Uniqueness of equilibrium strategies in dynamic mean-variance problems with random coefficients

TL;DR

This paper investigates the uniqueness of open-loop equilibrium strategies in dynamic mean-variance portfolio problems with random coefficients, providing new conditions and revealing that randomness in risk-free rates ensures unique strategies.

Contribution

It develops a unified method for both deterministic and random coefficient cases, extending previous results and establishing new necessity conditions for equilibrium strategy existence.

Findings

Unique equilibrium strategies depend on initial wealth when risk-free rate is random.

New necessity conditions for equilibrium strategies are established.

Random risk-free rates lead to unique strategies even with constant risk aversion.

Abstract

This paper is concerned with the uniqueness issue of open-loop equilibrium investment strategies of dynamic mean-variance portfolio selection problems with random coefficients. A unified method is developed to treat both the problems with deterministic risk-free return rate, state-dependent risk aversion, and that with full random coefficients, constant risk aversion. To do so, some new necessity conditions for the existence of equilibrium investment strategies are established, which considerably extends the analogue in [9] with distinctive ideas. Some new interesting facts are revealed. For example, if risk-free return rate is random, it is shown that there exists a unique open-loop equilibrium investment strategy relying on initial wealth, even when risk aversion is merely a constant but not state-dependent.