non-Lipschitz Condition Research Articles

Strong existence and uniqueness to a class of nonlinear SPDEs driven by Gaussian colored noises

In this paper, we establish the strong existence and uniqueness to the following stochastic partial differential equation: ut(x)=u0(x)+12∫0tΔus(x)ds+∫0t∫Rg(us(x),∇us(x))dF(s,y),where F(s,x) is a Gaussian noise on R+×R that is white in time and colored in space. This gives an affirmative answer to the open problem in Gomez et al. (2013) for certain non-Lipschitz condition of g.

Stochastic averaging principle for differential equations with non-Lipschitz coefficients driven by fractional Brownian motion

In this paper, we are concerned with the stochastic averaging principle for stochastic differential equations (SDEs) with non-Lipschitz coefficients driven by fractional Brownian motion (fBm) of the Hurst parameter [Formula: see text]. We define the stochastic integrals with respect to the fBm in the integral formulation of the SDEs as pathwise integrals and we adopt the non-Lipschitz condition proposed by Taniguchi (1992) which is a much weaker condition with wider range of applications. The averaged SDEs are established. We then use their corresponding solutions to approximate the solutions of the original SDEs both in the sense of mean square and of probability. One can find that the similar asymptotic results are suitable for those non-Lipschitz SDEs with fBm under different types of stochastic integrals.

Some results in stochastic functional integro-differential equations with infinite delays

In this paper, we study the results on averaging principle and stability of mild solutions for stochastic functional integro-differential equation with non-Lipschitz condition. We establish the result by the method of successive approximation and Bihari's inequality under the theory of resolvent operators. Finally, an example is provided for demonstration.

Some results in stochastic functional integro-differential equations with infinite delays

In this paper, we study the results on averaging principle and stability of mild solutions for stochastic functional integro-differential equation with non-Lipschitz condition. We establish the result by the method of successive approximation and Bihari's inequality under the theory of resolvent operators. Finally, an example is provided for demonstration.

Mild solutions of local non-Lipschitz neutral stochastic functional evolution equations driven by jumps modulated by Markovian switching

ABSTRACTIn this article, we initiate a study on neutral stochastic functional evolution equations driven by jumps modulated by Markovian switching in real separable Hilbert spaces. Our goal here is to derive the existence and uniqueness of mild solutions to equations of this class under local non-Lipschitz condition proposed by Taniguchi [J. Math. Anal. Appl. 340:(2009)197–208] by means of stopping time technique and Banach fixed-point theorem. The results obtained here generalize the main results from Luo and Taniguchi [Stoch. Dyn. 9:(2009)135–152] and Jiang and Shen [Comput. Math. Appl. 61:(2011)1590–1594]. Finally, an example is worked out to illustrate the obtained results.

On the non-Lipschitz stochastic differential equations driven by fractional Brownian motion

In this paper, we use a successive approximation method to prove the existence and uniqueness theorems of solutions to non-Lipschitz stochastic differential equations (SDEs) driven by fractional Brownian motion (fBm) with the Hurst parameter $H\in(\frac{1}{2},1)$ . The non-Lipschitz condition which is motivated by a wider range of applications is much weaker than the Lipschitz one. Due to the fact that the stochastic integral with respect to fBm is no longer a martingale, we definitely lost good inequalities such as the Burkholder-Davis-Gundy inequality which is crucial for SDEs driven by Brownian motion. This point motivates us to carry out the present study.

Fuzzy stochastic differential equations of decreasing fuzziness: Non-Lipschitz coefficients

We study fuzzy stochastic differential equations driven by multidimensional Brownian motion with solutions of decreasing fuzziness. The drift and diffusion coefficients are random. Under a non-Lipschitz condition, the existence and pathwise uniqueness of solutions to such the equations are proven. The solutions are considered to be fuzzy stochastic processes. The main result is obtained with a help of a sequence of approximate solutions that converge to a desired unique local solution with trajectories having decreasing fuzziness. A parallel assertion for solutions to fuzzy stochastic differential equations of increasing fuzziness is stated as well. We indicate that our considerations of fuzzy stochastic differential equations of decreasing fuzziness can be applied to examine non-Lipschitz set-valued stochastic differential equations with solutions being set-valued stochastic processes.

Existence and stability solutions of nonlinear stochastic functional partial integrodifferential equations with infinite delay

In this paper, we are focused upon the results on existence, uniqueness and stability of mild solution for stochastic functional integrodifferential equations with non-Lipschitz condition. The theory of resolvent operator is utilised to exhibit the existence of these mild solutions. The results are obtained by using the method of successive approximation and Bihari's inequality. Finally, an example is provided for demonstration.

Time discretization of FBSDE with polynomial growth drivers and reaction–diffusion PDEs

In this paper, we undertake the error analysis of the time discretization of systems of Forward-Backward Stochastic Differential Equations (FBSDEs) with drivers having polynomial growth and that are also monotone in the state variable. We show with a counter-example that the natural explicit Euler scheme may diverge, unlike in the canonical Lipschitz driver case. This is due to the lack of a certain stability property of the Euler scheme which is essential to obtain convergence. However, a thorough analysis of the family of $\theta$-schemes reveals that this required stability property can be recovered if the scheme is sufficiently implicit. As a by-product of our analysis, we shed some light on higher order approximation schemes for FBSDEs under non-Lipschitz condition. We then return to fully explicit schemes and show that an appropriately tamed version of the explicit Euler scheme enjoys the required stability property and as a consequence converges. In order to establish convergence of the several discretizations, we extend the canonical path- and first-order variational regularity results to FBSDEs with polynomial growth drivers which are also monotone. These results are of independent interest for the theory of FBSDEs.

Mild solutions of local non-Lipschitz stochastic evolution equations with jumps

By estimating the coefficients functions in the stochastic energy equality, the existence and uniqueness of mild solutions to stochastic evolution equations (SEEs) under local non-Lipschitz condition proposed by Taniguchi with jumps are proved here. The results of Taniguchi (2009) are generalized and improved as a special case of our theory. It should be pointed that the proof for SEEs with jumps is certainly not a straightforward generalization of that for SEEs without jumps and some new techniques are developed to cope with the difficulties due to the Poisson random measures.

Generalized fractional BSDE with non Lipschitz coefficients

In this work, we deal with a generalized backward stochastic differential equation driven by a fractional Brownian motion. We essentially prove an existence and uniqueness result under non-Lipschitz condition on the generator by help of an iterated scheme on a suitable sequence.

Existence and stability of solutions to non-Lipschitz stochastic differential equations driven by Lévy noise

In this paper, successive approximation method is applied to investigate the existence and uniqueness of solutions to stochastic differential equations (SDEs) driven by Lévy noise under non-Lipschitz condition which is a much weaker condition than Lipschitz one. The stability of solutions to non-Lipschitz SDEs driven by Lévy noise is also considered, and the stochastic stability is obtained in the sense of mean square.

Approximation properties for solutions to non-Lipschitz stochastic differential equations with Lévy noise

In this paper, we consider the non-Lipschitz stochastic differential equations and stochastic functional differential equations with delays driven by Levy noise, and the approximation theorems for the solutions to these two kinds of equations will be proposed respectively. Non-Lipschitz condition is much weaker condition than the Lipschitz one. The simplified equations will be defined to make its solutions converge to that of the corresponding original equations both in the sense of mean square and probability, which constitute the approximation theorems. Copyright © 2014 John Wiley & Sons, Ltd.

Anticipated backward stochastic differential equations with jumps under the non-Lipschitz condition

This paper deals with a class of anticipated backward stochastic differential equations with Poisson jumps (ABSDEJs). We first show that there is a duality between anticipated backward stochastic differential equations with jumps and stochastic differential delay equations with jumps (SDDEJs). Then, we prove the existence and uniqueness of adapted solutions and Lp solutions for such ABSDEJs under the non-Lipschitz conditions as well as a comparison theorem is obtained through constructing some iterative equations which are different from iterative equations in Peng and Yang (2009).

Stochastic simplified bardina turbulent model: existence of weak solution

In this paper, we consider the stochastic version of the 3D Bardina model arising from the turbulent flows of fluids. We obtain the existence of probabilistic weak solution for the model with the non-Lipschitz condition.

Existence and Stability Results for Second-Order Stochastic Equations Driven by Fractional Brownian Motion

Fractional Brownian motion has been widely used to model a number of phenomena in diverse fields of science and engineering. In this article, we investigate the existence, uniqueness and stability of mild solutions for a class of second-order nonautonomous neutral stochastic evolution equations with infinite delay driven by fractional Brownian motion (fBm) with Hurst parameter H ∈ (1/2, 1) in Hilbert spaces. More precisely, using semigroup theory and successive approximation approach, we establish a set of sufficient conditions for obtaining the required result under the assumption that coefficients satisfy non-Lipschitz condition with Lipschitz condition being considered as a special case. Further, the result is deduced to study the second-order autonomous neutral stochastic equations with fBm. The results generalize and improve some known results. Finally, as an application, stochastic wave equation with infinite delay driven by fractional Brownian motion is provided to illustrate the obtained theory.

Existence and uniqueness, attraction for stochastic age-structured population systems with diffusion and Poisson jump

In this paper, the problems about the existence and uniqueness, attraction for strong solution of stochastic age-structured population systems with diffusion and Poisson jump are considered. Under the non-Lipschitz condition with the Lipschitz condition being considered as a special case, the existence and uniqueness for such systems is firstly proved by using the Burkholder-Davis-Gundy inequality (B-D-G inequality) and Itô’s formula. And then by using a novel inequality technique, some sufficient conditions ensuring the existence for the domain of attraction are established. As another by-product, the exponential stability in mean square moment of strong solution for such systems can be also discussed.

Successive approximation of neutral stochastic functional differential equations with infinite delay and Poisson jumps

We establish results concerning the existence and uniqueness of solutions to neutral stochastic functional differential equations with infinite delay and Poisson jumps in the phase space C((-∞,0];R) under non-Lipschitz condition with Lipschitz condition being considered as a special case and a weakened linear growth condition on the coefficients by means of the successive approximation. Compared with the previous results, the results obtained in this paper is based on a other proof and our results can complement the earlier publications in the existing literatures.

Existence, uniqueness and stability of the solution to neutral stochastic functional differential equations with infinite delay under non-Lipschitz conditions

Choosing space as the phase space, the existence, uniqueness and stability of the solution to neutral stochastic functional differential equations with infinite delay (short for INSFDEs) are studied in this paper. Under non-Lipschitz condition, weakened linear growth condition and contractive condition, the existence-and-uniqueness theorem of the solution to INSFDEs by means of the Picard iteration, Doob’s martingale inequalities, Gronwall’s inequality and Bihari’s inequality is obtained. Furthermore, the continuous dependence of the solutions on the initial value to INSFDEs are derived. MSC:65C30, 60H10.

A note on the solutions of neutral SFDEs with infinite delay

The main aim of this paper is to discuss the existence and uniqueness of solution of the neutral stochastic functional differential infinite delay equations under a non-Lipschitz condition and a weakened linear growth condition. Furthermore, an estimate for the error between approximate solution and accurate solution is given. MSC:60H05, 60H10.