Restricted Path Characteristic Function Determines the Law of Stochastic Processes

Abstract: A central question in rough path theory is characterising the law of stochastic processes. It is established in [I. Chevyrev $&$ T. Lyons, Characteristic functions of measures on geometric rough paths, $\textit{Ann. Probab.}$ $\textbf{44}$ (2016), 4049--4082] that the path characteristic function (PCF), $\textit{i.e.}$, the expectation of the unitary development of the path, uniquely determines the law of the unparametrised path. We show that PCF restricted to certain subspaces of sparse matrices is sufficient to achieve this goal. The key to our arguments is an explicit algorithm -- as opposed to the nonconstructive approach in [I. Chevyrev $ &$ T. Lyons, $\textit{op. cit.}$] -- for determining a generic element $X$ of the tensor algebra $\mathcal{T}\left(\mathbb{R}^d\right) = \bigoplus_{n=0}^\infty\left(\mathbb{R}^d\right)^{\otimes n}$ from its moment generating function. Our only assumption is that $X$ has a nonzero radius of convergence, which relaxes the condition of having an infinite radius of convergence in the literature. As applications of the above theoretical findings, we propose the restricted path characteristic function distance (RPCFD), a novel distance function for probability measures on the path space that offers enormous advantages for dimension reduction. Its effectiveness is validated via hypothesis testing on fractional Brownian motions, thus demonstrating the potential of RPCFD in generative modeling for synthetic time series generation.