48 problems

The $1/e$-strategy waits until time $1/e$ and then accepts the first candidate who is best so far. In certain two-person games, a decision maker faces an adversary trying to minimi…

We consider a spectrally negative Lévy process $X$, a stopping time $\theta$, and the expected penalty … Assume that $g$ is negative and increasing on $(-\infty,0]$ and zero on…

In the two-sided secretary game, suppose there are $U$ men and $U$ women, each player meets $N$ partners over $N$ rounds, and preferences satisfy universal rank symmetry: if a man…

In the cooperative two-sided secretary game, let $N$ be the number of rounds and let $R_N(1)$ denote the expected $N$-rank of a player entering the game, under an optimal common st…

Markovian value-function conjecture. In view of the Markov structure, a solution $(V_t)_{t\geqslant 0}$ to this recursion must have the form

Let $b^{\star}$ denote the optimal extraction boundary, and let $\lambda_{\mathpzc{p}}>0$ be the positive-jump intensity parameter. Monotonicity conjecture. The mapping … is increa…

PINN alternative conjecture. For higher-dimensional problems where finite-difference schemes are no longer applicable, the PINN approach will be a good alternative for solving JCtr…

Let $T\in[0,\infty)$, let $\mathcal{C}_T$ be the continuation region of the optimal stopping problem, and let $b_T$ denote its stopping boundary. Assume that the discount functions…

Let $\lambda$ denote the temperature parameter controlling the weight on exploration, and consider the learned option prices obtained from the stopping and control procedures. For…

Let $P(\mu,f,u)$ be the optimal-stopping problem considered in the paper, with admissible target measures $\nu$ constrained by the initial data $\mu$, obstacle $\f$, and cost \u.…

Let $\beta_1,\beta_2\in(0,1)$ be regularity parameters, and consider payoff functions satisfying a margin condition with parameter $\beta_1$ or having regularity $\beta_2$. A strat…

Let $v^{\mathrm{FB}}$ be the first-best value function, let $\bar{F}$ be the barrier, let $\delta$ be the accident parameter, and let $v^{\mathrm{FB}}_0$ denote the initial value.…

Let $\{F_n\}$, $\{G_n\}$ and $\{H_n\}$ be the sequences of real functions and let $g_n$, $h_n$, $g$, $h$ and $\Theta$ be as in the preceding pointwise convergence conjecture. Assum…

Let $X$ be the market-condition process with inaccessible lower endpoint $\alpha$, let $M$ be its running maximum, and let $V(x,m)$ be the value function on the state space…

Let $X$ be the market-condition process, let $M$ be its running maximum, and let $V(x,m)$ be the value function. Let $b(m)$ denote the free boundary, with $min[underline m,\beta)$,…

Let an optimal stopping problem be given, with utility estimated using Monte Carlo simulation of the underlying process. Broadie and Glasserman's conjecture. There are no general u…

Let $S$ be the $n$-ray spider process, let $d_t$ denote the sum of the lengths of its visited segments along the rays, and let $\tau$ be an integrable stopping time. Define $c_n$ t…

Let $\overline{w}_n$ denote the finite-horizon value function for stopping times bounded by $n\in\mathbb{N}$, and let $u$ denote the corresponding infinite-horizon value function.…

Let $n$ numbers be drawn independently from $n$ different known distributions and randomly ordered before observation. A no-superstars assumption means that none of the distributio…

Given a pair $(\gamma,\Lambda)\in (\mathbb T^r,\mathbb L)$, let $\beta$ be the intensity of the account holder's randomized stopping strategy, with stopping intensity process…

Optimal exercise-time conjecture. The sequence of times

Let $X=(X_n)_{n\in\mathbb{N}}$ be a random walk with iid. increments taking discrete values with positive probability, and let $C$ denote the continuation set of the stopping probl…

Optimal-strategy conjecture. After observing $I_{k}=1$, the player whose turn it is should give up his turn to his opponent if and only if $p_{i,k}<\frac{1}{2}$ for all $i>k$. This…

For $\alpha>0$, let … Here $c_{\alpha}$ is the asymptotic expected value associated with the $\alpha$-percentile rule. Gaither–Ward continuity conjecture. The function…

Let $U$ be the value function defined in equation (U), and let denote the corresponding boundary condition. Boundary condition conjecture. The boundary condition holds for $U$ in.…