31 October 2023: Yuga Iguchi (supervisors: Alexandros Beskos and Samuel Livingstone)

Parameter Estimation with Increased Precision for Elliptic and Hypo-elliptic Diffusions
Parametric inference of Stochastic Differential Equation (SDE) from discrete time observations has been studied over recent decades and an important topic in applications, where one of main difficulties reduces to (in general) unavailability of transition density and exact simulation of SDEs. Thus, established approaches usually invoke a conditionally Gaussian time-discretisation for the approximation of the intractable transition dynamics. In this work, for a wide class of multi-dimensional SDEs characterised as elliptic and hypo-elliptic ones, we develop discretisation schemes with a higher order distributional convergence and produce a novel small-time expansion for the density of the schemes to form a proxy for the true intractable SDE transition density. Then, we establish a collection of analytic results for likelihood-based parameter estimates obtained via the formed proxies, thus providing a theoretical framework that showcases advantages from the use of the developed schemes for SDE calibration. We present numerical results from carrying out classical or Bayesian inference, for both elliptic and hypo-elliptic SDEs. This is a joint work with Alex Beskos and Matt Graham.