The seminar organised by the Department of Statistics & OR entitled 'Inference of sparsely observed diffusion processes using stable proposals' will be held on Friday 17 May at 12:00 noon in Room 602, Maths7 Physics Building (MP6020.
The speaker is Dr David Suda
Abstract
We study the problem of statistical inference for sparsely observed diffusion processes using stable proposals. Firstly though, we need to introduce maximum likelihood estimation based approaches, which have been a popular way for tackling the estimation problem for diffusion processes. These, however, can be a bit problematic since the transition density is not always known. Consequently, the true diffusion bridge of such processes cannot be obtained in closed form either.
One way of approaching this problem is via approximating diffusions. Originally, this was done by substituting the unknown transition density with, say, an Euler approximating density (other approximating densities have also been proposed). However the importance sampling approach for approximating such densities, using diffusion bridges as sampling proposals, has picked up some traction since the seminal paper by Durham and Gallant in 2002. We shall also look at the diffusion bridge proposed by Delyon and Hu in 2006, and a special case of the general diffusion bridge proposed in a paper by Schauer, van der Muelen and van Zanten, as recent as 2017. The latter is based on the Ornstein-Uhlenbeck (OU) process, and we shall henceforth refer to it as the OU proposal. There are other proposals, which we shall not delve into in detail. While most of these methods perform well when diffusions processes are frequently observed, so far there has not been any literature which looks at how robust these methods can be when observations are sparse.
We typically simulate diffusion bridges in discrete time. However, these are a discretisation of a continuous time construct and we must therefore be careful that the probability measure of the proposal bridge is absolutely continuous (via the Girsanov theorem) with respect to the probability measure of the true diffusion bridge in order to make our approach valid as we let our discretisation term tend to zero. There are theoretical results to ensure this is the case, and these shall also be mentioned in this presentation and play an important role in devising the main assumptions for our theoretical results. However, once we have taken care of the preliminary technical details, the main aim of this study is that of looking at some theoretical results concerning stability of these proposals as the distance between observations increases. If such proposals are stable, then these can also be implemented within the context of sparsely observed diffusions. We prove results which show that the proposal by Durham and Gallant is generally not stable, but under certain conditions, the proposal by Delyon and Hu and the OU proposal stable.
This talk is based on the frequentist approach for estimating diffusions. However, we shall also give an undetailed overview of a Bayesian approach to estimating diffusions, which is a data augmentation algorithm that includes Metropolis-Hastings type steps. Since we propose paths from a proposal bridge, and then accept or reject them via the resulting probability, stability when dealing with sparse observations is also desirable as it increases the acceptance probability. Finally, since empirical evidence shows that stability may exist beyond the realm of the results which we prove, we suggest possibilities where further research can be made in this area.