Statistics Seminar – Dr. Frederi Viens: “Parameter estimation for a partially observed Ornstein-Uhlenbeck process with long-memory noise”

Speaker Dr. Frederi Viens (Purdue University)
Date            Thursday, October 02, 2014
Time            4:00 PM – 5:00 PM
Location        269 Everitt

Abstract: Generically, an Ornstein-Uhlenbeck (OU) process is the Gaussian solution $X$ of a stochastic differential equation of the form $dX = -a X dt + dW$, where $a$ is a positive drift parameter (the negative sign turns the term $-a X dt$ into a mean-reversion term) and $W$ is a Gaussian noise term, typically with stationary increments which need not be independent. We consider a system of two such equations, where one equation is autonomous and is driven by a long-memory noise (specifically a fractional Gaussian noise), while the other equation’s noise term is the solution of the first equation. The question is to estimate the two drift parameters based on a single trajectory of the second process. A popular method for this type of task is to use a joint least-squares estimator drawing on continuous-time observation, which happens to coincide with a maximum likelihood estimator in and only in the case of white noise. We show that this approach does work, by proving strong consistency and asymptotic normality for increasing horizon asymptotics, by appealing to some Malliavin calculus computations. Two problems with this approach are that in practice observations are typically discrete, and that the use of Malliavin calculus is non-trivial. We address the first of these two problems by proposing various ways of discretizing the continuous-time estimators, which can be interpreted as generalized methods of moments. Strong consistency and asymptotic normality, for in-fill and increasing horizon asymptotics, are proved by showing how these discrete estimators are related to the continous-time estimator. We study the question of how the partial observation problem affects the speed of convergence and the restrictions on the spacing of observations. For the second issue, that of avoiding the use of the Malliavin calculus, work is currently in progress. This work is joint with Prof. Khalifa Es-sebaiy from the National School of Applied Sciences in Marrakech, Morocco. A third problem with our work and indeed with a large portion of the research to date on statistic of stochastic processes with long memory, is that the issue of estimating the memory length is conveniently swept under the rug. Time allowing, we will briefly mention two techniques for studying this problem: one, jointly with Prof. Alexandra Chronopoulou, which is an application of calibration using high-frequency data in quantitative finance, and another, with Prof. Bo Li, which applies the power of Bayesian estimation in paleoclimatology where calibration cannot rely on high-frequency data.