  In an equity market model with "Knightian" uncertainty regarding the relative
risk and covariance structure of its assets, we characterize in several ways
the highest return relative to the market that can be achieved using
nonanticipative investment rules over a given time horizon, and under any
admissible configuration of model parameters that might materialize. One
characterization is in terms of the smallest positive supersolution to a fully
nonlinear parabolic partial differential equation of the
Hamilton--Jacobi--Bellman type. Under appropriate conditions, this smallest
supersolution is the value function of an associated stochastic control
problem, namely, the maximal probability with which an auxiliary
multidimensional diffusion process, controlled in a manner which affects both
its drift and covariance structures, stays in the interior of the positive
orthant through the end of the time-horizon. This value function is also
characterized in terms of a stochastic game, and can be used to generate an
investment rule that realizes such best possible outperformance of the market.
