  Standard Markovian optimal stopping problems are consistent in the sense that
the first entrance time into the stopping set is optimal for each initial state
of the process. Clearly, the usual concept of optimality cannot in a
straightforward way be applied to non-standard stopping problems without this
time-consistent structure. This paper is devoted to the solution of
time-inconsistent stopping problems with the reward depending on the initial
state using an adaptation of Strotz's consistent planning. More precisely, we
give a precise equilibrium definition --- of the type subgame perfect Nash
equilibrium based on pure Markov strategies. In general, such equilibria do not
always exist and if they exist they are in general not unique. We, however,
develop an iterative approach to finding equilibrium stopping times for a
general class of problems and apply this approach to one-sided stopping
problems on the real line. We furthermore prove a verification theorem based on
a set of variational inequalities which also allows us to find equilibria. In
the case of a standard optimal stopping problem, we investigate the connection
between the notion of an optimal and an equilibrium stopping time. As an
application of the developed theory we study a selling strategy problem under
exponential utility and endogenous habit formation.
