A class of stochastic differential equations(SDEs) driven by semimartingale with non-Lipschitz coefficients was studied.By using Gronwall inequality,the non-confluence of solutions is proved under the general conditions.
In this paper, we generalize Gronwall lemma to the case with time lags and use them to study delay controlled systems. For delay controlled systems associated with C_o-semigroup and analytic semigroup, we obtain the e...In this paper, we generalize Gronwall lemma to the case with time lags and use them to study delay controlled systems. For delay controlled systems associated with C_o-semigroup and analytic semigroup, we obtain the existences of mild solutions and optimals control. Lastly, an example is given to illustrate our abstract results.展开更多
In this paper,we present an efficient method of two-grid scheme for the approximation of two-dimensional nonlinear parabolic equations using an expanded mixed finite element method.We use two Newton iterations on the ...In this paper,we present an efficient method of two-grid scheme for the approximation of two-dimensional nonlinear parabolic equations using an expanded mixed finite element method.We use two Newton iterations on the fine grid in our methods.Firstly,we solve an original nonlinear problem on the coarse nonlinear grid,then we use Newton iterations on the fine grid twice.The two-grid idea is from Xu's work[SIAM J.Numer.Anal.,33(1996),pp.1759–1777]on standard finite method.We also obtain the error estimates for the algorithms of the two-grid method.It is shown that the algorithm achieve asymptotically optimal approximation rate with the two-grid methods as long as the mesh sizes satisfy h=O(H^((4k+1)/(k+1))).展开更多
Let M = {M<sub>z</sub>, z∈R<sub>+</sub><sup>2</sup>} be a continuous square integrable martingale and A = {A<sub>z</sub>, z∈ R<sub>+</sub><sup>2</...Let M = {M<sub>z</sub>, z∈R<sub>+</sub><sup>2</sup>} be a continuous square integrable martingale and A = {A<sub>z</sub>, z∈ R<sub>+</sub><sup>2</sup>} be a continuous adapted increasing process. Consider the following stochastic partial differential equations in the plane: dX<sub>z</sub>=α（z, X<sub>z</sub>）dM<sub>2</sub>+β（z,X<sub>z</sub>）dA<sub>z</sub>, z∈R<sub>+</sub><sup>2</sup>, X<sub>z</sub>=Z<sub>z</sub>, z∈R<sub>+</sub><sup>2</sup>, where R<sub>+</sub><sup>2</sup>=[0,+∞)×[0,+∞) and R<sub>+</sub><sup>2</sup> is its boundary, Z is a continuous stochastic process on R<sub>+</sub><sup>2</sup>. We establish a new theorem on the pathwise uniqueness of solutions for the equation under a weaker condition than the Lipschitz one. The result concerning the one-parameter analogue of the problem we consider here is immediate （see [1, Theorem 3.2]）. Unfortunately, the situation is much more complicated for two-parameter process and we believe that our result is the first one of its kind and is interesting in itself. We have proved the existence theorem for the equation in.展开更多
基金National Natural Science Foundation of China(No.71171003)Natural Science Foundation of Anhui Province of China(No.090416225)Natural Science Foundation of Universities of Anhui Province of China(No.KJ2010A037)
文摘A class of stochastic differential equations(SDEs) driven by semimartingale with non-Lipschitz coefficients was studied.By using Gronwall inequality,the non-confluence of solutions is proved under the general conditions.
基金supported by the National Natural Sciences Foundation of China (No.29362003) and ScienceCommittee of Guizhou province.
文摘In this paper, we generalize Gronwall lemma to the case with time lags and use them to study delay controlled systems. For delay controlled systems associated with C_o-semigroup and analytic semigroup, we obtain the existences of mild solutions and optimals control. Lastly, an example is given to illustrate our abstract results.
基金supported by Guangdong Province Universities and Colleges Pearl River Scholar Funded Scheme(2008)National Science Foundation of China 10971074+1 种基金the National Basic Research Program under the Grant 2005CB321703Hunan Provincial Innovation Foundation For Postgraduate CX2009B119。
文摘In this paper,we present an efficient method of two-grid scheme for the approximation of two-dimensional nonlinear parabolic equations using an expanded mixed finite element method.We use two Newton iterations on the fine grid in our methods.Firstly,we solve an original nonlinear problem on the coarse nonlinear grid,then we use Newton iterations on the fine grid twice.The two-grid idea is from Xu's work[SIAM J.Numer.Anal.,33(1996),pp.1759–1777]on standard finite method.We also obtain the error estimates for the algorithms of the two-grid method.It is shown that the algorithm achieve asymptotically optimal approximation rate with the two-grid methods as long as the mesh sizes satisfy h=O(H^((4k+1)/(k+1))).
基金Supported by the National Science Foundationthe Postdoctoral Science Foundation of China
文摘Let M = {M<sub>z</sub>, z∈R<sub>+</sub><sup>2</sup>} be a continuous square integrable martingale and A = {A<sub>z</sub>, z∈ R<sub>+</sub><sup>2</sup>} be a continuous adapted increasing process. Consider the following stochastic partial differential equations in the plane: dX<sub>z</sub>=α（z, X<sub>z</sub>）dM<sub>2</sub>+β（z,X<sub>z</sub>）dA<sub>z</sub>, z∈R<sub>+</sub><sup>2</sup>, X<sub>z</sub>=Z<sub>z</sub>, z∈R<sub>+</sub><sup>2</sup>, where R<sub>+</sub><sup>2</sup>=[0,+∞)×[0,+∞) and R<sub>+</sub><sup>2</sup> is its boundary, Z is a continuous stochastic process on R<sub>+</sub><sup>2</sup>. We establish a new theorem on the pathwise uniqueness of solutions for the equation under a weaker condition than the Lipschitz one. The result concerning the one-parameter analogue of the problem we consider here is immediate （see [1, Theorem 3.2]）. Unfortunately, the situation is much more complicated for two-parameter process and we believe that our result is the first one of its kind and is interesting in itself. We have proved the existence theorem for the equation in.
