Non-Arbitrage up to Random Horizon for Semimartingale Models

This paper addresses the question of how an arbitrage-free semimartingale
model is affected when stopped at a random horizon. We focus on
No-Unbounded-Profit-with-Bounded-Risk (called NUPBR hereafter) concept, which
is also known in the literature as the first kind of non-arbitrage. For this
non-arbitrage notion, we obtain two principal results. The first result lies in
describing the pairs of market model and random time for which the resulting
stopped model fulfills NUPBR condition. The second main result characterises
the random time models that preserve the NUPBR property after stopping for any
market model. These results are elaborated in a very general market model, and
we also pay attention to some particular and practical models. The analysis
that drives these results is based on new stochastic developments in
semimartingale theory with progressive enlargement. Furthermore, we construct
explicit martingale densities (deflators) for some classes of local martingales
when stopped at random time.