Consumption-investment problems of a relaxed investor with partial and insider information

Abstract

In a financial market consisting of a risk-free asset and several risky assets, an investor with logarithmic or power utility functions aims to maximize the expected utility of terminal wealth and intermediate consumption. The price dynamics of the risky assets follow a geometric Brownian motion where the investor cannot observe the drift. However, the investor has additional information about the stock prices (insider information). Therefore, the consumption-investment strategy has to be adapted with respect to the filtration generated by the observations of the stock prices and the insider information. In this thesis, we model the unknown drift using a Bayesian approach and the insider information using a random variable (drift, terminal stock prices or terminal value of the Brownian motion) whose noisy value the insider investor knows at time zero. Since continuously trading is not possible in reality, we discretize the model by restricting the consumption-investment decisions of the investor to an equidistant time grid with mesh size h > 0. Using a filltering recursion we are able to apply the theory of Markov Decision Processes in order to solve the discrete-time consumption-investment problem with partial and insider information. To compare the three differerent types of insider information we define the value of the insider information as a certainty equivalent. A numerical example lets us suppose that an investor prefers information about the stock prices rather than information about the drift or the Brownian motion. Since deriving optimal closed form solutions is difficult, we investigate also the continuous-time consumption-investment problem and present here an optimal strategy in explicit form. From this solution we construct for small h a good strategy for the discrete-time investor with logarithmic utility functions.