Preference-fitting Framework: Elicited Utility Function and PHARA Approximation

The utility function plays a core role in portfolio selection, but its specific form is typically hard to elicit. We propose a definition of the elicited utility function and develop a preference-fitting method to obtain it. Basically, we use intuitive probability-wealth pairs to derive a fitted terminal wealth, a fitted portfolio and a fitted utility function, which converge to the optimal terminal wealth, the optimal portfolio and the elicited utility function of the investor, respectively. Specifically, we first establish a bijection between the utility functions and the terminal wealth functions, based on which we construct the fitted terminal wealth, and then obtain the fitted portfolio and the fitted utility function through the martingale-duality method. Next, we develop a piecewise hyperbolic absolute risk aversion (abbr. PHARA) utility approximation method, and verify the convergences in various senses: almost surely, $L^r$, uniform, etc. We demonstrate two applications of our method: obtaining asymptotically explicit portfolios and handling portfolio selection under Value-at-Risk (abbr. VaR) constraints, thereby illustrating its advantages including intuitiveness, analytical tractability, and ability to circumvent the Lagrange multiplier.