🤖 AI Summary
This study addresses the problem of optimal dynamic quoting at the individual order level in a limit order book, rather than merely controlling aggregate trading rates. By developing a stochastic dynamic programming framework that incorporates signal-dependent drift, price impact, inventory risk, and execution risk, the authors combine Hamilton-Jacobi-Bellman equations, point-process modeling of fill intensity, CARA utility, and a triangular finite-dimensional structural approach to derive, for the first time, explicit analytical solutions for four canonical execution objectives. The resulting fully explicit value functions and optimal quoting strategies not only reveal intrinsic connections among different objectives and enable asymptotic analysis over long horizons, but also demonstrate well-posedness and practical implementability through numerical experiments, highlighting the critical role of signal-dependent drift in shaping optimal execution strategies.
📝 Abstract
This paper develops a unified explicit solution theory for optimal execution through sequential limit-order placement in a limit order book. Rather than controlling only the trading speed of a metaorder, we determine how individual limit orders should be quoted over time. The model incorporates signal-dependent drift, price impact, inventory risk, and execution risk, with fills modeled by point processes whose intensities depend on the submitted quotes. We formulate four execution criteria: expected terminal wealth, expected terminal wealth with running inventory penalty, CARA utility of terminal wealth, and CARA utility with running inventory penalty. For general price-impact and inventory-penalty functions, we derive the corresponding HJB equations and show that all four problems reduce to a triangular finite-dimensional structure which can be solved explicitly, leading to fully explicit value functions and optimal quotes across all cases. We also prove well-posedness, admissibility, and verification results. The explicit formulas reveal connections between quoting strategies under different criteria, support long-horizon asymptotic analysis, and show numerically that signal-dependent drift can substantially affect optimal execution.