| Stochastic differential equations(SDEs)have come to play an important role in many branches of science and industry,such as biology,chemistry,mechanics,economics,finance,etc.For more than 30 years,lots of scholars have devoted to the study of numerical methods for solving SDEs and stochastic delay differential equations(SDDEs),and have achieved plenty of remarkable results.Since many SDEs and SDDEs in application are complex and nonlinear,numerical methods for solving nonlinear SDEs and SDDEs have attracted more and more attention in recent decade.In this paper,we study the convergence of numerical methods for nonlinear SDDEs under some weak non-global Lipschitz assumptions.In Chapter 1,we give the background of this topic,and review the existing results in literature.In Chapter 2,we introduce basic definitions and notations,theorems,and inequalities commonly used in theoretical derivation.Furthermore,we present the non-global Lipschitz assumption for the drift coefficient and the diffusion coefficient for the nonlinear SDDEs,and prove the boundedness of the analytic solution.In Chapter 3,we first give the fundamental theorem on convergence of one-step explic-it methods for solving autonomous nonlinear SDDEs under the given non-global Lipschitz assumption,that is,the relationship of convergence between the local truncated error and the global error in the sense of p-th moments.Second,we propose a balanced Euler scheme for nonlinear SDDEs and present the boundedness of numerical solutions.By using the fun-damental theorem,we study the convergence of numerical solutions of the balanced Euler scheme.Under the given non-global Lipschitz conditions,we prove that the balanced Euler scheme is of a half order convergence in the sense of p-th moment.In Chapter 4,we use the numerical examples to test the accuracy and convergence order of the balanced Euler scheme.The numerical results are consistent with the theoretical results. |
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