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Excursion theory and local times for Bessel and Brownian diffusions: with applications to credit risk

Zhu, Xiaolin (2020) Excursion theory and local times for Bessel and Brownian diffusions: with applications to credit risk. PhD thesis, The London School of Economics and Political Science (LSE).

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Abstract

By means of excursion theory, the evolution of a continuous Markov process satisfying regularity assumptions is analysed in terms of its behaviour between visits to a recurrent point, for instance the point zero in the state space of Brownian and Bessel diffusions of type reflecting at the origin. As a preliminary conclusion, a sample path of the process can be reconstructed by the excursions away from zero of random finite lengths and the time spent at visits to zero. These two together constitute the core of the work in this thesis. With respect to the zero-free intervals, we study the duration of the excursion in process away from zero by time t, namely the age process, of a Bessel process instantaneously reflected at the origin. The main contribution of our work is the development of a hybrid structural-reduced form model with an endogenous intensity defined by the age process. This model provides a framework for assessing default probabilities within a circumstance of very limited information, assuming that some statistics about a firm are not observable but the time points when they reach certain level are. Results presented include distributional properties for the default time and level as a joint stopping process, by which we discover a decomposition theorem that contributes to exact schemes for simulating the default process. A counting process for monitoring consecutive arrivals of some event driven by the same intensity is also established. Main aspects to be addressed are the properties and the derivations of distributional quantities concerning the interarrival times, the arrival of the nth event and the associated counting process. With respect to the zero set, we construct a continuous family of functionals for the part of time spent at the origin by the age process, namely the local time at zero. It is a well known fact that there is no unified representation for the local time of Markov process, as it can be approximated as a limit of various processes describing the behaviour of trajectories of the underlying process. That being so, the focus and efforts are put on the certain properties of the limit processes served as the approximations, and on the first and second order limit theorems for the convergences to the local time.

Item Type: Thesis (PhD)
Additional Information: © 2020 Xiaolin Zhu
Library of Congress subject classification: H Social Sciences > HB Economic Theory
H Social Sciences > HD Industries. Land use. Labor > HD61 Risk Management
Q Science > QA Mathematics
Sets: Departments > Statistics
Supervisor: Dassios, Angelos and Xing, Hao
URI: http://etheses.lse.ac.uk/id/eprint/4226

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