Risk averse asymptotics in a Black–Scholes market on a finite time horizon

We consider the optimal investment and consumption problem in a Black–Scholes market, if the target functional is given by expected discounted utility of consumption plus expected discounted utility of terminal wealth. We investigate the behaviour of the optimal strategies, if the relative risk aversion tends to infinity. It turns out that the limiting strategies are: do not invest at all in the stock market and keep the rate of consumption constant!