Time--consistent investment under model uncertainty: the robust forward criteria

TL;DR

This paper introduces robust forward criteria for time-consistent, ambiguity-averse investment strategies, combining dual characterizations and explicit solutions for specific utility and penalty functions, enhancing model uncertainty handling.

Contribution

It develops a novel framework for robust, time-consistent investment under model uncertainty, including dual characterizations and explicit solutions for non-volatile criteria.

Findings

Dual problem formulation simplifies the search for optimal strategies.

Explicit solution for logarithmic utility with quadratic penalty.

Optimal policy is a fractional Kelly strategy based on estimated market risk.

Abstract

We combine forward investment performance processes and ambiguity averse portfolio selection. We introduce the notion of robust forward criteria which addresses the issues of ambiguity in model specification and in preferences and investment horizon specification. It describes the evolution of time-consistent ambiguity averse preferences. We first focus on establishing dual characterizations of the robust forward criteria. This offers various advantages as the dual problem amounts to a search for an infimum whereas the primal problem features a saddle-point. Our approach is based on ideas developed in Schied (2007) and Zitkovic (2009). We then study in detail non-volatile criteria. In particular, we solve explicitly the example of an investor who starts with a logarithmic utility and applies a quadratic penalty function. The investor builds a dynamical estimate of the market price of risk $\hat \lambda$ and updates her stochastic utility in accordance with the so-perceived elapsed market opportunities. We show that this leads to a time-consistent optimal investment policy given by a fractional Kelly strategy associated with $\hat \lambda$. The leverage is proportional to the investor's confidence in her estimate $\hat \lambda$.