Statistička teorija uzročnosti, stohastičke diferencijalne jednačine i svojstvo martingalne reprzentacije

Abstract

One of the important and basic goals of science is to establish cause-e®ect re- lations between events. Many discussions were about the concept of causality and how it can be measured. The concept of Granger's causality (Granger, 1969) is very well known in economy and it can be applied in researches. Granger's de¯nition of causality is based on the idea that the present and the future cannot e®ect the past. About the concept of causality have been discussed for a very long time in all areas of science. In last decade we are dealing with a signi¯cant progress. Today, a concept of causality have a wide application in physical, biological andsocial sciences, history, medicine, especially in epidemiology, economy and etc. The area of research of this Phd dissertation is statistical theory of causality and its application on weak solutions of stochastic di®erential equations and martingale representation property. It have been shown that this concept of causality is equivalent with the concept of weak uniqueness of weak solutions for the stochastic di®erential equations and extremal solutions of the martingale problem. This concept of causality can be characterized with stopping times and its connection with extremal solution of the stopped martingale problem can be proved, as well as with locally unique weak local solutions. The concept of causality can be related to the theory of martingales, too. Namely, this concept can be connected with the preservation of the martingale property, orthogonal martingales, stable subspaces as well as with martingale representation property, which have an application, especially in ¯nancial mathematics.