A financial market with delay driven by reflected Brownian motion
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. 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.