A General Theory of Paths: Signatures, Jump Lifts, and Expected Signatures of Self-Exciting Processes

Battle Count: 54

The paper is primarily a theoretical mathematics contribution to path signature theory and stochastic process algebra. Its direct relevance to quantitative trading is moderate: Hawkes processes model clustered event arrivals (order flow, trade executions, defaults) central to market microstructure. The cross-area directionality result provides a signature-native lead-lag statistic for event systems. The expected signature closure gives finite-dimensional ODE systems for moment computation of self-exciting clocks. However, the paper does not develop trading strategies, pricing models, or direct market applications. It provides foundational algebraic tools that could underpin future quantitative finance work.

The theoretical framework requires deep knowledge of tensor algebras, Hopf algebras, free Lie algebras, rough path theory, and stochastic calculus. The Hawkes closure involves constructing finite-dimensional linear ODE systems with explicit matrix entries (11×11 at level two). Numerical validation requires exact Ogata thinning simulation and Monte Carlo estimation. The algebraic identities (Hoffman exponential, shuffle relations, antipode) are non-trivial to implement correctly. However, the final computational objects (matrix exponentials, ODE solutions) are tractable once the framework is understood.

The paper includes a reproducibility script validating algebraic identities, Hawkes formulas, Hopf square, Marcus-Itô gap, identifiability reconstruction, and cross-area sign experiment. Numerical validation uses exact Ogata thinning for Hawkes simulation with specified parameters (µ=0.5, α=0.8, β=1.0, T=10). However, the script itself is not linked via a public repository URL in the extract.