In this paper we discuss two-stage Miistein methods for solving Ito stochastic differential equations （SDEs）. Six fully explicit methods （TSM 1 -- TSM 6） are given in this paper. Their order of strong convergence ...In this paper we discuss two-stage Miistein methods for solving Ito stochastic differential equations （SDEs）. Six fully explicit methods （TSM 1 -- TSM 6） are given in this paper. Their order of strong convergence is proved. The stability properties and numerical results show the effectiveness of these methods in the pathwise approximation of Ito SDEs.展开更多
In this paper, we construct a composite Milstein method for nonlinear stochastic differential delay equations. Then we analyze the mean square stability for this method and obtain the step size condition under which t...In this paper, we construct a composite Milstein method for nonlinear stochastic differential delay equations. Then we analyze the mean square stability for this method and obtain the step size condition under which the composite Milstein method is mean square stable. Moreover, we get the step size condition under which the composite Milstein method is global mean square stable. A nonlinear test stochastic differential delay equation is given for numerical tests. The results of numerical tests verify the theoretical results proposed.展开更多
In this paper,a numerical method for solving nonlinear stochastic delay differential equations is proposed:two-step Milstein method.The mean square consistent and mean square convergence of the numerical method are st...In this paper,a numerical method for solving nonlinear stochastic delay differential equations is proposed:two-step Milstein method.The mean square consistent and mean square convergence of the numerical method are studied.Through the relevant derivation,the conditions that the coefficients need to be satisfied when the numerical method is mean-square consistent and mean-square convergent are obtained,and it is proved that the mean-square convergence order of the numerical method is 1.Finally,the theoretical results are verified by numerical experiments.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
文摘In this paper we discuss two-stage Miistein methods for solving Ito stochastic differential equations （SDEs）. Six fully explicit methods （TSM 1 -- TSM 6） are given in this paper. Their order of strong convergence is proved. The stability properties and numerical results show the effectiveness of these methods in the pathwise approximation of Ito SDEs.
基金Supported by National Natural Science Foundation of China(No.61272024)Anhui Provincial Natural Science Foundation(No.11040606M06)
文摘In this paper, we construct a composite Milstein method for nonlinear stochastic differential delay equations. Then we analyze the mean square stability for this method and obtain the step size condition under which the composite Milstein method is mean square stable. Moreover, we get the step size condition under which the composite Milstein method is global mean square stable. A nonlinear test stochastic differential delay equation is given for numerical tests. The results of numerical tests verify the theoretical results proposed.
基金supported by the National Natural Science Foundation of China(Grant No.12471391).
文摘In this paper,a numerical method for solving nonlinear stochastic delay differential equations is proposed:two-step Milstein method.The mean square consistent and mean square convergence of the numerical method are studied.Through the relevant derivation,the conditions that the coefficients need to be satisfied when the numerical method is mean-square consistent and mean-square convergent are obtained,and it is proved that the mean-square convergence order of the numerical method is 1.Finally,the theoretical results are verified by numerical experiments.
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(11961029,11701237)the National Social Science Foundation of China(21CTJ018)+1 种基金the Scientific Research Project of Education Department of Hubei Province(D20212202)the Natural Science Foundation of Jiangxi Province(20202BABL201007)。
