The aim of this work is to find a probabilistic measure which is closest in the sense of the minimal entropy to the set of measures that is spanned by a specific fixed set of generating measures. This problem is motivated by a question from mathematical finance. Each asset in the economy can be understood as a specific probabilistic measure and portfolios can be understood as linear combination of these measures known as mixture distributions. A market agent is assumed to represent another opinion in terms of a new probability distribution and the agent is assumed to maximize logarithmic utility function. The optimal solution for the utility maximizing agent is known to be a likelihood ratio of her subjective measure and the market measure represented by the assets. However, this optimization problem has an additional restriction, namely that the best portfolio must be a linear combination of the existing assets. It means that the agent tries to find the best proxy to the unrestricted optimal solution in terms of the linear combination of assets.
In the situation of the logarithmic utility function, the optimal solution should minimize the relative entropy between the unrestricted and restricted optimal solutions. For simplicity, the work should be restricted to discrete distribution where we can obtain some analytical solutions. More specifically, a model with 3 possible outcomes should be the easiest to study. The work should also study the mean variance approximation of the logarithmic utility function, where we expect to get analytical solutions.
References
Vecer, J.: Numeraire Invariance of the Logarithmic Utility Function, Working Paper (2022)
