4 problems

Let $L_{t}^{s,x}$ denote the local time of $\varphi(s)$ at level $x$ up to time $t$, and let $\tau^{-,j}_{\ell}(s,x)$ and $\theta^{-,j}_{\ell}(s,x)$ denote the successive downcross…

Fix $s\in\mathbb{R}_{+}$, $x\in\mathbb{R}_{+}$, and define $\tau^{-,j}_{\ell}(s,x)$ as the successive downcrossing times of $[x-\ell,x+\ell]$ by $\varphi(s)$. Let…

Fix $s\in\mathbb{R}_{+}$, $x\in\mathbb{R}_{+}$, and $\ell\in(0,\infty)$. Let $\varphi_t(s)$ be the process considered in the paper, let $\tau^{-,j}_{\ell}(s,x)$ be its $j$-th downc…

Downcrossing variance conjecture. For any $\boldsymbol{0}$-feasible process,