes1

We investigate a sufficient condition for pathwise uniqueness property for 1D stochastic differential equation driven by symmetric α-stable Lévy process, where α ∈ (1, 2).     

On a class of singular stochastic control problems driven by Lévy noise

A class of singular stochastic control problems whose value functions satisfy an invariance property was studied by Lasry and Lions (2000). They have shown that, within this class, any singular control problem is equivalent to the corresponding standard stochastic control problem. The equivalence is in the sense that their value functions are equal. In this work, we clarify their idea and extend their work to allow Lévy type noise. In addition, for the purpose of application, we apply our result to an optimal trade execution problem studied by Lasry and Lions (2007).     

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.     

Stability analysis of a stochastic Gilpin–Ayala model driven by Lévy noise

A stochastic one-dimensional Gilpin–Ayala model driven by Lévy noise is presented in this paper. Firstly, we show that this model has a unique global positive solution under certain conditions. Then sufficient conditions for the almost sure exponential stability and moment exponential stability of the trivial solution are established. Results show that the jump noise can make the trivial solution stable under some conditions. Numerical example is introduced to illustrate the results.     

Lévy driven CARMA generalized processes and stochastic partial differential equations

We give a new definition of a Lévy driven CARMA random field, defining it as a generalized solution of a stochastic partial differential equation (SPDE). Furthermore, we give sufficient conditions for the existence of a mild solution of our SPDE. Our model unifies all known definitions of CARMA random fields, and in particular for dimension 1 we obtain the classical CARMA process.     

Anticipative calculus for Lévy processes and stochastic differential equations*

We develop an anticipative calculus for Lévy processes with finite second moment for analysing anticipating stochastic differential equations. The calculus is based on the chaos expansion of square-integrable random variables in terms of iterated integrals with respect to the compensated Poisson random measure. We define a space of smooth and generalized random variables in terms of such chaos expansions, and present anticipative stochastic integration, the Wick product and the so-called 𝒮-transform. These concepts serve as tools for studying general Wick type stochastic differential equations with anticipative initial conditions. We apply the 𝒮-transform to find the unique solutions to a class of linear stochastic differential equations. The solutions can be expressed in terms of the Wick product.