Signature-Based Optimal Execution for Statistical Arbitrage with Path-Dependent Trading Signals

We develop a signature-based framework for optimal execution in statistical arbitrage strategies with path-dependent predictive signals. Both the alpha process and the trading speed are modelled as linear functionals of the truncated signature of a time-augmented market path, placing signal generation and execution on the same truncated signature basis. This allows the trading rule to react to the realised history of the signal while accounting for temporary impact, inventory exposure, terminal liquidation, and approximate dollar neutrality. The main contribution is a quadratic reduction theorem: within the class of signature-linear trading speeds, the restricted path-dependent execution problem becomes a finite-dimensional concave quadratic programme in the policy coefficients. After running synthetic experiments under a mean-reverting log-spread model, we find that the fitted policy achieves a higher return on turnover than a classical $z$-score threshold benchmark. We show how the same workflow can be deployed on a historical equity pairs-trading backtest, where the fitted signature policy again outperforms the benchmark in accounting terms.