The Relaxed Investor with Partial Information

We consider an investor in a financial market consisting of a riskless bond and several risky assets. The price processes of the risky assets are geometric Brownian motions where either the drifts are modeled as random variables assuming a constant volatility matrix or the volatility matrix is considered random and drifts are assumed to be constant. The investor is only able to observe the asset prices but not all the model parameters and hence information is only partial. A Bayesian approach is used with known prior distributions for the random model parameters. We assume that the investor can only trade at discrete-time points which are multiples of $h>0$ and investigate the loss in expected utility of terminal wealth which is due to the fact that the investor cannot trade and observe continuously. It turns out that in general a discretization gap appears, i.e., for $h\to 0$ the expected utility of the h-investor does not converge to the expected utility of the continuous investor. This is in contrast to results under full information in [L.C.G. Rogers, Finance Stoch., 5(2001), pp. 131–154]. We also present simple asymptotically optimal portfolio strategies for the discrete-time problem. Our results are illustrated by some numerical examples