  This work addresses a classic problem of online prediction with expert
advice. We assume an adversarial opponent, and we consider both the
finite-horizon and random-stopping versions of this zero-sum, two-person game.
Focusing on an appropriate continuum limit and using methods from optimal
control, we characterize the value of the game as the viscosity solution of a
certain nonlinear partial differential equation. The analysis also reveals the
predictor's and the opponent's minimax optimal strategies. Our work provides,
in particular, a continuum perspective on recent work of Gravin, Peres, and
Sivan (Proc SODA 2016). Our techniques are similar to those of Kohn and Serfaty
(Comm Pure Appl Math 2010), where scaling limits of some two-person games led
to elliptic or parabolic PDEs.
