Defence of dissertation in the field of mathematics, Zhe Chen, M.Sc.
On Pathwise Stochastic Integration of Processes with Unbounded Power Variation
Zhe Chen, M.Sc., will defend the dissertation "On Pathwise Stochastic Integration of Processes with Unbounded Power Variation" on 11 March 2016 at 12 noon in Aalto University School of Science, lecture hall M1, Otakaari 1, Espoo. The main result of the dissertation is derived from the new sufficient conditions under which pathwise integrals can be defined for stochastic processes with unbounded power variation.
Integration of stochastic processes has various applications in the real world, and Brownian motion is the most well-known process. However, Brownian motion has independent increments, which sometimes is not ideal for describing the complex phenomena. Therefore, other stochastic processes are being considered, for example, fractional Brownian motions. Fractional Brownian motion with a Hurst index H, where H is in (0, 1), is a standard Brownian motion when H=1/2. When H is greater 1/2, fractional Brownian motion possesses long-range dependence. Together with the property of self-similarity, fractional Brownian motion can be applied in different areas including financial time series, hydrology and telecommunications.
Because a fractional Brownian motion is not a semimartingale unless H=1/2, the classical integration theory based on semimartingales cannot be applied. There are different approaches to define stochastic integrals with respect to fractional Brownian motions. In this dissertation, pathwise integration has been considered. However, current pathwise integration theory cannot be applied to processes with unbounded power variation paths, which are the cases for most interesting processes.
This dissertation extends the theory of pathwise stochastic integration to cover a general class of unbounded power variation processes if certain assumptions are satisfied. It also studied integration of multidimensional processes and a new proof of change of variables formula.
Dissertation release (pdf)
Opponent: Professor Tommi Sottinen, University of Vaasa
Custos: Professor Lasse Leskelä, Aalto University School of Science, Department of Mathematics and Systems Analysis
Electronic dissertation: http://urn.fi/URN:ISBN:978-952-60-6672-1
School of Science, electronic dissertations: https://aaltodoc.aalto.fi/handle/123456789/52