The Value Of Information In Stochastic Control And Finance

We present an optimal portfolio problem with logarithmic utility in the following three cases: (i) The classical case, with complete information from the market available to the agent at all times. Mathematically this means that the portfolio process is adapted to the filtration of the underlying Brownian motion (or, more generally, the underlying Lévy process). (ii) The partial observation case, in which the trader has to base her portfolio choices on less information than . Mathematically this means that the portfolio process must be adapted to a filtration for all t. For example, this is the case if the trader can only observe the asset prices and not the underlying Lévy process. (iii) The insider case, in which the trader has some inside information about the future of the market. This information could for example be the price of one of the assets at some future time. Mathematically this means that the portfolio process is allowed to be adapted to a filtration for all t. In this case the associated stochastic integrals become anticipating, and it is necessary to explain what mathematical model it is appropriate to use and to clarify the corresponding anticipating stochastic calculus. We solve the problem in all these three cases and we compute the corresponding maximal expected logarithmic utility of the terminal wealth. Let us call these quantities , respectively. Then represents the loss of value due the loss of information in (ii), and is the value gained due to the inside information in (iii).