non-Lipschitz Coefficients Research Articles

Homeomorphic flows for multi-dimensional SDEs with non-Lipschitz coefficients

In this paper, we study the homeomorphic property of solutions of multi-dimensional stochastic differential equations with non-Lipschitz coefficients.

On the Burgers Equation with a Stochastic Stepping-Stone Noisy Term

We consider the one-dimensional Burgers equation perturbed by a white noise term with Dirichlet boundary conditions and a non-Lipschitz coefficient. We obtain existence of a weak solution proving tightness for a sequence of polygonal approximations for the equation and solving a martingale problem for the weak limit.

Well-Posedness in Sobolev Spaces for Second-Order Strictly Hyperbolic Equations with Nondifferentiable Oscillating Coefficients

The goal of this paper is to study well-posedness to strictly hyperbolic Cauchyproblems with non-Lipschitz coefficients with low regularity with respect to timeand smooth dependence with respect to space variables. The non-Lipschitz conditionis described by the behaviour of the time-derivative of coefficients. This leads to a classification of oscillations, which has a strong influence on the loss of derivatives. To study well-posednesswe propose a refined regularizing technique. Two steps of diagonalizationprocedure basing on suitable zones of the phase spaceand corresponding nonstandard symbol classes allow to applya transformation corresponding to the effect of loss of derivatives.Finally, the application of sharp Garding's inequality allows to derive a suitable energy estimate. From this estimatewe conclude a result about C∞well-posedness of the Cauchy problem.

A class of stochastic differential equations with non-Lipschitzian coefficients: pathwise uniqueness and no explosion

A new result for the pathwise uniqueness of solutions of stochastic differential equations with non-Lipschitzian coefficients is established. Furthermore, we prove that the solution has no explosion under the growth ξlog ξ. To cite this article: S. Fang, T. Zhang, C. R. Acad. Sci. Paris, Ser. I 337 (2003).

Stochastic flows for SDEs with non-Lipschitz coefficients

We prove the bicontinuity and homeomorphic property of solutions of stochastic differential equations driven by infinite many Brownian motions and with non-Lipschitz coefficients.

On a type of stochastic differential equations driven by countably many Brownian motions

Consider the one-dimensional SDE X t=x+∑ i=1 ∞ ∫ 0 t σ i(X s) dW s i+ ∫ 0 t b(X s) ds , where W i is an infinite sequence of independent standard Brownian motions, i=1,2,3,… . We prove two theorems on the existence and uniqueness of solutions with non-Lipschitz coefficients, and give a non-contact property and a strong comparison theorem for solutions.

Construction of Parametrix to Strictly Hyperbolic Cauchy Problems with Fast Oscillations in NonLipschitz Coefficients†

The goal of this article is to construct parametrix to strictly hyperbolic Cauchy problems with nonLipschitz coefficients depending on space and time. The nonLipschitz condition is described by the behavior of the time-derivative of coefficients. This leads to a classification of oscillations, where fast oscillations represent the critical case. To study this critical case we propose a refined perfect diagonalization procedure basing on suitable zones of the phase space and corresponding nonstandard symbol classes. After this diagonalization procedure we construct the parametrix in several steps. Here the perfect diagonalization helps to understand, that only a finite loss of derivatives appears for solutions valued in Sobolev spaces. We point out where this loss comes from. From construction of parametrix we conclude a result about C ∞-well posedness of the Cauchy problem. †Dedicated to the memory of Professor Sigeru Mizohata.

On a stochastic Burgers equation with Dirichlet boundary conditions

We consider the one‐dimensional Burgers equation perturbed by a white noise term with Dirichlet boundary conditions and a non‐Lipschitz coefficient. We obtain existence of a weak solution proving tightness for a sequence of polygonal approximations for the equation and solving a martingale problem for the weak limit.

On Convergence of Population Processes in Random Environments to the Stochastic Heat Equation with Colored Noise

We consider the stochastic heat equation with a multiplicative colored noise term on the real space for dimensions greater or equal to 1. First, we prove convergence of a branching particle system in a random environment to this stochastic heat equation with linear noise coefficients. For this stochastic partial differential equation with more general non-Lipschitz noise coefficients we show convergence of associated lattice systems, which are infinite dimensional stochastic differential equations with correlated noise terms, provided that uniqueness of the limit is known. In the course of the proof, we establish existence and uniqueness of solutions to the lattice systems, as well as a new existence result for solutions to the stochastic heat equation. The latter are shown to be jointly continuous in time and space under some mild additional assumptions.

On solutions of backward stochastic differential equations with jumps and with non-Lipschitzian coefficients in Hilbert spaces and stochastic control

Existence and uniqueness is established for solutions to backward stochastic differential equations with jumps and non-Lipschitzian coefficients in Hilbert space. The results are used to solve some special types of optimal stochastic control problems with respect to certain BSDEs with jumps in Hilbert space. The existence of some non-Lipschitzian optimal controls is also established.

On the stochastic integral equations with non-lipschitz coefficients

Consider the stochastic integral equation (S.I.E.) where f satisfies some non-Lipschitz condition and H,Z are F t -semimartingales, continuous or discontinuous, on some probability space (Ω,F,{F t } t∈R + ,P). We prove that if f satisfies Condition H 1 or H 2 (defined in Sec. 0), then both the existence and the uniqueness of the solutions of 1 hold.

Approximations to mild solutions of stochastic semilinear equations with non-Lipschitz coefficients

In the present paper, using a Picard Type method of approximation, we investigate the global existence of mild solutions for a class of Ito Type stochastic differential equations whose coefficients satisfy conditions more general than the Lipschitz and linear growth ones.

Gevrey-well-posedness for weakly hyperbolic operators with non-regular coefficients

We prove that the Cauchy problem for a class of weakly hyperbolic equations with non-Lipschitz coefficients is well-posed in Gevrey spaces.

Pathwise uniqueness for a SDE with non-Lipschitz coefficients

We consider the ordinary stochastic differential equation dX=−cX dt+ 2(1−|X| 2) dB on the closed unit ball E in R n . While it is easy to prove existence and distribution uniqueness for solutions of this SDE for each c⩾0, pathwise uniqueness can be proved by standard methods only in dimension n=1 and in dimensions n⩾2 if c=0 or if c⩾2 and the initial condition is in the interior of E. We sharpen these results by proving pathwise uniqueness for c⩾1. More precisely, we show that for X 1, X 2 solutions relative to the same Brownian motion, the function t↦|X 1(t)−X 2(t)| 2+| 1−|X 1(t)| 2 − 1−|X 2(t)| 2 | 2 is almost surely nonincreasing. Whether or not pathwise uniqueness holds in dimensions n⩾2 for 0< c<1 is still open.

On the existence and uniqueness of solutions to stochastic differential equations of mixed Brownian and Poissonian sheet type

Let X z=X 0+ ∫ R z f(ξ) dM ξ+ ∫ R z u(ξ) dA ξ+ ∫ R z α(ξ)Λ( dξ)+ ∫ R z β(ξ)N( dξ) be a two-parameter semimartingale, where M is a continuous martingale, Λ is the character of the Poisson point measure Y, N=Y−Λ , we prove that f( X z ) is expressible as such a sum once again via the partial differentiation formula, where f is a twice continuously differentiable function. Then, we prove a new theorem on the existence and uniqueness of solutions to the mixed Brownian and Poissonian sheet type stochastic differential equations with non-Lipschitz coefficients by applying the partial differentiation formula.

On solutions of backward stochastic differential equations with jumps, with unbounded stopping times as terminal and with non-lipschitz coefficients, and probabilistic interpretation of quasi-linear elliptic type integro-differential equations

The existence and uniqueness of solutions to backward stochastic differential equations with jumps and with unbounded stopping time as terminal under the non-Lipschitz condition are obtained. The convergence of solutions and the continuous dependence of solutions on parameters are also derived. Then the probabilistic interpretation of solutions to some kinds of quasi-linear elliptic type integro-differential equations is obtained.

On Quasi-linear parabolic SPDEs with non-Lipschitz coefficients

On Quasi-linear parabolic SPDEs with non-Lipschitz coefficients

On the pathwise uniqueness of solutions of stochastic differential equations

We give an example of a stochastic differential equation with non-Lipschitz coefficients for which the pathwise uniqueness of solutions holds

Local Uniqueness in the Cauchy Problem for Second Order Elliptic Equations with Non-Lipschitzian Coefficients

We show the local uniqueness of the Cauchy problem for the second order elliptic operators whose coefficients of the principal part are real-valued and continuous with some modulus of continuity. These coefficients are not necessarily lipschitz continuous. The proof is given by drawing the Carleman estimates with a weight attached to the modulus of continuity.

Adapted solutions of backward stochastic differential equations with non-Lipschitz coefficients

In this paper we shall establish a new theorem on the existence and uniqueness of the adapted solution to a backward stochastic differential equation under a weaker condition than the Lipschitz one.