Stability Analysis Of Numerical Methods For Stochastic Differential Equations

Stochastic differential equations have been widely used in applications of economics, biology, physics, electronics, wireless communication, etc. Since the analytical solution can hardly be obtained for most stochastic differential equations, numerical analysis have aroused a lot of attention. In this Ph.D thesis, we study the moment and almost sure stability of numerical methods for stochastic differential equations.In Chapter 1, the background and developments of the problems, some notations, as-sumptions, preparations and main works are briefly introduced.In Chapter 2, for scalar linear SDE, five numerical methods including a forward derivative-free scheme (FDFS), the split-step backward Euler (SSBE) method, Heun scheme, Milstein method and a first-order Runge-Kutta method involving the Ito coefficient are considered. It is proved that these methods correctly preserve almost sure and small-moment exponential stability of the analytical solution for sufficiently small stepsizes. Then we study the stability of the FDFS, a backward derivative-free scheme (BDFS) and the SSBE method for nonlinear stochastic differential equations. Furthermore, the results are extended to nonlinear SDEs with multidimensional noise.In Chapter 3, we study the delay-dependent stability of numerical methods for nonlin-ear stochastic delay differential equations(SDDEs). It is proved that the Euler-Maruyama method can reproduce the delay-dependent exponential stability of the analytical solution to a class of SDDEs when the stepsize is appropriately small. Then we extend the results to stochastic differential equations with time varying delays.In Chapter 4, the delay-dependent exponential stability of the Backward Euler method has been discussed. For the problems in Chapter 3, it is proved that the Backward Euler method is mean square exponentially stable for any stepsize.In Chapter 5, the delay-dependent asymptotic and exponential stability of the stochas-tic theta method have been investigated. By a special treatment of the numerical scheme, we prove that:(1) whenθ∈[0,1/2) and the stepsize is appropriately small, the stochastic theta method maintains mean square asymptotic stability of the analytical solution for a class of SDDEs; (2)whenθ∈[1/2,1], the method is mean square asymptotically stable for every stepsize. Then by another approach, we further study the exponential stability of this method.In Chapter 6, under a delay-dependent stability condition, in the case of constrained mesh, i.e. the stepsize is a submultiple of the delay, it is proved that the Milstein method is mean-square and almost surely exponentially stable for appropriately small stepsize.In Chapter 7, two split-step forward methods, the drifting split-step Euler (DRSSE) and diffused split-step Euler (DISSE) methods are considered for nonlinear SDDEs. By the technique of semimartingale convergence theorem, the DRSSE and DISSE are proved to reproduce almost sure exponential stability of the exact solution under the linear growth condition. Furthermore, we study the delay-dependent stability of the split-step backward Euler method.In Chapter 8, we are concerned with mean square stability of the stochastic theta method for nonlinear neutral stochastic delay differential equations (NSDDEs). We con-clude that (1) whenθ∈[1/2,1], the stochastic theta method is mean square asymptotically stable for every stepsize; (2)whenθ∈[0,1/2), the stochastic theta method is mean square asymptotically stable for sufficiently small stepsize.Numerical examples are given to illustrate our theory.