  In this paper, we present a family of a control-stopping games which arise
naturally in equilibrium-based models of market microstructure, as well as in
other models with strategic buyers and sellers. A distinctive feature of this
family of games is the fact that the agents do not have any exogenously given
fundamental value for the asset, and they deduce the value of their position
from the bid and ask prices posted by other agents (i.e. they are pure
speculators). As a result, in such a game, the reward function of each agent,
at the time of stopping, depends directly on the controls of other players. The
equilibrium problem leads naturally to a system of coupled control-stopping
problems (or, equivalently, Reflected Backward Stochastic Differential
Equations (RBSDEs)), in which the individual reward functions (or, reflecting
barriers) depend on the value functions (or, solution components) of other
agents. The resulting system, in general, presents multiple mathematical
challenges due to the non-standard form of coupling (or, reflection). In the
present case, this system is also complicated by the fact that the continuous
controls of the agents, describing their posted bid and ask prices, are
constrained to take values in a discrete grid. The latter feature reflects the
presence of a positive tick size in the market, and it creates additional
discontinuities in the agents reward functions (or, reflecting barriers).
Herein, we prove the existence of a solution to the associated system in a
special Markovian framework, provide numerical examples, and discuss the
potential applications.
