A fundamental theorem of asset pricing for continuous time large financial markets in a two filtration setting

Abstract

We present a version of the fundamental theorem of asset pricing (FTAP) for continuous time large financial markets with two filtrations in an $L^p$-setting for $ 1 \leq p < \infty$. This extends the results of Yuri Kabanov and Christophe Stricker \cite{KS:06} to continuous time and to a large financial market setting, however, still preserving the simplicity of the discrete time setting. On the other hand it generalizes Stricker's $L^p$-version of FTAP \cite{S:90} towards a setting with two filtrations. We do neither assume that price processes are semi-martigales, (and it does not follow due to trading with respect to the \emph{smaller} filtration) nor that price processes have any path properties, neither any other particular property of the two filtrations in question, nor admissibility of portfolio wealth processes, but we rather go for a completely general (and realistic) result, where trading strategies are just predictable with respect to a smaller filtration than the one generated by the price processes. Applications range from modeling trading with delayed information, trading on different time grids, dealing with inaccurate price information, and randomization approaches to uncertainty.