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

We present a surprisingly simple 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 Kabanov and Stricker in [“The Dalang–Morton–Willinger theorem under delayed and restricted information,” in In Memoriam: Paul-André Meyer, Springer, 2006, pp. 209–213] to continuous time and to a large financial market setting while, however, still preserving the simplicity of the discrete time setting. On the other hand, it generalizes Stricker's $L^p$-version of FTAP [Ann. Inst. H. Poincaré Probab. Statist., 26 (1990), pp. 451–460] towards a setting with two filtrations. We do not assume that price processes are semimartingales (and it does not follow due to trading with respect to the smaller filtration) or have any specific path properties. The two filtrations in question can also be completely general, and we do not require admissibility of portfolio wealth processes. We 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 include modeling trading with delayed information, trading on different time grids, dealing with inaccurate price information, and randomization approaches to uncertainty, which will be dealt with elsewhere.