Strong Convergence Of Implicit One Step Method For Stochastic Delay Differential Equations

The thesis deals with the consistency and convergence of numerical methods of stochastic delay differential equations. As important mathematic models, stochastic delay differential equations are applied widely in many fields such as economics, biology and medicine. Because the explicit solutions of stochastic delay differential equations can hardly be obtained, investigating appropriate numerical methods and studying the properties of the numerical solutions are very important both in theory and in application.Chapter one introduces the major work of the thesis.Chapter two reviews the basic knowledge of stochastic calculus and Ito-Taylor expansion.Chapter three introduces the main work of the paper. Firstly, basic knowledge of the analytical solutions and numerical solutions of stochastic delay differential equations are presented. Secondly, consistency and convergence of theθ-method of stochastic delay differential equations are investigated. It is proved that under the condition theθ-method is consistent if the coefficients of the equations satisfy the global Lipschitz condition and the linear growth condition, the derivatives of drift coefficients are uniformly bounded and initial conditions satisfy H(o|¨)lder continuity; It is proved that theθ-method is convergent if the coefficients of the equations satisfy the global Lipschitz condition and the linear growth condition, and initial conditions satisfy H(o|¨)lder continuity, the order of convergence is 0.5. Lastly, convergence of implicit strong Taylor approximation is investigated.