Title: Strong and Weak Equilibria for Time-Inconsistent Stochastic Control in Continuous Time,
Abstract: A new definition of continuous-time equilibrium controls is introduced. As
opposed to the standard definition, which involves a derivative-type operation,
the new definition parallels how a discrete-time equilibrium is defined, and
allows for unambiguous economic interpretation. The terms "strong equilibria"
and "weak equilibria" are coined for controls under the new and the standard
definitions, respectively. When the state process is a time-homogeneous
continuous-time Markov chain, a careful asymptotic analysis gives complete
characterizations of weak and strong equilibria. Thanks to Kakutani-Fan's
fixed-point theorem, general existence of weak and strong equilibria is also
established, under additional compactness assumption. Our theoretic results are
applied to a two-state model under non-exponential discounting. In particular,
we demonstrate explicitly that there can be incentive to deviate from a weak
equilibrium, which justifies the need for strong equilibria. Our analysis also
provides new results for the existence and characterization of discrete-time
equilibria under infinite horizon.