PARAMETER ESTIMATION IN STOCHASTIC DIFFERENTIAL EQUATIONS

By ABIGAIL WILSON

CONTENTS PARAMETRIC STOCHASTIC DI?ERENTIAL EQUATIONS PART I CONTINUOUS SAMPLING RATES OF WEAK CONVERGENCE OF ESTIMATORS IN HOMOGENEOUS DI?USIONS INTRODUCTION BERRY-ESSEEN BOUNDS FOR ESTIMATORS IN THE ORNSTEIN- UHLENBECK PROCESS RATES OF CONVERGENCE IN THE BERNSTEIN-VON MISES THEOREM FOR ERGODIC DI?USIONS RATES OF CONVERGENCE OF THE POSTERIOR DISTRIBUTIONS IN ERGODIC DI?USIONS BERRY-ESSEEN BOUND FOR THE BAYES ESTIMATOR EXAMPLE: HYPERBOLIC DI?USION MODEL LARGE DEVIATIONS OF ESTIMATORS IN HOMOGENEOUS DI?USIONS MODEL, ASSUMPTIONS AND PRELIMINARIES LARGE DEVIATIONS FOR THE MAXIMUM LIKELIHOOD ESTIMATOR LARGE DEVIATIONS FOR BAYES ESTIMATORS EXAMPLES LOCAL ASYMPTOTIC MIXED NORMALITY FOR NONHOMOGENEOUS DI?USIONS ASYMPTOTICS OF THE MAXIMUM LIKELIHOOD ESTIMATOR THE BERNSTEIN-VON MISES TYPE THEOREM AND ASYMPTOTICS OF BAYES ESTIMATORS ASYMPTOTICS OF MAXIMUM PROBABILITY ESTIMATOR BAYES AND SEQUENTIAL ESTIMATION IN STOCHASTIC PDES LONG TIME ASYMPTOTICS BERNSTEIN-VON MISES THEOREM ASYMPTOTICS OF BAYES ESTIMATORS SEQUENTIAL ESTIMATION SEQUENTIAL MAXIMUM LIKELIHOOD ESTIMATION EXAMPLE SPECTRAL ASYMPTOTICS MODEL AND PRELIMINARIES THE BERNSTEIN-VON MISES THEOREM BAYES ESTIMATION EXAMPLE: STOCHASTIC HEAT EQUATION MAXIMUM LIKELIHOOD ESTIMATION IN FRACTIONAL DI?USIONS FRACTIONAL STOCHASTIC CALCULUS MAXIMUM LIKELIHOOD ESTIMATION IN DIRECTLY OBSERVED FRACTIONAL DI?USIONS MAXIMUM LIKELIHOOD ESTIMATION IN PARTIALLY OBSERVED FRACTIONAL DI?USIONS PART II DISCRETE SAMPLING APPROXIMATE MAXIMUM LIKELIHOOD ESTIMATION IN NONHOMOGENEOUS DI?USIONS MODEL, ASSUMPTIONS AND DE?NITIONS ACCURACY OF APPROXIMATIONS OF THE ITˆO AND FS INTEGRALS ACCURACY OF APPROXIMATIONS OF THE LOG-LIKELIHOOD FUNCTION ACCURACY OF APPROXIMATIONS OF THE MAXIMUM LIKELIHOOD ESTIMATE EXAMPLE: CHAN-KARLOYI-LONGSTA?-SANDERS MODEL SUMMARY OF TRUNCATED DISTRIBUTIONS RATES OF WEAK CONVERGENCE OF ESTIMATORS IN THE ORNSTEIN-UHLENBECK PROCESS NOTATIONS AND PRELIMINARIES BERRY-ESSEEN TYPE BOUNDS FOR AMLE1 BERRY-ESSEEN TYPE BOUNDS FOR AMLE2 BERRY-ESSEEN TYPE BOUNDS FOR APPROXIMATE MINIMUM CONTRAST ESTIMATORS BERRY-ESSEEN BOUNDS FOR APPROXIMATE BAYES ESTIMATORS LOCAL ASYMPTOTIC NORMALITY FOR DISCRETELY OBSERVED HOMOGENEOUS DI?USIONS WEAK CONVERGENCE OF THE APPROXIMATE LIKELIHOOD RATIO RANDOM FIELDS ASYMPTOTICS OF APPROXIMATE ESTIMATORS AND BERNSTEIN-VON MISES TYPE THEOREMS EXAMPLE: LOGISTIC DI?USION ESTIMATING FUNCTION FOR DISCRETELY OBSERVED HOMOGENEOUS DI?USIONS 10 1 INTRODUCTION 10 2 RATE OF CONSISTENCY 10 3 BERRY-ESSEEN BOUND 10 4 EXAMPLES