Efficient Estimating Functions for Stochastic Differential Equations

PhD defense ved Nina Munkholt Jakobsen.

The overall topic of the thesis is approximate martingale estimating function-based estimation for solutions of stochastic differential equations, sampled at high frequency. Focus lies on the asymptotic properties of the estimators.

 The first part deals with diffusions observed over a fixed time interval. Rate optimal and efficient estimators are obtained for a one-dimensional diffusion parameter. Stable convergence in distribution is used to achieve a practically applicable Gaussian limit distribution for suitably normalised estimators. In a simulation example, the limit distributions of an efficient and an inefficient estimator are compared graphically.

 The second part concerns diffusions with finite-activity jumps, observed over an increasing interval with terminal sampling time going to infinity. Asymptotic distribution results are derived for consistent estimators of a general multidimensional parameter. Conditions for rate optimality and efficiency of estimators of drift-jump and diffusion parameters are given in special cases. These conditions extend the pre-existing conditions applicable to continuous diffusions, and impose much stronger requirements on the estimating functions in the presence of jumps. Certain implications of these conditions are discussed, as is a heuristic notion of how efficient estimating functions might be constructed, thus setting the stage for further research.

Academic advisor: Michael Sørensen,Professor, University of Copenhagen, Denmark

Assessment Committee:

Susanne Ditlevsen (chairman) Professor, University of Copenhagen, Denmark

Arnaud Gloter Professor, Université d’Évry Val d’Essonne, France

Hiroki Masuda Associate Professor, Kyushu University, Japan