| Abstract: |
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. |