Analysis and Probability Seminar

The Muckenhoupt (Ap) condition and the existence of an optimal martingale measure in optimal investment.

Dmitry Kramkov (Carnegie Mellon University)

Friday, October 14, 2016 1:30 pm
MONT 414

An unpleasant qualitative feature of the general theory of optimal investment is that the dual optimizer may not correspond to the density of a martingale measure. The existence of such an optimal martingale measure is often desirable. For instance, in pricing theory, this existence is equivalent to the uniqueness of marginal utility based prices for all bounded contingent claims.

In this paper, we show that the optimal martingale measure exists if one can find p>1$p>1$ such that

(i) There is a martingale measure whose density process satisfies the Muckenhoupt (Ap) condition from BMO spaces;

(ii) The relative risk-aversion of the utility function is bounded below by 1/q=1−1/p$1/q=1-1/p$ and above by a constant.

We construct a counterexample showing that the lower bound in (ii) is sharp.

The presentation is based on a joint paper with Kim Weston.