In this paper,we study the concept of split monotone variational inclusion problem with multiple output sets.We propose a new relaxed inertial iterative method with self-adaptive step sizes for approximating the solut...In this paper,we study the concept of split monotone variational inclusion problem with multiple output sets.We propose a new relaxed inertial iterative method with self-adaptive step sizes for approximating the solution of the problem in the framework of Hilbert spaces.Our proposed algorithm does not require the co-coerciveness nor the Lipschitz continuity of the associated single-valued operators.Moreover,some parameters are relaxed to accommodate a larger range of values for the step sizes.Under some mild conditions on the control parameters and without prior knowledge of the operator norms,we obtain strong convergence result for the proposed method.Finally,we apply our result to study certain classes of optimization problems and we present several numerical experiments to demonstrate the implementability of the proposed method.Several of the existing results in the literature could be viewed as special cases of our result in this paper.展开更多
In this paper,we study the strong convergence of a jump-adapted implicit Milstein method for a class of jump-diffusion stochastic differential equations with non-globally Lipschitz drift coefficients.Compared with the...In this paper,we study the strong convergence of a jump-adapted implicit Milstein method for a class of jump-diffusion stochastic differential equations with non-globally Lipschitz drift coefficients.Compared with the regular methods,the jump-adapted methods can significantly reduce the complexity of higher order methods,which makes them easily implementable for scenario simulation.However,due to the fact that jump-adapted time discretization is path dependent and the stepsize is not uniform,this makes the numerical analysis of jump-adapted methods much more involved,especially in the non-globally Lipschitz setting.We provide a rigorous strong convergence analysis of the considered jump-adapted implicit Milstein method by developing some novel analysis techniques and optimal rate with order one is also successfully recovered.Numerical experiments are carried out to verify the theoretical findings.展开更多
This paper establishes a local limit theorem for solutions of backward stochastic differential equations with Mao＇s non-Lipschitz generator, which is similar to the limit theorem obtained by [3] under the Lipschitz a...This paper establishes a local limit theorem for solutions of backward stochastic differential equations with Mao＇s non-Lipschitz generator, which is similar to the limit theorem obtained by [3] under the Lipschitz assumption.展开更多
A class of stochastic differential equations with random jump magnitudes( SDEwRJMs) is investigated. Under nonLipschitz conditions,the convergence of semi-implicit Euler method for SDEwRJMs is studied. The main purpos...A class of stochastic differential equations with random jump magnitudes( SDEwRJMs) is investigated. Under nonLipschitz conditions,the convergence of semi-implicit Euler method for SDEwRJMs is studied. The main purpose is to prove that the semi-implicit Euler solutions converge to the true solutions in the mean-square sense. An example is given for illustration.展开更多
We prove that Euler＇s approximations for stochastic differential equations driven by infinite many Brownian motions and with non-Lipschitz coefficients converge almost surely. Moreover, the rate of convergence is obt...We prove that Euler＇s approximations for stochastic differential equations driven by infinite many Brownian motions and with non-Lipschitz coefficients converge almost surely. Moreover, the rate of convergence is obtained.展开更多
In this paper, we first study the existence and uniqueness of solutions to the stochastic differential equations driven by fractional Brownian motion with non-Lipschitz coefficients. Then we investigate the explosion ...In this paper, we first study the existence and uniqueness of solutions to the stochastic differential equations driven by fractional Brownian motion with non-Lipschitz coefficients. Then we investigate the explosion time in stochastic differential equations driven by fractional Browmian motion with respect to Hurst parameter more than half with small diffusion.展开更多
In this paper, we are interested in solving multidimensional backward stochastic differential equations(BSDEs) with a new kind of non-Lipschitz coefficients. We establish an existence and uniqueness result of the L^p(...In this paper, we are interested in solving multidimensional backward stochastic differential equations(BSDEs) with a new kind of non-Lipschitz coefficients. We establish an existence and uniqueness result of the L^p(p > 1) solutions, which includes some known results as its particular cases.展开更多
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,solutions of the following non-Lipschitz stochastic differential equations driven by G-Brownian motion：Xt=x＋∫^t0b（s,w,Xs）ds＋∫^t0h（s,ω,Xs）ds＋∫^t0σ（s,ω,Xs）dBs are constructed.It is shown th...In this paper,solutions of the following non-Lipschitz stochastic differential equations driven by G-Brownian motion：Xt=x＋∫^t0b（s,w,Xs）ds＋∫^t0h（s,ω,Xs）ds＋∫^t0σ（s,ω,Xs）dBs are constructed.It is shown that they have the cocycle property.Moreover,under some special non-Lipschitz conditions,they are bi-continuous with respect to t,x.展开更多
This article proves the existence and uniqueness of solution to two-parameter stochastic Volterra equation with non-Lipschitz coefficients and driven by Brownian sheet, where the main tool is Bihari＇s inequality in t...This article proves the existence and uniqueness of solution to two-parameter stochastic Volterra equation with non-Lipschitz coefficients and driven by Brownian sheet, where the main tool is Bihari＇s inequality in the plane. Moreover, we also discuss the time regularity property of the solution by Kolmogorov＇s continuity criterion.展开更多
For a stochastic differential equation with non-Lipschitz coefficients, we construct, by Euler scheme, a measurable flow of the solution, and we prove the solution is a Markov process.
A projected subgradient method for solving a class of set-valued mixed variational inequalities （SMVIs） is proposed when the mapping is not necessarily Lipschitz. Under some suitable conditions, it can be proven tha...A projected subgradient method for solving a class of set-valued mixed variational inequalities （SMVIs） is proposed when the mapping is not necessarily Lipschitz. Under some suitable conditions, it can be proven that the sequence generated by the method can strongly converge to the unique solution to the problem in the Hilbert spaces.展开更多
In this article, we first introduce g-expectation via the solution of backward stochastic differential equation（BSDE in short） with non-Lipschitz coefficient, and give the properties of g-expectation, then we establ...In this article, we first introduce g-expectation via the solution of backward stochastic differential equation（BSDE in short） with non-Lipschitz coefficient, and give the properties of g-expectation, then we establish a general converse comparison theorem for backward stochastic differential equation with non-Lipschitz coefficient.展开更多
The existence and uniqueness of solutions to the multivalued stochastic differential equations with non-Lipschitz coefficients are proved, and bicontinuous modifications of the solutions are obtained.
基金Supported by National Research Foundation(NRF)of South Africa Incentive Funding for Rated Researchers(Grant No.119903)。
文摘In this paper,we study the concept of split monotone variational inclusion problem with multiple output sets.We propose a new relaxed inertial iterative method with self-adaptive step sizes for approximating the solution of the problem in the framework of Hilbert spaces.Our proposed algorithm does not require the co-coerciveness nor the Lipschitz continuity of the associated single-valued operators.Moreover,some parameters are relaxed to accommodate a larger range of values for the step sizes.Under some mild conditions on the control parameters and without prior knowledge of the operator norms,we obtain strong convergence result for the proposed method.Finally,we apply our result to study certain classes of optimization problems and we present several numerical experiments to demonstrate the implementability of the proposed method.Several of the existing results in the literature could be viewed as special cases of our result in this paper.
基金supported by the National Natural Science Foundation of China(Grant Nos.11901565,12071261,11831010,11871068)by the Science Challenge Project(No.TZ2018001)by National Key R&D Plan of China(Grant No.2018YFA0703900).
文摘In this paper,we study the strong convergence of a jump-adapted implicit Milstein method for a class of jump-diffusion stochastic differential equations with non-globally Lipschitz drift coefficients.Compared with the regular methods,the jump-adapted methods can significantly reduce the complexity of higher order methods,which makes them easily implementable for scenario simulation.However,due to the fact that jump-adapted time discretization is path dependent and the stepsize is not uniform,this makes the numerical analysis of jump-adapted methods much more involved,especially in the non-globally Lipschitz setting.We provide a rigorous strong convergence analysis of the considered jump-adapted implicit Milstein method by developing some novel analysis techniques and optimal rate with order one is also successfully recovered.Numerical experiments are carried out to verify the theoretical findings.
基金the National Natural Science Foundation of China(No.10671205)China Postdoctoral Science Foundation(No.20060400158)973 Program of China(No.2007CB814901)
文摘This paper establishes a local limit theorem for solutions of backward stochastic differential equations with Mao＇s non-Lipschitz generator, which is similar to the limit theorem obtained by [3] under the Lipschitz assumption.
基金National Natural Science Foundations of China(Nos.11401261,11471071)Qing Lan Project of Jiangsu Province,China(No.2012)+2 种基金Natural Science Foundation of Higher Education Institutions of Jiangsu Province(No.13KJB110005)the Grant of Jiangsu Second Normal University(No.JSNU-ZY-02)the Jiangsu Government Overseas Study Scholarship,China
文摘A class of stochastic differential equations with random jump magnitudes( SDEwRJMs) is investigated. Under nonLipschitz conditions,the convergence of semi-implicit Euler method for SDEwRJMs is studied. The main purpose is to prove that the semi-implicit Euler solutions converge to the true solutions in the mean-square sense. An example is given for illustration.
基金Supported by National Natural Science Foundation of China (Grant Nos.10901065 and 11271013)Fundamental Research Funds for the Central Universities,Huazhong University of Science and Technology (Grant No.2012QN028)
文摘We prove that Euler＇s approximations for stochastic differential equations driven by infinite many Brownian motions and with non-Lipschitz coefficients converge almost surely. Moreover, the rate of convergence is obtained.
基金Supported in part by National Natural Science Foundation of China (Grant Nos. 10901065,60934009)
文摘In this paper, we first study the existence and uniqueness of solutions to the stochastic differential equations driven by fractional Brownian motion with non-Lipschitz coefficients. Then we investigate the explosion time in stochastic differential equations driven by fractional Browmian motion with respect to Hurst parameter more than half with small diffusion.
基金Supported by the Fundamental Research Funds for the Central Universities(No.2017XKQY98)
文摘In this paper, we are interested in solving multidimensional backward stochastic differential equations(BSDEs) with a new kind of non-Lipschitz coefficients. We establish an existence and uniqueness result of the L^p(p > 1) solutions, which includes some known results as its particular cases.
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 Science Foundation of China(No.11001051)
文摘In this paper,solutions of the following non-Lipschitz stochastic differential equations driven by G-Brownian motion：Xt=x＋∫^t0b（s,w,Xs）ds＋∫^t0h（s,ω,Xs）ds＋∫^t0σ（s,ω,Xs）dBs are constructed.It is shown that they have the cocycle property.Moreover,under some special non-Lipschitz conditions,they are bi-continuous with respect to t,x.
基金supported by NSF (10971076 and 11061032) of ChinaScience and Technology Research Projects of Hubei Provincial Department of Education (Q20132505)
文摘This article proves the existence and uniqueness of solution to two-parameter stochastic Volterra equation with non-Lipschitz coefficients and driven by Brownian sheet, where the main tool is Bihari＇s inequality in the plane. Moreover, we also discuss the time regularity property of the solution by Kolmogorov＇s continuity criterion.
文摘For a stochastic differential equation with non-Lipschitz coefficients, we construct, by Euler scheme, a measurable flow of the solution, and we prove the solution is a Markov process.
基金supported by the Key Program of National Natural Science Foundation of China(No.70831005)the National Natural Science Foundation of China(No.10671135)the Fundamental Research Funds for the Central Universities(No.2009SCU11096)
文摘A projected subgradient method for solving a class of set-valued mixed variational inequalities （SMVIs） is proposed when the mapping is not necessarily Lipschitz. Under some suitable conditions, it can be proven that the sequence generated by the method can strongly converge to the unique solution to the problem in the Hilbert spaces.
基金Foundation item： Supported by the＇Natured Science Foundation of the Edudation Department of Jiangsu Province（06KJD110092）
文摘In this article, we first introduce g-expectation via the solution of backward stochastic differential equation（BSDE in short） with non-Lipschitz coefficient, and give the properties of g-expectation, then we establish a general converse comparison theorem for backward stochastic differential equation with non-Lipschitz coefficient.
基金supported by the National Natural Science Foundation of China (No.10871215).
文摘The existence and uniqueness of solutions to the multivalued stochastic differential equations with non-Lipschitz coefficients are proved, and bicontinuous modifications of the solutions are obtained.
