99 problems

A Lévy process is a stochastic process with stationary and independent increments, starting at zero, and stochastically continuous. Hunt's hypothesis (H) is the assertion that ever…

Monotonicity conjecture. The function $\Sigma^2(k)$ is monotone increasing in $r$.

Constant-barrier conjecture. If the optimal strategy is of constant-barrier type, then the value function should have the form

Càdlàg regularity conjecture. The càdlàg property should hold under much weaker conditions than condition (ii).

Conditional functional limit conjecture. The conditional distribution of

For a Lévy–Student process, the reduced increment law tends to a normal law for large times when the underlying distribution has finite variance, while at finite large times its as…

Let $p(x,t\mid\nu_0)$ be the marginal density of a Lévy–Student process at time $t$, represented as a mixture of Student densities $f(x\mid\nu,\delta)$ with mixing distribution…

A Lévy–Student process is the process whose one-dimensional marginal density at time $t$ is denoted by $p(x,t\mid\nu_0)$, with $\nu_0$ the initial Student index. Let…

The p-adic Lévy–Khintchine conjecture. $p$-adic fields have the Lévy–Khintchine property.

Let $D$ be a domain, let $(X_t)$ be a Markov process of jump type, and let the representation formula referred to in the source be denoted by … generalizes to a wide class of Marko…

Tail asymptotic conjecture. The same identity

Boundedness conjecture. The condition

Let $X$ be a pure-jump Lévy process, let $\beta$ be its Blumenthal–Getoor index, and let $p>0$. Finite $p$-variation conjecture. The process $X$ has finite $p$-variation provided…

Bertoin–Yor conjecture. The absence of positive jumps in $\xi$ is necessary for $X_\xi$ to be moment-determinate.

An ICRT is an inhomogeneous continuum random tree with parameter $ctheta$. Mixtures of ICRTs are obtained by first sampling $ctheta$ according to a distribution and then, condition…

Consider the $cmathbb R$-tree $(cmathcal{T},d)$ obtained from the line-breaking construction with intensity process $ctilde{\sigma}$, together with its limiting uniform measure…

Let $\xi=(\xi_s)_{s\geq 0}$ be the Lévy process in the factorisation … and let $\overline{F}_\Psi(x)=\mathbb{P}(I_\Psi>x)$ denote the tail of the exponential functional $I_\Psi$. R…

Let $I_\Psi$ be the exponential functional of a potentially killed Lévy process with Lévy measure tail $\overline{\Pi}(0)$ and Gaussian coefficient $\sigma^2$. Infinite differentia…

Consider the fractional stochastic Navier–Stokes system studied in the paper, but replace its Gaussian white-noise forcing by discontinuous forcing generated by a Lévy process with…

Let a controlled Lévy process start from an initial surplus, and consider a double-barrier strategy with periodic dividend payments above an upper barrier and continuous capital in…

Extension conjecture. The results of Dareiotis, Kumar and related authors on the tamed Euler method should remain valid for every

Asymptotic approximation conjecture. The sequence $\{\mathbf{I}_{3,2}^{(n)}\}_{n\geq 1}$ can be asymptotically approximated by the sequence…

Fourier–Hahn–Lévy conjecture. If $f:\mathbb{Z}\to\mathbb{Z}$ is tractable, then the $f$-moment can be estimated to within a constant factor with a Lévy-Tower, either directly or af…

For a function $f$, its $f$-moment is the quantity … A function is tractable if its moment can be estimated to within a constant factor using a small sketch. Lévy–Khintchine repres…

For a function $f$ on the frequency domain, an $f$-moment is the quantity … A function is tractable if its moment can be estimated to within a constant factor using…