The Minimal Entropy Martingale Measure and Hedging in Incomplete Markets

The intent of these essays is to study the minimal entropy martingale measure, to examine some new martingale representation theorems and to discuss its related Kunita-Watanabe decompositions. Such problems arise in mathematical finance for an investor who is confronted with the issues of pricing and hedging in incomplete markets. We adopt the standpoint of a ra­ tional investor who principally endeavours to maximize her expected exponential utility. Resolving this issue within a semimartingale framework leads to a non-trivial martingale problem equipped with an equation between random variables but not processes. It is well known that utility maximization admits a dual formulation: maximizing ex­ pected utility is equivalent to minimizing some sort of distance to the physical probability measure. In our setting, this is compatible to finding the entropy minimizing martingale measure whose density process can be written in a particular form. This minimal entropy martingale model has an information theoretic interpretation: if the physical probability measure encapsulates some information about how the market behaves, pricing financial instruments with respect to this entropy minimizer corresponds to selecting a martingale measure by adding the least amount of information to the physical model. We present a method of solving the non-trivial martingale problem within models which exhibit stochastic compensator. Several martingale representation theorems are estab­ lished to derive an apparent entropy equation. We then verify that the conjectured mar­ tingale measure is indeed the entropy minimizer.