Linear Multistep Methods For Solving Nonlinear Stochastic Differential Equations

Stochastic differential equations(SDEs)have a variety of applications in many disciplines such as biology,physics and finance.Since most of SDEs that describe the behaviors of com-plex systems are nonlinear,it is not easy to obtain closed-form analytical solutions for these equations.Developing numerical methods for these equations are needed.Under certain nonlinear assumptions,we study the strong convergence of a specific stochas-tic linear multi-step methods(LMMs)for nonlinear SDEs.The main contributions of the dis-sertation include the following two parts:For the SDEs with the drift coefficients satisfying one-sided Lipschitz condition and the diffusion coefficients satisfying the global Lipschitz condition,we propose a type of stochas-tic LMMs and derive the corresponding consistency conditions.Using the consistency condi-tions,we prove a version of the fundamental mean-square convergence theorem under certain constraints on the step-size and the parameters of the method,and we also show that the con-vergence order is the same as the consistency order.Following the convergence theorem,we further prove that the numerical solutions from a two-step LMM is moment-bounded and the or-der of strong convergence of the method is 1/2.In computational experiment,we demonstrate that the stochastic Adams-Moulton2 scheme is the two-step LMM that satisfies the assumptions we made in the dissertation,and the order of strong convergence is indeed 1/2.For the Cox-Ingersoll-Ross(CIR)model with the coefficients satisfying 2????~2,we first perform the Lamperti transformation on the original equation to make the drift and diffu-sion terms satisfy the local and global Lipschitz conditions,respectively.We then solve the transformed CIR model using the proposed stochastic two-step LMM with certain assump-tions on coefficients and initial values,in order to preserve the positivity of numerical so-lution.We also prove that the two-step method can produce numerical solutions,which are moment-bounded and convergent with order one in mean-square sense.Numerical results for the stochastic Adams-Moulton2 scheme agree well with our theoretical results.In addition,the comparison of four different numerical methods shows that the proposed stochastic Adams-Moulton2 is more accurate than the Adams-Moulton2 Milstein scheme,implicit EM scheme,and double implicit Milstein scheme.