Asymptotic analysis of forward performance processes in incomplete markets and their ill-posed HJB equations

TL;DR

This paper performs an asymptotic analysis of forward performance processes in incomplete markets with stochastic factors, deriving explicit formulas for optimal portfolios and analyzing the ill-posed HJB equations involved.

Contribution

It introduces a novel asymptotic expansion approach for ill-posed HJB equations in forward investment problems with stochastic factors, providing explicit leading and first-order correction terms.

Findings

Explicit formulas for leading order and correction terms of forward performance processes

Analysis of the impact of stochastic factors on optimal portfolios

Justification of the asymptotic expansion with remainder estimates

Abstract

We consider the problem of optimal portfolio selection under forward investment performance criteria in an incomplete market. The dynamics of the prices of the traded assets depend on a pair of stochastic factors, namely, a slow factor (e.g. a macroeconomic indicator) and a fast factor (e.g. stochastic volatility). We analyze the associated forward performance SPDE and provide explicit formulae for the leading order and first order correction terms for the forward investment process and the optimal feedback portfolios. They both depend on the investor's initial preferences and the dynamically changing investment opportunities. The leading order terms resemble their time-monotone counterparts, but with the appropriate stochastic time changes resulting from averaging phenomena. The first-order terms compile the reaction of the investor to both the changes in the market input and his recent performance. Our analysis is based on an expansion of the underlying ill-posed HJB equation, and it is justified by means of an appropriate remainder estimate.