This thesis focuses on exploring and solving several problems based on partiallyobserved diffusion models. The thesis has two parts.... Show moreThis thesis focuses on exploring and solving several problems based on partiallyobserved diffusion models. The thesis has two parts.
In the first part we present a tractable sufficient condition for the consistency
of maximum likelihood estimators (MLEs) in partially observed diffusion models,
stated in terms of stationary distributions of the associated test processes, under the
assumption that the set of unknown parameter values is finite. We illustrate the
tractability of this sufficient condition by verifying it in the context of a latent price
model of market microstructure. Finally, we describe an algorithm for computing
MLEs in partially observed diffusion models and test it on historical data to estimate
the parameters of the latent price model.
In the second part we provide a thorough analysis of the particle filtering
algorithm for estimating the conditional distribution in partially observed diffusion
models. Specifically, we focus on estimating the distribution of unobserved processes
using observed data. The algorithm involves several steps and assumptions, which are
described in detail. We also examine the convergence of the algorithm and identify
the sufficient conditions under which it converges. Finally, we derive an explicit
upper bound of the convergence rate of the algorithm, which depends on the set of
parameters and the choice of time frequency. This bound provides a measure of the
algorithm’s performance and can be used to optimize its parameters to achieve faster
convergence. Show less
