Abstract
The signature of a parametric curve is a sequence of tensors whose entries are iterated integrals. This construction is central to the theory of rough paths in stochastic analysis. It is examined here through the lens of algebraic geometry. We introduce varieties of signature tensors for both deterministic paths and random paths. For the former, we focus on piecewise linear paths, on polynomial paths, and on varieties derived from free nilpotent Lie groups. For the latter, we focus on Brownian motion and its mixtures.
Highlights
A path is a continuous map X : [0, 1] → Rd
The main contribution of this paper is a new bridge between applied algebraic geometry and stochastic analysis, where rough paths are encoded in signature tensors
We present explicit formulas for the entries of the signature tensors for two special paths in Rd
Summary
A path is a continuous map X : [0, 1] → Rd. Unless otherwise stated, we assume that the coordinate functions X1, X2, . . . , Xd are (piecewise) continuously differentiable functions. Theorem 3.3 shows that the signature images for polynomial paths of degree m agree with those for piecewise linear paths with m segments This result does not generalize to higher order tensors. The order k component of the nilpotent Lie group defines a variety that contains the signature tensors of all deterministic paths. The main contribution of this paper is a new bridge between applied algebraic geometry and stochastic analysis, where rough paths are encoded in signature tensors. Varieties of such tensors offer a concise representation of geometric data seen in numerous applications
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