| Summary: | Suppose that γ and σ are two continuous bounded variation paths which
take values in a finite-dimensional inner product space V. The recent papers respectively introduced the truncated and the untruncated signature kernel
of γ and σ, and showed how these concepts can be used in classification and prediction tasks involving multivariate time series. In this paper, we introduce signature
kernels K
γ,σ
φ
indexed by a weight function φ which generalise the ordinary signature kernel. We show how K
γ,σ
φ
can be interpreted in many examples as an average
of PDE solutions, and thus we show how it can be estimated computationally using
suitable quadrature formulae. We extend this analysis to derive closed-form formulae for expressions involving the expected (Stratonovich) signature of Brownian
motion. In doing so we articulate a novel connection between signature kernels
and the notion of the hyperbolic development of a path, which has been a broadly
useful tool in the recent analysis of the signature. As
applications we evaluate the use of different general signature kernels as a basis for
non-parametric goodness-of-fit tests to Wiener measure on path space.
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