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.     

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     

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.     

Random attractor for stochastic lattice dynamical systems with α-stable Lévy noises

The present paper is devoted to the existence of a random attractor for stochastic lattice dynamical systems with α-stable Lévy noises.     

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.     

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.     

Global solutions of stochastic 2D Navier-Stokes equations with Lévy noise

In this paper, we prove the global existence and uniqueness of the strong and weak solutions for 2D Navier-Stokes equations on the torus perturbed by a Lévy process. The existence of invariant measure of the solutions are proved also. This work was supported by National Basic Research Program of China (Grant No. 2006CB8059000), Science Fund for Creative Research Groups (Grant No. 10721101), National Natural Science Foundation of China (Grant Nos. 10671197, 10671168), Science Foundation of Jiangsu Province (Grant Nos. BK2006032, 06-A-038, 07-333) and Key Lab of Random Complex Structures and Data Science, Chinese Academy of Sciences     

Dynamic programming principle for stochastic control problems driven by general Lévy noise

We extend the proof of the dynamic programming principle (DPP) for standard stochastic optimal control problems driven by general Lévy noise. Under appropriate assumptions, it is shown that the DPP still holds when the state process fails to have any moments at all.     

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.     

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.     

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.     

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.     

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.     

Large deviations for the optimal filter of nonlinear dynamical systems driven by Lévy noise

In this paper, we focus on the asymptotic behavior of the optimal filter where both signal and observation processes are driven by Lévy noises. Indeed, we study large deviations for the case where the signal-to-noise ratio is small by considering weak convergence arguments. To that end, we first prove the uniqueness of the solution of the controlled Zakai and Kushner–Stratonovich equations. For this, we employ a method which transforms the associated equations into SDEs in an appropriate Hilbert space. Next, taking into account the controlled analogue of Zakai and Kushner–Stratonovich equations, respectively, the large deviation principle follows by employing the existence, uniqueness and tightness of the solutions.     

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).     

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.     

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.     

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.