<A General Theory of Paths: Signatures, Jump Lifts, and Expected Signatures of Self-Exciting Processes>
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摘要
This paper develops a path-first theory using signatures as universal coordinates for deterministic paths, rough paths, jumps, and path-valued random variables. It introduces the geometricity defect principle, characterizes Itô vs Marcus lifts, and delivers finite-dimensional Hawkes closures for affine and scalar cases, along with a directional cross-area statistic. The results link shuffle, Marcus–Itô, and signature kernels to the geometry of the free nilpotent group, and establish identifiability and large-deviation implications for path-law representations. [page::0]
