9 problems

Modified Lyons–Sidorova conjecture. One has

Perturbation invariance conjecture. If $w$ is $(\rho,\gamma)$-irregular, then $w^\varphi$ is also $(\rho,\gamma)$-irregular.

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…

Conjecture. More generally, adaptive SDE solvers should converge provided that the scheme does not introduce Lévy area bias.

Let $(B_t)_{t\geq 0} = (B_t^1, \dots, B_t^d)_{t\geq 0}$ be a $d$-dimensional Brownian motion and let $V$ be an invertible $d\times d$ matrix. For $X=VB$, write…

Signature asymptotic conjecture. If $\gamma$ is tree-reduced, then

Let the Euler and Milstein approximation results for the linear stochastic differential equation be understood in the rough path setting, with the error estimates described by the…

Let $\tau$ be a rooted tree with size $|\tau|$, and let $q_\gamma(\tau)$ be defined recursively by … Here $\gamma\in(1/(n+1),1)$ and $\tau!$ denotes the tree factorial. Asymptotic…