Before revision:   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.
After revision:  We allow the stock price dynamics to depend on both a continuous process (Brownian motion) and a jump process (Poisson random measure). We believe that jumps and delays are essential in order to get more realistic financial market models. 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.