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Abstract:
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Many new developments in the filed of probability and statistics focus on finding
causal connections between observed processes. This leads to considering dependence relations
and investigating how the past influence the present and the future. The well known concept
of Granger (1969) causality is closely related to the idea of local dependence introduced by
Schweder (1970). Granger studied time series, while Schweder considered Markov chains. The
concept was later extended to more general stochastic processes by Mykland (1986). All this
concepts incorporate the time into consideration dependence.
The dissertation consist of four chapters. New results are presented in the fourth chap-
ter. The main aim of this doctoral dissertation is to determine di↵erent concepts of stochastic
predictability using the well known tool of conditional independence. Follow Granger’s idea,
relationships between family of sigma - algebras (filtrations) and between processes in continuous ti-
me were considered since continuous time models dependence represent the first step in various
applications, such as in finance, econometric practice, neuroscience, epidemiology, climatology,
demographic, etc. In this dissertation the concept of dependence between stochastic processes
and filtration is introduced. This concept is named causal predictability since it is focused on
prediction. Some major characteristics of the given concept are shown and connections with
known concept of dependence are explained. Finally, the concept of causal predictability is
applied to the processes of di↵usion type, more precisely, to the uniqueness of weak solutions
of Ito stochastic di↵erential equations and stochastic di↵erential equations with driving semi-
martingales. Also, the representation theorem in terms of causal predictability is established
and numerous examples of applications of the given concept are presented such as application
in financial mathematics in the view of modeling default risk, in Bayesian statistics.
The idea for the future might be to deal with the case of progressive stochastic predictability,
i.e. the generalization of stochastic predictability from fixed time to stopping time. |