Hamilton-Jacobi Equations with Semilinear Costs and State Constraints, with Applications to Large Deviations in Games

Abstract

We characterize solutions of a class of time-homogeneous optimal control problems with semilinear running costs and state constraints as maximal viscosity subsolutions to Hamilton-Jacobi equations and show that optimal solutions to these problems can be constructed explicitly. We present applications to large deviations problems arising in evolutionary game theory.

Large deviations and Stochastic stability in Population Games

“…In fact [52] identifies a singular arc as the optimal solution in the logit dynamics. More generally, the recent paper [48] investigates the arising optimal control problem with the help of Hamilton-Jacobi equations and provides a characterization of solutions in terms of obstacle problems.…”

“…If the set O is understood from the context, we can suppress them from the definitions of these two value functions. It is then easy to see that rad(x, O) corresponds to the target problem, and corad(O, x) to the source problem, respectively, studied in [48]. They provide a PDE characterization of the radius and coradius of the Lipschitz continuous subsolutions of the Hamilton-Jacobi equation…”

“…where U is a compact convex subset of T X (x). The exact statement is as follows: 48]). The radius x → rad(x, T) is the maximal Lipschitz continuous function R : X → R satisfying…”

“…With the help of this PDE characterization, [48] go further in deriving a powerful verification theorem which turns out to be a convenient analytical tool to solve non-trivial examples. In the next sections we illustrate this in the context of threestrategy games.…”

“…Using the verification theorem of [48], [1] showed that the optimal exit paths under probit choice may lead to new qualitative features. In particular, [1] showed that these paths may feature indirect exit i.e., the optimal exit paths from some initial conditions in the basin of attraction may involve the agents switching back to the status quo equilibrium strategy as shown in Figure 1, panel (B) and Figure 2.…”