Non-Arbitrage Under Additional Information for Thin Semimartingale
Models

This paper completes the two studies undertaken in
\cite{aksamit/choulli/deng/jeanblanc2} and
\cite{aksamit/choulli/deng/jeanblanc3}, where the authors quantify the impact
of a random time on the No-Unbounded-Risk-with-Bounded-Profit concept (called
NUPBR hereafter) when the stock price processes are quasi-left-continuous (do
not jump on predictable stopping times). Herein, we focus on the NUPBR for
semimartingales models that live on thin predictable sets only and the
progressive enlargement with a random time. For this flow of information, we
explain how far the NUPBR property is affected when one stops the model by an
arbitrary random time or when one incorporates fully an honest time into the
model. This also generalizes \cite{choulli/deng} to the case when the jump
times are not ordered in anyway. Furthermore, for the current context, we show
how to construct explicitly local martingale deflator under the bigger
filtration from those of the smaller filtration.