This paper addresses the question of how an arbitrage-free semimartingale model is affected when stopped at a random horizon or when an honest time is incorporated. Precisely, 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. Herein, we prove that any quasi-left-continuous process satisfying NUPBR, will preserve the NUPBR when one stops it at any random horizon or when one incorporates a%DIFDELCMD < {\it %%% specific honest time. For the general case of semimartingale, we provide necessary and sufficient conditions for the NUPBR to be preserved. Precisely, we elaborate two types of results. For a fixed semimartingale and a random time, we provide necessary and sufficient conditions on the process and the random time for which the non-arbitrage concept remains valid . The second type of results consists of giving necessary and sufficient conditions on the random time for which the non-arbitrage is preserved for any process. Our class of honest times ---that we consider in the paper--- is much larger than the class of all stopping times, and plays an important r\^ole in classifying random times. The crucial stochastic tool that drives our analysis lies in the%DIFDELCMD < {\it %%% optional stochastic integral (or%DIFDELCMD < {\it %%% compensated stochastic integral ) that was introduced in early eighties .
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. %DIFDELCMD < {\it %%% 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 %DIFDELCMD < {\it %%% %DIFDELCMD < {\it %%% 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 .
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or when an honest time is incorporated. Precisely, we
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Herein, we prove that any quasi-left-continuous process satisfying NUPBR, will preserve the NUPBR when one stops it at any random horizon or when one incorporates a
D
specific
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honest time. For the general case of semimartingale, we provide necessary and sufficient conditions for the NUPBR to be preserved. Precisely, we elaborate two types of results. For a fixed semimartingale and a random time, we provide necessary and sufficient conditions on the process and the
For this non-arbitrage notion, we obtain two principal results. The first result lies in describing the pairs of market model and
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non-arbitrage concept remains valid
resulting stopped model fulfills NUPBR condition
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type of results consists of giving necessary and sufficient conditions on
main result characterises
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for which the non-arbitrage is preserved for any process. Our class of honest times ---that we consider in the paper--- is much larger than the class of all stopping times, and plays an important r\^ole in classifying random times. The crucial stochastic tool that drives our analysis lies in the
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optional stochastic integral
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(or
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compensated stochastic integral
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) that was introduced in early eighties
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