Absolutely continuous optimal martingale measures

Summary

In this paper, we consider the problem of maximizing the expected utility of terminal wealth in the framework of incomplete financial markets. In particular, we analyze the case where an economic agent, who aims at such an optimization, achieves infinite wealth with strictly positive probability. By convex duality theory, this is shown to be equivalent to having the minimal-entropy martingale measure Q^{^} non-equivalent to the historical probability P (what we call the absolutely-continuous case). In this anomalous case, we no longer have the representation of the optimal wealth as the terminal value of a stochastic integral, stated in Schachermayer [9] for the case of Q^{^} ∼ P (i.e. the equivalent case). Nevertheless, we give an approximation of this terminal wealth through solutions to suitably-stopped problems, solutions which still admit the integral representation introduced in [9]. We also provide a class of examples fitting to the absolutely-continuous case.