INFORMATION, MODEL PERFORMANCE, PRICING AND TRADING MEASURES IN INCOMPLETE MARKETS

In the incomplete market setting, we define a generalized Kullback-Leibler relative entropy in terms of an investor's expected utility. We motivate, from an economic point of view, this quantity — the relative U-entropy. Relative U-entropy measures the discrepancy from a set of pricing measures to a single probability measure. We show that the relative U-entropy shares a number of important properties with the usual Kullback-Leibler relative entropy, and establish the link between this quantity and the pricing measure corresponding to the least favorable market completion. We also describe an economic performance measure for probabilistic models that may be used by an investor in an incomplete market setting. We then introduce a statistical learning paradigm suitable for investors who learn models and base investment decisions, in an incomplete market, on these models.