The Convergence Rate of Stochastic Tracking with Application to Optimal Execution

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Highly relevant to quantitative trading. The paper directly addresses optimal execution with transient price impact, a core problem in algorithmic trading. The explicit, implementable strategy Q_epsilon provides a practical alternative to solving the regularized optimization problem. The convergence rate O(sqrt(epsilon)) quantifies the cost of regularizing trading strategies to avoid information leakage. The framework handles stochastic terminal inventory, which is realistic for institutional trading desks managing order flow. The Hilbert-space approach provides new theoretical tools for execution problems.

The explicit strategy Q_epsilon (equation 5.19-5.20) is relatively simple to implement: it requires computing an exponential moving average of the unregularized optimal inventory Q0 (which itself requires the terminal-target martingale Xi_t = E[Xi_T | F_t]), followed by a terminal bridge. However, computing Q0 requires the full Obizhaeva-Wang solution (Theorem 4.2) with time-varying parameters, and the terminal-target martingale requires conditional expectations. The mathematical framework (Hilbert spaces, semimartingale calculus) is sophisticated, but the final feedback rule is explicit and does not require solving PDEs or BSDEs.

The paper is purely theoretical with complete mathematical proofs. All results are self-contained with explicit formulas and constructive strategies. The explicit strategy Q_epsilon (equation 5.19-5.20) is directly implementable. However, no numerical code or computational experiments are provided beyond Figure 1. Mathematical verification is straightforward for experts in stochastic control.