47 problems

Dubins–Émery–Yor conjecture. The martingale $M$ has the same law as its Lévy transform $\widehat{M}$ if and only if its DDS Brownian motion and $\langle M,M\rangle$ are independent…

Let $h$ be a martingale, and let $A_1$ and $A_2$ be operators on martingales. Suppose that $A_i$ and $h$ belong to some subsets of operators and martingales, respectively, which ma…

Let $H:\operatorname{\mathbb{R}}\times\operatorname{\mathbb{R}}_+\to\operatorname{\mathbb{R}}$ be a Borel function, and let $N$ be a continuous local martingale with…

Yor's classification conjecture. The only $(N,\overline{N})$-harmonic functions are

An observable for SLE$_2$ is a process associated with the evolving SLE domain that is a martingale; in the examples discussed, such observables include positive harmonic functions…

The discussion concerns finitary versions of the classical proofs for fluctuation bounds, applied to supermartingales rather than only martingales. Approximation-bound conjecture.…

Let the multitype branching Brownian motion be the model under consideration, and let condition denote the integrability condition defined earlier in the paper for the derivative m…

Let $\{d\mu_t\}_{t\in[0,T)}$ be a differentiable one-parameter family of probability marginals with $T>0$ and $\mu_s\preceq\mu_t$ for $s\leq t$, and let $\rho_t$ denote the density…

Conditional-regret conjecture. Behind all concentration results derived by constructing an appropriate non-negative (super)martingale and then applying Ville's inequality, there ex…

Let $d$ be the dimension, let $cmathcal H(d)$ denote the Hankel subspace, and let $p^*=\max\{p,p/(p-1)\}$ be the conjugate exponent. Write $C(\mathcal H(d),p)$ for the optimal…

Let $W_\infty^{(x)}$ and $D_\infty^{(x)}$ denote the truncated martingales started from position $x$, and let $\kappa$ be the parameter appearing in the tail behavior of…

Assumption-free quarter-rate conjecture. The same conclusion of Theorem 1/4+ε for variance holds without those assumptions.

Let $(X_i,\mathcal{F}_i)_{i\geq 0}$ be a supermartingale difference sequence, and let $x,y,v\geq 0$. Write $(u)_+=\max\{u,0\}$ and let $\widetilde{\Pi}_{v^2/y}$ denote the random v…

A martingale is a stochastic process with the defining martingale properties, and conditional expectation is the associated probabilistic operation. The conjecture concerns represe…

An almost-supermartingale is a stochastic process satisfying the relevant approximate supermartingale conditions; the paper also considers the associated quantitative complexity of…

Let $\mathcal{P}^{\rm mar}$ and $\mathcal{Q}^{\rm mar}$ be the marginal null and alternative families, with $\mathcal{P}=\{P^\infty:P\in\mathcal{P}^{\rm mar}\}$ and…

Let $\bm\pi(\bm\omega\mid\cdot)$ be the conditional-probability specification associated with the signals and assignments described above, and let the resulting extreme martingales…

Let $X^{\sigma^\star}$ be an optimally controlled weak process obtained by switching between radial motion and tangential motion according to its radial position, with switching bo…

Let $D=B_R(0)\subset\mathbb{R}^d$, let $0<\rho<R$, and define the step cost … The value function is the infimum of the expected accumulated cost up to the exit time from $D$. On th…

Let $D=B_R(0)subset\mathbb{R}^d$ with $R>0$ and $d\geq2$, and let $X^\sigma$ be a controlled martingale whose volatility takes values in the admissible control set. Call the two co…

Let $\mathrm{T}\colon S^{d-1}\to\mathbb{R}^{\ell}$ be sufficiently smooth, at least Hölder continuous, and weakly cancelling: … For $\alpha\in(0,d)$, define … let…

Let $\mathcal{N}_t^2$ be the set of type-$2$ particles at time $t$, with positions $X_u(t)$, and let $\overline Z_\infty$ denote the previously defined limit of the relevant multit…

Let $\mathcal{Y}^*=\bigcup_{k\in\mathbb{N}\setminus\{0\}}\mathcal{Y}^k$ be the set of nonempty finite strings over $\mathcal{Y}$, let $\{Y_k:k\in\mathbb{N}\}$ be the Markov chain a…

Let $b$ be a centred Lévy $L^p$-process, let $M_p(t)=\tau(|b_t|^p)$, and let $\tilde{\beta}_p$, $\tilde{\beta}_p$ and the quantities $\tilde{\beta}_p$ be as in the preceding growth…

Let $X$) be a separable Banach space. For integers $N,M\geq 0$, let $(f_{n,m}:0\leq n\leq N,0\leq m\leq M)$ be an $X$-valued martingale on $(\Omega,\mathcal{F},\mathbb{P})$ with re…