Title:

Pvariation calculation and statistical inference for the discretely observed differential equation driven by fractional Brownian motion

The dissertation is centred at the topic of parameter estimation for fractional stochastic differential equations. Three projects are included, and they all contribute to the problem of parameter estimation motivated by rough path theory. The first project derives an algorithm for the calculation of the pvariation of a piecewise linear path. One thing to be noticed is that the definition of pvariation is different from the definition in It^o's sense where the mesh of the partition goes to zero. The algorithm first transforms the path into a weighted graph and then exploits the optimal substructure. The second one studies the method of the construction of approximate maximum likelihood estimators (MLEs) for discretely observed case by applying it to fractional Ornstein Uhlenbeck (OU) process. Parameter estimation for fractional processes is in the early development, especially for discretely observed cases. Most methods follow a strategy which approximates the exact likelihood function, and the method we study constructs the MLEs by building the exact likelihood function from an approximate model. The method is constructed under the frame work of rough path theory which makes it has a wide application even for the very rough paths. The final project relates to the second one. We study an inverse algorithm in the final project which allows us to calculate the driving force fXtg given observations fYtg. The difficulty of the inverse problem we considered lies in the mismatch between data and model. We need to calculate the piecewise linear driving force of the approximate model, given the observations generated from the original continuous partial differential equation. The algorithm is motivated by the property of the signature of a path, that is, the signature is invariant by adding tree like paths. Thanks to the algorithm, the method in project two can be implemented to stochastic differential equations without analytical solutions, and we conduct numerical experiments on several such differential equations.
