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Optional Processes and their Applications in Mathematical Finance, Risk Theory and Statistics

  • Author / Creator
    Pak, Andrey
  • This thesis is dedicated to the study of the general class of random processes, called optional processes, and their various applications in Mathematical Finance, Risk Theory, and Statistics.

    First, different versions of a comparison theorem and a uniqueness theorem for a general class of optional stochastic differential equations are stated and proved using a local time approach. Furthermore, these results are applied to the pricing of financial derivatives.

    Second, the estimates of N. V. Krylov for distributions of stochastic integrals by means of Lebesgue norm of a measurable function are well-known and are widely used in the theory of stochastic differential equations and controlled diffusion processes. These estimates are generalized for optional semimartingales. After that, they are applied to extend the change of variables formula for a general class of functions from Sobolev space. It is also shown how to use the obtained estimates for the investigation of mean-square convergence of solutions of optional SDE's.

    Furthermore, an optional semimartingale risk model for the capital process of a company is introduced and exhaustively investigated. A general approach to the calculation of ruin probabilities of such models is shown and supported by diverse examples.

    The main object of the final part of this thesis is a general regression model in an optional setting – when an observed process is an optional semimartingale depending on an unknown parameter.
    The cases when the model consists of a one-dimensional and a multi-dimensional unknown parameter are studied separately. The main results include the proof of strong consistency of least squares estimates and the property of fixed accuracy of sequential least squares estimates. It is expected that the proposed general regression models will further be developed and applied in the context of modern mathematical finance.

  • Subjects / Keywords
  • Graduation date
    Fall 2021
  • Type of Item
    Thesis
  • Degree
    Doctor of Philosophy
  • DOI
    https://doi.org/10.7939/r3-brh5-qa27
  • License
    This thesis is made available by the University of Alberta Libraries with permission of the copyright owner solely for non-commercial purposes. This thesis, or any portion thereof, may not otherwise be copied or reproduced without the written consent of the copyright owner, except to the extent permitted by Canadian copyright law.