On the analyticity of the value function in optimal investment and stochastically dominant markets

TL;DR

This paper investigates the conditions under which the value function in optimal investment problems is analytic, linking utility functions and market models through stochastic dominance, and provides explicit derivative formulas.

Contribution

It identifies specific classes of utilities and market models ensuring analyticity of the value function and constructs counterexamples illustrating when analyticity fails.

Findings

Analyticity holds for utilities with completely monotonic inverse marginals and markets with maximal stochastic dominance.

Explicit formulas for derivatives and optimizers of the value function are derived.

Infinite-order stochastic dominance is equivalent to second-order dominance within supermartingale deflators.

Abstract

We study the analyticity of the value function in optimal investment with expected utility from terminal wealth and the relation to stochastically dominant financial models. We identify both a class of utilities and a class of semi-martingale models for which we establish analyticity. Specifically, these utilities have completely monotonic inverse marginals, while the market models have a maximal element in the sense of infinite-order stochastic dominance. We construct two counterexamples, themselves of independent interest, which show that analyticity fails if either the utility or the market model does not belong to the respective special class. We also provide explicit formulas for the derivatives, of all orders, of the value functions as well as their optimizers. Finally, we show that for the set of supermartingale deflators, stochastic dominance of infinite order is equivalent to the apparently stronger dominance of second order.