Control-Stopping Games for Market Microstructure and Beyond

In this paper, we present a family of control-stopping games that 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), 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 because of the nonstandard 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.

Local martingales with two reflecting barriers

We give an account of the characteristics that result from reflecting a drifting local martingale (i.e. the sum of a local martingale and a multiple of its quadratic variation process) in 0 and b > 0. We present conditions which guarantee the existence of finite moments of what is required to keep the reflected process within its boundaries. Also, we derive an associated law of large numbers and a central limit theorem which apply when the input is continuous. Similar results for integrals of the paths of the reflected process are also presented. These results are in close agreement to what has previously been shown for Brownian motion.

Local martingales with two reflecting barriers

We give an account of the characteristics that result from reflecting a drifting local martingale (i.e. the sum of a local martingale and a multiple of its quadratic variation process) in 0 and b > 0. We present conditions which guarantee the existence of finite moments of what is required to keep the reflected process within its boundaries. Also, we derive an associated law of large numbers and a central limit theorem which apply when the input is continuous. Similar results for integrals of the paths of the reflected process are also presented. These results are in close agreement to what has previously been shown for Brownian motion.

Nash Equilibrium Payoffs for Stochastic Differential Games with two Reflecting Barriers

In this paper we study Nash equilibrium payoffs for nonzero-sum stochastic differential games with two reflecting barriers. We obtain an existence and a characterization of Nash equilibrium payoffs for nonzero-sum stochastic differential games with nonlinear cost functionals defined by doubly controlled reflected backward stochastic differential equations with two reflecting barriers.

Nash Equilibrium Payoffs for Stochastic Differential Games with two Reflecting Barriers

In this paper we study Nash equilibrium payoffs for nonzero-sum stochastic differential games with two reflecting barriers. We obtain an existence and a characterization of Nash equilibrium payoffs for nonzero-sum stochastic differential games with nonlinear cost functionals defined by doubly controlled reflected backward stochastic differential equations with two reflecting barriers.

On the Dynamics of Semimartingales with Two Reflecting Barriers

We consider a semimartingaleXwhich is reflected at an upper barrierTand a lower barrierS, whereSandTare also semimartingales such thatTis bounded away fromS. First, we present an explicit construction of the reflected process. Then we derive a relationship in terms of stochastic integrals linking the reflected process and the local times at the respective barriers toX,S, andT. This result reveals the fundamental structural properties of the reflection mechanism. We also present a few results showing how the general relationship simplifies under additional assumptions onX,S, andT, e.g. if we takeX,S, andTto be independent martingales (which satisfy some extra technical conditions).

On the Dynamics of Semimartingales with Two Reflecting Barriers

We consider a semimartingale X which is reflected at an upper barrier T and a lower barrier S, where S and T are also semimartingales such that T is bounded away from S. First, we present an explicit construction of the reflected process. Then we derive a relationship in terms of stochastic integrals linking the reflected process and the local times at the respective barriers to X, S, and T. This result reveals the fundamental structural properties of the reflection mechanism. We also present a few results showing how the general relationship simplifies under additional assumptions on X, S, and T, e.g. if we take X, S, and T to be independent martingales (which satisfy some extra technical conditions).