Variance optimality in constrained and unconstrained stochastic differential games

The purpose of this paper is to extend the variance optimality criterion to the settings of constrained and unconstrained two-person stochastic differential games. We give conditions that ensure the existence of Nash equilibria with minimal variance. This criterion is a complement to that of the average long-run expected reward of an ergodic Markov process. Our main contribution is that we give sufficient conditions to define and ensure the existence of relaxed strategies that optimize the limiting variance of a constrained performance index of each player in the continuous-time framework. To the best of our knowledge, this theoretical task has been approached only for unconstrained Markov decision problems in a discrete-time context. The applications of our research range from predator–prey systems to actuarial paradigms and cell growth modeling in patients diagnosed with cancer.