Asymptotic analysis of long-term investment with two illiquid and correlated assets

Abstract

We consider a long-term portfolio choice problem with two illiquid and correlated assets, which is associated with an eigenvalue problem in the form of a variational inequality. The eigenvalue and the free boundaries implied by the variational inequality correspond to the portfolio's optimal long-term growth rate and the optimal trading strategy, respectively. After proving the existence and uniqueness of viscosity solutions for the eigenvalue problem, we perform an asymptotic expansion in terms of small correlations and obtain semi-analytical approximations of the free boundaries and the optimal growth rate. Our leading order expansion implies that the free boundaries are orthogonal to each other at four corners and have C1 regularity. We propose an efficient numerical algorithm based on the expansion, which proves to be accurate even for large correlations and transaction costs. Moreover, following the approximate trading strategy, the resulting growth rate is very close to the optimal one.