Stability And Convergence Of Numerical Methods For Stochastic Pantogragh Differential Equations

As important mathematical models, stochastic differential equations aregenerally applied in many fields such as geography, chemistry, control science,finance, biology and neural network. There are theoretical significance and realapplication value to study the property of solution for stochastic differentialequations.Stability and convergence of numerical methods play important role instochastic differential equations. Results on mean-square and almost surelystability are a lot, especially on exponent stability, But there are so many equationswhich do not satisfy exponent stability, So studying general rate stability is ofimportance. As the analytical solution is generally unable to solute, or the explicitsolution is very complicated, so it is significant to solve the numerical solution ofequations with numerical method and researching the convergence of numericalmethod is very important.This paper studies non-linear stochastic pantograph differential equations, themain research contents of two parts.Chapter3studies the polynomial asymptotic properties of the exact solutionof non-linear stochastic pantograph differential equations. The sufficientconditions of mean-square polynomial stability and almost surely polynomialstability are given.The numerical solution of non-linear stochastic pantograph differentialequations is studied in chapter4. We use the semi-implicit Euler method to solvethe approximate solution of non-linear stochastic pantograph differential equations.Under the global-lipschitz condition, the consistence and convergence of themethod are proved. The order of consistence in average sense is1.5; the order inmean-square sense is1; the order of convergence in mean-square sense is0.5.