Implicit numerical methods for stiff stochastic differential equations and numerical simulations of stochastic models

In this thesis efficient implicit numerical methods are constructed for solving stochastic differential equations and numerical simulations are presented for stochastic models in environmental modelling and mathematical finance. Two types of stochastic differential equations are discussed in this thesis: Ito and Stratonovich. For Ito stochastic differential equations, the composite Euler method and implicit Taylor series methods are derived which have very good stability properties. For Stratonovich stochastic differential equations, all of the work is along the line of stochastic Runge-Kutta methods. The work in this direction includes two-stage diagonally implicit Runge-Kutta methods, two-stage composite Runge-Kutta methods, predictor-corrector methods of Runge-Kutta type and implicit Runge-Kutta methods based on splitting. The designed implicit methods in this thesis have very good stability properties and are suitable for solving stiff equations in both the deterministic and stochastic components. In addition, a stochastic model for solute transport in porous media and a financial market model are discussed. Numerical simulations of these stochastic models are presented.