8 problems

Modified Lyons–Sidorova conjecture. One has

Let $\Gamma$ be a torsion-free group. Consider the maps … and … defined by the corresponding $L^2$-signature and ordinary signature constructions. Torsion-free group-ring signature…

Let $\sigma_2(q/p)$ denote the genus-two signature at a rational number $q/p$. For coprime odd integers $0<q<p$, consider the transformed fraction $q/(2q+p)$. Modular transformatio…

Lyons-Sidorova conjecture. The logarithmic signature of $\gamma$ has infinite R.O.C. if and only if $\gamma$ is a line segment. The conjecture proposes that, among continuous tree-…

A path $gamma$ in $\mathbb{R}^d$ has bounded variation, with length $L(gamma)$. Let $\mathbf{S}(gamma)$ be its signature, let $\mathbf{S}_k(gamma)$ denote its degree-$k$ tail, and…

Let $(S_t)_{t\ge 0}$ satisfy the Black–Scholes dynamics … where $\mu\in\mathbb{R}$, $\gamma$ is deterministic, and $(W_t)_{t\ge 0}$ is a standard Brownian motion. Consider one mode…

Let $\gamma$ be a tree-reduced continuous path with bounded variation, and let $g$ be its signature, with $g_n$ denoting the $n$-th term and $\|\cdot\|_{\mathrm{proj}}$ the project…

A weakly geometric rough path is a rough path taking values in the corresponding weakly geometric rough path space, with its signature defined by the associated multiplicative func…