We study a financial market where the risky asset is modelled by a  stochastic differential equation driven by a partially reflected Brownian motion . 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 for an explanation and discussion of this model. This corresponds to a local time term in the equation for the asset price. As already pointed out by Karatzas Shreve (see also Jarrow Protter ) this allows for arbitrages in the market.  \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 and Hida-Malliavin 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 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 KS (see also Jarrow \& Protter \mbox{%DIFAUXCMD JP ) this allows for arbitrages in the market. The model is also relevant for high frequency trading issues. See e.g. Lachapelleet al \mbox{%DIFAUXCMD 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.
A
Brownian motion driven
R
driven by a partially reflected Brownian motion
with a local time in the drift term
R
Protter
\& Protter \mbox{%DIFAUXCMD JP
D
This corresponds to a local time term in the equation for the asset price.
R
Shreve
\& Shreve \mbox{%DIFAUXCMD KS
R
Protter
\& Protter \mbox{%DIFAUXCMD JP
A
The model is also relevant for high frequency trading issues. See e.g. Lachapelle
A
et al
A
\mbox{%DIFAUXCMD LLLL
D
and Hida-Malliavin