Multi-agent dynamic financial portfolio management: a differential game approach

In this paper we extend the classical Merton model to the case of several players and we propose a multi-agent Merton-type portfolio optimization model formulated by means of differential games. Each player is price-taker and invests on the market in order to maximize her inter-temporal utility. On the market there are p different assets, of them are risky and their price are generated by a geometric Brownian motion, while one is risk-free. We discuss the non-cooperative and cooperative case and show the existence of a closed-form solution when the class of CRRA utility functions is assumed for both infinite and finite time horizon. We perform a numerical study which illustrates that a cooperative strategy is preferable to a non-cooperative one as it allows for the adoption of optimal strategies from both players. Non-cooperation, instead, leads to sub-optimal solutions.