| The research conducted for this dissertation addresses two different problems from computational science and mathematical modeling.; The first problem concerns the relaxation of the Efficient Market Hypothesis and the development of agent-based models that can help explain the consistent, but poorly understood, non-Gaussian statistical properties of real financial markets. A simple yet robust class of models is introduced in which the agents are driven both by rational considerations and also by less rational, but psychologically plausible, factors. It is shown that these models are capable of reproducing many of the important features of real markets. Furthermore, the results suggest that the two most significant deviations from Gaussian behavior in the price returns, namely 'fat tails' and volatility clustering, may have different causes allowing for future models that can more accurately reproduce them.; The second area of research is the analysis and development of efficient variable timestepping algorithms for the numerical solution of stochastic differential equations. The starting point is a previously introduced algorithm that uses a dual-error-control strategy. One local error estimate corresponds to the error in the drift and the other to the diffusion. It is proved that under certain modes of operation the algorithm is mean-square stable for a class of test problems with multiplicative noise. Additional error controls are then introduced to determine whether they result in improved performance on different test problems. The research demonstrates that algorithms based upon such error controls are feasible and can result in large efficiency and stability improvements over their fixed timestepping counterparts. |
