Stochastic Differential Portfolio Games

We study stochastic dynamic investment games in continuous time between two investors (players) who have available two different, but possibly correlated, investment opportunities. There is a single payoff function which depends on both investors’ wealth processes. One player chooses a dynamic portfolio strategy in order to maximize this expected payoff while his opponent is simultaneously choosing a dynamic portfolio strategy so as to minimize the same quantity. This leads to a stochastic differential game with controlled drift and variance. For the most part, we consider games with payoffs that depend on the achievement of relative performance goals and/or shortfalls. We provide conditions under which a game with a general payoff function has an achievable value, and give an explicit representation for the value and resulting equilibrium portfolio strategies in that case. It is shown that nonperfect correlation is required to rule out trivial solutions. We then use this general result to explicitly solve a variety of specific games. For example, we solve a probability maximizing game, where each investor is trying to maximize the probability of beating the other’s return by a given predetermined percentage. We also consider objectives related to the minimization or maximization of the expected time until one investor’s return beats the other investor’s return by a given percentage. Our results allow a new interpretation of the market price of risk in a Black-Scholes world. Games with discounting are also discussed as are games of fixed duration related to utility maximization.