A Peano-Like Theorem for Stochastic Differential Equations with Nonlocal Sample Dependence

The existence of a not-necessarily-unique strong solution for a stochastic differential equations with nonlocal sample dependence is established under the assumption that the coefficients satisfy an asymptotically local boundedness condition in addition to continuity. The proof is by an Euler-like construction of approximations. These equations include mean-field stochastic differential equations, but the nonlocal sample dependence can be more general than just the dependence on moments of the solution.     

Well-posedness of SDEs with drifts in mixed-norm spaces and driven by mixed-noises

We prove the strong well-posedness for stochastic differential equations driven by partial α-stable noise and partial Brownian noise. The drift coefficients belong to certain mixed-norm spaces, which generalize the known results even for stochastic differential equations with single noise cases.     

Existence and Uniqueness of Solutions to Time-delays Stochastic Fractional Differential Equations with Non-Lipschitz Coefficients

In this paper, we consider the existence and uniqueness of solutions to time-varying delays stochastic fractional differential equations (SFDEs) with non-Lipschitz coefficients. By using fractional calculus and stochastic analysis, we can obtain the existence result of solutions for stochastic fractional differential equations.     

Existence results and the moment estimate for nonlocal stochastic differential equations with time-varying delay

This paper considers a class of nonlocal stochastic differential equations with time-varying delay whose coefficients are dependent on the pth moment. By applying the fixed point theorem, the existence and uniqueness of the solution of nonlocal stochastic differential delay equations is studied. Also, a class of moment estimates of solutions is considered. The results are a generalization and continuation of the recent results on this issue. An example is provided to illustrate the effectiveness of our results.     

On approximations of the Euler-Peano scheme for multivalued stochastic differential equations

It is known that a unique strong solution exists for multivalued stochastic differential equations under the Lipschitz continuity and linear growth conditions. In this paper we apply the Euler-Peano scheme to show that existence of weak solution and pathwise uniqueness still hold when the coefficients are random and satisfy one-sided locally Lipschitz continuous and an integral condition (i.e. Krylov's conditions put forward in On Kolmogorov's equations for finite-dimensional diffusions, Stochastic PDE's and Kolmogorov Equations in Infinite Dimensions (Cetraro, 1998), Lecture Notes in Math., 1715, Springer, Berlin, 1999, pp. 1–63). When the coefficients are nonrandom and possibly discontinuous but only satisfy some integral conditions, the sequence of solutions of the Euler-Peano scheme converges weakly, and the limit is a weak solution of the corresponding MSDE. As a particular case, we obtain a global semi-flow for stochastic differential equations reflected in closed, convex domains.     

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.     

Stochastic partial differential equations with gradient driven by space-time fractional noises

We establish a class of stochastic partial differential equations (SPDEs) driven by space-time fractional noises, where we suppose that the drfit term contains a gradient and satisfies certain non-Lipschitz condition. We prove the strong existence and uniqueness and joint Hölder continuity of the solution to the SPDEs.     

非线性随机分数阶微分方程Euler方法的弱收敛性

论文首先证明了非线性随机分数阶微分方程解的存在唯一性,然后构造了数值求解该方程的Euler方法,并证明了当方程满足一定约束条件时,该方法是弱收敛的.特别地,当分数阶α=0时,该方程退化为非线性随机微分方程,所获结论与现有文献中的相关结论是一致的;当α≠0,且初值条件为齐次时,所获结论可视为现有文献中线性随机分数阶微分方...     

Stochastic Retarded Evolution Equations: Green Operators,Convolutions, and Solutions

Abstract

In this work, we shall investigate solution (strong, weak and mild) processes and relevant properties of stochastic convolutions for a class of stochastic retarded differential equations in Hilbert spaces. We introduce a strongly continuous one-parameter family of bounded linear operators which will completely describe the corresponding deterministic systematical dynamics with time delays. This family, which constitutes the fundamental solutions (Green's operators) of our stochastic retarded systems, is applied subsequently to define mild solutions of the stochastic retarded differential equations considered. The relations among strong, weak and mild solutions are explored. By virtue of a strong solution approximation method, Burkholder–Davis–Gundy's type of inequalities for stochastic convolutions are established.     

Jump type stochastic differential equations with non-Lipschitz coefficients: Non-confluence,Feller and strong Feller properties,and exponential ergodicity

This paper considers multidimensional jump type stochastic differential equations with super linear and non-Lipschitz coefficients. After establishing a sufficient condition for nonexplosion, this paper presents sufficient local non-Lipschitz conditions for pathwise uniqueness. The non-confluence property for solutions is investigated. Feller and strong Feller properties under local non-Lipschitz conditions are investigated via the coupling method. Sufficient conditions for irreducibility and exponential ergodicity are derived. As applications, this paper also studies multidimensional stochastic differential equations driven by Lévy processes and presents a Feynman–Kac formula for Lévy type operators.     

Well-posedness and long time behavior of singular Langevin stochastic differential equations

In this paper, we study damped Langevin stochastic differential equations with singular velocity fields. We prove the strong well-posedness of such equations. Moreover, by combining the technique of Lyapunov functions with Krylov’s estimate, we also establish exponential ergodicity for the unique strong solution.     

一类常微分方程解的存在唯一性及其应用

提出并证明了一类常微分方程解的存在唯一性成立的一个充要条件,并给出了多项式形式增长函数的一列上界.最终将此结果应用到证明一类倒向随机微分方程的唯一解问题.     

A study of a class of stochastic differential equations with non-Lipschitzian coefficients

We study a class of stochastic differential equations with non-Lipschitz coefficients. A unique strong solution is obtained and the non confluence of the solutions of stochastic differential equations is proved. The dependence with respect to the initial values is investigated. To obtain a continuous version of solutions, the modulus of continuity of coefficients is assumed to be less than |x-y| log Finally a large deviation principle of Freidlin-Wentzell type is also established in the paper.     

On existence and uniqueness of solutions to uncertain backward stochastic differential equations简

This paper is concerned with a class of uncertain backward stochastic differential equations （UBSDEs） driven by both an m-dimensional Brownian motion and a d-dimensional canonical process with uniform Lipschitzian coefficients. Such equations can be useful in mod- elling hybrid systems, where the phenomena are simultaneously subjected to two kinds of un- certainties： randomness and uncertainty. The solutions of UBSDEs are the uncertain stochastic processes. Thus, the existence and uniqueness of solutions to UBSDEs with Lipschitzian coeffi- cients are proved.     

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.     

Pathwise uniqueness of multi-dimensional stochastic differential equations with H?lder diffusion coefficients

We extend Yamada-Watababe’s criterion [J. Math. Kyoto Univ., 1971, 11: 553–563] on the pathwise uniqueness of one-dimensional stochastic differential equations to a special class of multi-dimensional stochastic differential equations.     

SOLUTION TO BSDE WITH NONHOMOGENEOUS JUMPS UNDER LOCALLY LIPSCHITZIAN CONDITION

In this paper, we investigate the existence and uniqueness of the solution to a quasilinear backward stochastic differential equation with Poisson jumps. By introducing a series of approximate equations, we can show that BSDE has a unique adapted solution.     

Existence and uniqueness of stochastic differential equations with random impulses and Markovian switching under non-lipschitz conditions

In the paper, stochastic differential equations with random impulses and Markovian switching are brought forward, where the so-called random impulse means that impulse ranges are driven by a series of random variables and impulse times are a random sequence, so these equations extend stochastic differential equations with jumps and Markovian switching. Then the existence and uniqueness of solutions to such equations are investigated by employing the Bihari inequality under non-Lipschtiz conditions.     

Local existence and non-explosion of solutions for stochastic fractional partial differential equations driven by multiplicative noise

In this paper we prove the local existence and uniqueness of solutions for a class of stochastic fractional partial differential equations driven by multiplicative noise. We also establish that for this class of equations adding linear multiplicative noise provides a regularizing effect: the solutions will not blow up with high probability if the initial data is sufficiently small, or if the noise coefficient is sufficiently large. As applications our main results are applied to various types of SPDE such as stochastic reaction–diffusion equations, stochastic fractional Burgers equation, stochastic fractional Navier–Stokes equation, stochastic quasi-geostrophic equations and stochastic surface growth PDE.     

Existence of global solutions and invariant measures for stochastic differential equations driven by Poisson type noise with non-Lipschitz coefficients

The purpose of this paper is twofold. Firstly, we investigate the problem of existence and uniqueness of solutions to stochastic differential equations with one sided dissipative drift driven by semi-martingales. Secondly, we investigate the problem of existence of an invariant measure for such equations when the coefficients are time independent.