A Game Theoretical Approach to Homothetic Robust Forward Investment Performance Processes in Stochastic Factor Models

Abstract

This paper studies an optimal forward investment problem in an incomplete market with model uncertainty, in which the underlying stocks depend on the correlated stochastic factors. The uncertainty stems from the probability measure chosen by an investor to evaluate the performance. We obtain directly the representation of the homothetic robust forward performance processes in factor-form by combining the zero-sum stochastic differential game and ergodic BSDE approach. We also establish the connections with the risk-sensitive zero-sum stochastic differential games over an infinite horizon with ergodic payoff criteria, as well as with the classical robust expected utilities for long time horizons. Finally, we give an example to illustrate that our approach can be applied to address a type of robust forward investment performance processes with negative realization processes.

<i>G</i> -forward performance process and representation of homothetic case via ergodic quadratic <i>G</i> -BSDE

“…In addition to the aforementioned drawbacks, the model can be ambiguous for investors, namely, it is always difficult to specify a certain probability measure in the real market. This type of uncertainty is known as Knightian uncertainty, introduced by Knight [26]. Motivated by this, Källblad, Obłój and Zariphopoulou [23] proposed the notion of robust forward criteria, which are pairs of dynamic utilities and penalty functions, and established the dual characterization of them.…”