Equilibrium strategies for the mean-variance investment problem over a random horizon

We study equilibrium feedback strategies for a dynamic mean-variance problem of investing in a risky financial market. We assume the time horizon is random, and we consider both discretetime and continuous-time frameworks. The random time horizon is assumed to have a distribution that is independent of the underlying asset processes. By applying stochastic control theory, we derive the extended Hamilton{Jacobi{Bellman (HJB) system of equations under both discrete-time and continuous-time frameworks. Furthermore, in the continuous-time framework, we allow the risk aversion coefficient to depend on the state variable in a natural way. We explicitly obtain equilibrium feedback strategies in some cases, and we find that equilibrium feedback strategies over a geometrically or exponentially distributed random horizon are not necessarily unique. Moreover, we prove that the equilibrium feedback strategy under zero riskless interest rate is identical to the equilibrium feedback strategy for a finite, fixed time horizon. This result implies that, if the time horizon is random, the equilibrium feedback strategy loses the status of optimality among constant strategies. This phenomenon occurs because the players at each point in time are noncooperative, local optimizers. We also compare our results with those for the fixed horizon and for the standard Merton problem of maximizing expected utility of terminal wealth.