6 Conclusions
Different approaches to examine asset price processes exist. On the one hand,
due to the availability of financial markets data, an enormous amount of time
series properties have been well documented by recent empirical studies. On
the other hand many theoretical papers establish necessary characteristics
of asset price processes to be consistent with an equilibrium. Unfortunately
the gap between empirically well documented stylized facts and theoretically
explainable facts is still vast. The purpose of this paper was on the one hand
to introduce new mathematical tools for the analysis of asset price processes
and to give a general context in which the behavior of asset price processes
can be studied. On the other hand we wanted to show that the discrep-
ancy between empirical and theoretical findings may result from the fact
that one important source of risk is neglected. In this paper we have consid-
ered this risk, i.e. that even the risk of an asset is unknown and therefore
risky, too. In this case the information process has stochastic volatility. With
this generalization of usual information processes a foundation for stochastic
volatility models of asset prices has been established. Further it has been
shown that the still prevalent assumption of zero correlation between asset
prices and their volatility is not sensible and hence we gave a foundation for
option pricing models allowing for correlation between asset prices and their
volatility. Further, we have shown that under resonable assumptions the cor-
relation between asset prices and their volatility is negative. Finally we have
shown the close relationship between the volatility process of the information
process and the risk premia of an asset. Because of this dependence of the
risk premia on the volatility process a theoretical foundation can be given
for financial market phenomena.
Our approach offers numerous avenues for future research. More research
should be devoted to the information process and its volatility. Because
of the established coherence between information processes and asset price
processes this is no more a purely theoretical task. Characteristics of in-
formation processes can be deduced from asset price processes. Hence, it is
possible to investigate empirically whether asset price processes are consis-
tent with the strong assumptions usually made on information processes.
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