NSF DMS-9704732 Computationally Tractable Estimation Methods for Markov Processes Peter W. Glynn Stanford University Markov process theory provides a rich analytic and probablistic structure which is intrinsically natural from a modelling perspective. Much of the literature on inference for continuous- time Markov processes assumes that the process has been observed continuously over some time interval. However, in practice, many of the data sets available involve observing only discrete ``snapshots'' of the process. Existing theory on parameter estimation in this setting often results in computationally prohibitive methodologies. This research addresses the issue of computationally tractable parameter estimation for discretely observed continuous-time Markov processes. Many of the results utilize Monte Carlo simulation to achieve tractability. The second fundamental question examined in this research is that of error propagation through a stochastic system. When the output performance measure of the stochastic system cannot be evaluated in closed form the output measure is simulated from the modelled system. To address this situation, the investigators develop estimators and their sampling properties for functionals of the stochastic system. Statistical models have always proven a powerful tool for purposes of modelling, understanding and predicting complex systems. This research advances the frontier of statistical models for dependent observations. For example, the statistical methods of this research are applicable to the diverse areas of modelling levels of pollutants and contaminants in air, soil and water which evolve over time and/or space; forecasting changes in the stock market; and predicting or assessing demand for the development of an optimal communications network. The use of statistical models for purposes of modelling complex systems has been limited in the past due to the state of computing power; a state which has certainl y improved in recent years. In this research, the investigators develop a framework for practical implementation of advanced statistical methodologies which capitalizes fully on the high performance computing available today. The research addresses the issue of modelling under partially observed information. For example, the electrical or computer engineer may use such models to assess network status when only partial information is available on the state of the system; in predicting air quality for a given region, observations of pollutant levels are made at sites irregularly located over the region and often at irregular points in time. The theoretical constructs necessary to implement the models in the more common scenario when only partial information is available are presented. Additionally, this research involves error assessment of the predictions or output performance measure of an estimated complex system again capitalizing on the availability of high powered computing.
