172 problems

Let $V(x_1,x_2)$ be the value function for two collaborating business lines, and write $x=x_1+x_2$ and $g(x)=V(x_1,x_2)$. The reduced HJB equation is … with $g(0)=0$. Barrier-strat…

Let $h(t,m;s)$ denote the solution of the PDE … with terminal condition $h(s,m;s)=1$. Quadratic-exponential form conjecture. The solution has the form … This ansatz is intended to…

We consider a single-queue system with one fast server and two identical slow servers. Jobs arrive according to a Poisson process with rate $\lambda$; the fast server has exponenti…

Logarithmic ansatz conjecture. The optimal value function is conjectured to have the form

Waiting-region threshold conjecture. There exists a function $\Gamma(x,t)$ such that

Quadratic value-function conjecture. We conjecture that $V(t,x)$ is quadratic in $x$:

Let $U(x,y)=\log x+h(y)$ on $\mathbb{R}_+\times\mathbb{R}$, where $h$ is bounded and continuous with $m\leq h(y)\leq 1$ for some $m\in(0,1)$. Let $V(x,y)$ be the value function of…

Affine decoupling conjecture. According to the terminal condition of the filtering forward-backward stochastic differential equation, one has

The stochastic setting considered here concerns model predictive control under uncertainty; the relevant turnpike property describes the tendency of optimal trajectories to remain…

Optimal asset barrier conjecture. The optimal barrier for the assets is $\beta_1^*$ as defined above. This is stated as a conjecture in the paper, and no resolution is supplied in…

Boundedness of the proxy approximation. There exists a constant $C>0$ such that, for a sufficiently fine state-space discretization,

Time-dependent Riccati feedback conjecture. The optimal control retains the same feedback form as in the time-independent case, with $\bm{\chi}^{\mathbb{X}}$ solving the Riccati eq…

Let $p_{01}$, $\rho$, $\kappa$, and $\beta$ be parameters satisfying … A discounted single-project problem is PCL-indexable when the conditions $\textup{(PCLI1)--(PCLI3)}$ hold on…

Let $(\zeta_i(k))_{i,k=1}^n$ be independent or identically distributed random variables, as appropriate, and let $V^n$ denote the optimal online vector-balancing cost. Let…

Single-agent robust mean field control limit conjecture. In the limit $N\to+\infty$, the problem should become a robust control problem involving a single agent, with a cost depend…

Let $V^n$ and $U^n$ be the system-reliability and survivor-count value functions defined for $x=(x_1,\ldots,x_n)\in\mathbb R^n_+$ by … and … where $\mathcal A_n$ is the set of prog…

Linear value-function conjecture. The function

Let $g$ be the reduced value function, let $w_0$ and $b^*$ be switching points with $w_0<b^*$, let $K_1,K_2,K_4$ be constants, and let $v$ be the function appearing in the middle-r…

Let $g$ be the reduced value function, let $a_2$ be the dividend coefficient for the second business line, and let $b^*$ be the barrier level. Smooth-pasting derivative conjecture.…

Generalized convex-growth existence and uniqueness conjecture. Under these assumptions, the system admits a unique solution $u\in\mathcal{A}_g^k$. This conjecture proposes extendin…

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…

The delayed forward stochastic control problem involves a controlled forward system with state delays, while problem (O) is the corresponding control problem considered in the pape…

Consider the two-stage queueing system with service-time parameters $m_1$ and $m_2$, and suppose Assumption holds. At the relevant decision states, let the number of jobs waiting a…

A stochastic control system may be unstable in its deterministic form yet become stabilized by stochastic perturbations; this phenomenon is described as almost sure exponential sta…

Single-exponential-mode conjecture. Either $C^{b}$ vanishes identically or $C^{l}$ vanishes identically. In particular,