We study a financial market where the risky asset is modelled by a Brownian motion driven stochastic differential equation with a local time in the drift term.This models a situation where the asset price is partially controlled by a company which intervenes when the price is reaching a certain lower barrier in order to prevent it from going below that barrier . See e.g. Jarrow \& Protter \mbox{%DIFAUXCMD JP  for an explanation and discussion of this model  . As already pointed out by Karatzas \& Shreve \mbox{%DIFAUXCMD KS0pt%DIFAUXCMD (see also Jarrow \& Protter \mbox{%DIFAUXCMD JP }0pt%DIFAUXCMD )}  }  this allows for arbitrages in the market. The model is also relevant for high frequency trading issues. See e.g. Lachapelle\textit{et al \mbox{%DIFAUXCMD \cite{LLLL }\hspace{0pt}%DIFAUXCMD and the references therein}  . In this paper we consider the case when there is a delay \theta> 0 in the information flow available for the trader. Using white noise calculus we compute explicitly the optimal consumption rate and portfolio in this case and we show that the maximal value is finite as long as \theta> 0. This implies that there is no arbitrage in the market in that case. However, when \theta goes to 0, the value goes to infinity. This is in agreement with the above result that is an arbitrage when there is no delay  }  .
We study a financial market where the risky asset is modelled by a geometric It\^o-L\'evy process, with a singular drift term.This can for example model a situation where the asset price is partially controlled by a company which intervenes when the price is reaching a certain lower barrier . See e.g. Jarrow Protter JP for an explanation and discussion of this model in the Brownian motion case . As already pointed out by Karatzas 0pt%DIFAUXCMD (see also Jarrow \& Protter \mbox{%DIFAUXCMD JP }0pt%DIFAUXCMD )} Shreve KS} (in the Brownian motion case), this allows for arbitrages in the market. \textit{ }\hspace{0pt}%DIFAUXCMD and the references therein} However, the situation in the case of jumps is not clear. Moreover, it is not clear what happens if there is a delay in the system . In this paper we consider a jump diffusion market model with a singular drift term modelled as the local time of a given process, and with a delay \theta> 0 in the information flow available for the trader. Using white noise calculus we compute explicitly the optimal consumption rate and portfolio in this case and we show that the maximal value is finite as long as \theta> 0. This implies that there is no arbitrage in the market in that case. However, when \theta goes to 0, the value goes to infinity. This is in agreement with the above result that is an arbitrage when there is no delay . Our model is also relevant for high frequency trading issues. See e.g. Lachapelle et al LLLL} and the references therein .
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Brownian motion driven stochastic differential equation with a local time in the
geometric It\^o-L\'evy process, with a singular
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models
can for example model
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in order to prevent it from going below that barrier
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\& Protter \mbox{%DIFAUXCMD JP
Protter
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JP
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in the Brownian motion case
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\& Shreve \mbox{%DIFAUXCMD KS
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Shreve
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KS
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(in the Brownian motion case),
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The model is also relevant for high frequency trading issues. See e.g. Lachapelle
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et al
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\mbox{%DIFAUXCMD \cite{LLLL
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However, the situation in the case of jumps is not clear. Moreover, it is not clear what happens if there is a delay in the system
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the case when there is a
a jump diffusion market model with a singular drift term modelled as the local time of a given process, and with a
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. Our model is also relevant for high frequency trading issues. See e.g. Lachapelle et al
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LLLL
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and the references therein