Tail Asymptotics of the Signature of various stochastic processes and its connection to the Quadratic Variation

Abstract: The signature of a path is a sequence, whose $n$-th term contains $n$-th order iterated integrals of the path. These iterated integrals of sample paths of stochastic processes arise naturally when studying solutions of differential equation driven by those processes. This paper extends the work of Boedihardjo and Geng (arXiv:1609.08111) who found a connection between the signature of a $d$-dimensional Brownian motion and the time elapsed, which equals to the quadratic variation of Brownian motion. We prove this connection for a more general class of processes, namely for It^o signature of semimartingales. We also establish a connection between the fWIS Signature of fractional Brownian motion and its Hurst parameter. We propose a conjecture for extension to multidimensional space.