Signature Inversion via Orthogonal Polynomials

In this thesis, we propose a method to reconstruct a path from its truncated signature. This is achieved through explicit recursive formulae for the coefficients of an orthogonal polynomial expansion of the path, represented as linear functionals of the signature. A key example of our approach is the application of ‘shift-and-scale’ Hermite polynomials, which facilitates the accurate point-wise recovery of the path with relatively small errors. Consequently, this leads to the successful implementation of point-wise signature inversion utilizing Hermite polynomials. The novel techniques proposed in this study offer an innovative insight into the field of signature inversion and potentially pave the way for more accurate and efficient computational methods.

BSc Dissertation, Imperial College London