Numerical Methods For Several Kinds Of Nonlinear Stochastic Differential Equations

Random phenomena are ubiquitous in nature,and it is inappropriate for people to describe such phenomena simply by deterministic differential equations.Stochastic differential equations can be well adapted to various random problems and have been widely used in many fields such as genetics,finance,chemical engineering and aerospace control etc.However,the exact solutions of most stochastic differential equations do not have explicit expressions,so it is necessary to construct numerical methods to solve them numerically.In this thesis,the convergence,stability and positivity of numerical methods for several kinds of nonlinear stochastic differential equations are discussed.This dissertation mainly studies the following contents:For highly nonlinear stochastic differential equations,two explicit two-step stochastic methods,namely projected two-step Euler method and projected two-step Milstein method are constructed.Under the global monotone condition,the mean square convergence of these methods is proved based on the stability and consistency properties of these methods.Besides,the mean square convergence orders of the projected two-step Euler method and the projected two-step Milstein method are shown to be 1/2 and 1 respectively.In particular,the global monotone condition allows the super-linear growth of the drift coefficient and the diffusion coefficient,so the mean-square convergence conclusions are suitable for stochastic differential equations with nonlinear drift and diffusion coefficients.Two kinds of explicit projected Euler type methods for solving stochastic differential equations with Markovian switching are constructed.Under the monotone condition and the polynomial growth condition,the convergence of these numerical methods is analyzed based on their local properties.In addition,these two schemes are applied to highly nonlinear equations(including stochastic ordinary differential equations and stochastic differential equations with Markovian switching)with small noise,and based on their stability and local truncation errors,the mean square convergence and convergence rates of these two schemes are obtained.The asymptotic boundedness,stability and strong convergence of the split step theta method for highly nonlinear neutral stochastic delay integro differential equations are studied.Under the generalized coercive condition,it is proved that when θ ∈[1/2,1].the split step theta numerical approximation strongly converge to the exact solution.In addition,if θ ∈(1/2,1],it is proved that the method can unconditionally maintain the mean square asymptotic boundedness and mean square exponential stability of the exact solution,and the mean-square asymptotic boundary and exponential decay rate of the exact solution can also be maintained when the step size is small enough.Based on the logarithmic transformation,an explicit positiveness preserving numer-ical method is constructed for a class of stochastic ordinary differential equations with positive solutions,and the almost surely convergence,Lq convergence and the corre-sponding convergence rates are obtained.The coefficients of the new process obtained by logarithmic transformation may grow exponentially,so the strong convergence of ex-plicit truncated Euler method for solving stochastic ordinary differential equations with exponential growth coefficients is also proved.