Inference and parameter estimation for diffusion processes

Diffusion processes provide a natural way of modelling a variety of physical and economic phenomena. It is often the case that one is unable to observe a diffusion process directly, and must instead rely on noisy observations that are discretely spaced in time. Given these discrete, noisy observations, one is faced with the task of inferring properties of the underlying diffusion process. For example, one might be interested in inferring the current state of the process given observations up to the present time (this is known as the filtering problem). Alternatively, one might wish to infer parameters governing the time evolution the diffusion process. In general, one cannot apply Bayes’ theorem directly, since the transition density of a general nonlinear diffusion is not computationally tractable. In this thesis, we investigate a novel method of simplifying the problem. The stochastic differential equation that describes the diffusion process is replaced with a simpler ordinary differential equation, which has a random driving noise that approximates Brownian motion. We show how one can exploit this approximation to improve on standard methods for inferring properties of nonlinear diffusion processes