Study On Numerical Methods For Solving Stochastic (Delay) Differential Equations

This thesis studies the construction of numerical methods of stochastic (delay)differential equations and analyzes the convergence, stability and precision of themethods proposed. There are six chapters in this thesis:Chapter1briefly introduces the background of stochastic (delay) differentialequations, the significance, history and status of the research on numerical algorithmsfor stochastic (delay) differential equations.Chapter2states some relative concepts in stochastic differential equations,including random process, stochastic integral, It formula, stochastic Taylorexpansion, etc.Chapter3summaries some common used numerical algorithms and theirconvergence, stability, and precision based on the stochastic Taylor expansion.Chapter4presents two classes of three-stage semi-implicit stochasticRunge-Kutta methods, YZP1and YZP2algorithms, with strong order1for theStratonovich stochastic differential equations from order conditions based on thecolored rooted trees theory. Theoretical analysis and numerical tests show that thesetwo new methods are superior both in stability and precision compared with theexisting algorithms.Chapter5gives a split-step forward Euler method (SSFE method) for nonlinearstochastic delay differential equation, proves that SSFE method is convergent in meansquare with orderγ=12, and obtains the sufficient condition of the mean squarestability of SSFE method. Numerical examples test the above theoretical results.