Stability And Convergence Of Several Classes Of Stochastic Numerical Methods

Stochastic differential equations(SDES) arise widely in many fields such as economics,biology,physics,automatic control etc.For a long time,many models that have been developed to describe physical phenomena have ignored stochastic effects because of the difficulty in solution both in terms of the lack of suitable numerical methods and also the non-availability of sufficiently powerful computers.Recently,there have been some results in developing numerical methods for the numerical solution of stochastic differential equations,and this has meant that more realistic models are capable of being solved.In this paper,firstly,I introduce the background of stochastic differential equations and give the important properties of analytical solutions,in which the theory of uniqueness and existence of the solution to stochastic differential equations is given and the moment qualities of the solution is discussed.As for the analytical solution,in general,nonlinear stochastic differential equations do not have explicit solutions.In practice,we can use approximate solutions.However,it is possible to find the explicit solutions to linear equations and the unique solution is presented. It is because of the complexity of the stochastic systems that the analytical solutions are rare for stochastic differential equations.Thus,the construction of numerical methods is very important.In the case of numerical methods for stochastic differential equations,the work is much less advanced than that in the case of ordinary differential equations.For the specific properties of stochastic systems,it is not possible to merely translate a deterministic numerical methods to a stochastic differential equation,instead,a very detailed analysis of order,stability,and error behaviour is needed in order to constuct suitably appropriate method.The simplest method is the explicit Euler method.Given the two paticular cases of stochastic differential equations,namely,multiplicative noise and additive noise,when both of the drift coefficient and diffusion coefficient satisfy the linear growth condition and Lipschitz condition,it is shown that the strong convergence orders of Euler method are 0.5 and 1.0 ,respectively.The stochastic Taylor expansions are obtained by generalizing the deterministic Taylor expansions and the Ito formula.They can be used directly in designing numerical schemes for stochastic differential equations.For example,the three Milstein methods in this paper are obtained from the truncated expansion of the stochastic Taylor expansions.I study a linear test equation with a multiplicative noise term and another with an additive noise term,and consider the A-stability,mean-square stability and T-stability of Milstein methods.I also plot mean-square stability regions for the case that the test equation has real parameters,and prove the T-stability is equivalent to the asymptotic stability property of a numerical methods when applied to the linear test equation.Finally,with respect to the two kinds of weak numerical methods,that is,the weak Euler method and the weak Milstein method,the convergence theory is given,and I study the stabilities of Milstein method such as M-stability,mean-square stability and T-stability,in which the definitions of mean-square stability and T-stability are the same as the ones in strong solutions,while the M-stability is discussed based on a newtype test equation.