22 problems

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…

The standard penalty-encoded QUBO approach submits both an objective and constraint penalties to a sparse-topology quantum processing unit (QPU), while a penalty-free approach samp…

Covariance-structure conjecture. This effect is related to the structure of the covariance matrix $V$ and its positive definiteness.

Let $c3_{risk}$ and $c3_{returns}$ denote the standard deviations of portfolio risk and returns, respectively, as defined in modern portfolio theory. Let $b6$ be a lower bound infl…

Exponential ansatz for the value function. Based on the terminal condition, the value function is conjectured to have the form

Let $U(x,y)=x^{\alpha}U(1,y)$ be the utility function, and let $V(x,y)$ be the value function, bounded with respect to $y$. Let $\rho$, $\tau$, $\gamma$, and $\alpha$ be the model…

Logarithmic-form conjecture. The value function has the form

In the single-factor model, let the market variance be $\sigma_M^2$, let the idiosyncratic variances be $\sigma_i^2$, let the nonnegative beta factors be $\beta_i$, and let…

Error-order conjecture. The expected coordinatewise error satisfies

Let $h(t,y;s)$ solve the Heston-model partial differential equation specified in the surrounding analysis, with terminal condition $h(s,y;s)=1$. Exponential-affine solution conject…

Optimal-strategy conjecture. The optimal strategy $u^*$ over the admissible set $\mathcal{A}$ is a constant strategy, and hence coincides with the optimal constant strategy $u_c^*$…

Variational characterization conjecture. The function $\hat{u}$ in the minimization problem defining the optimal strategy is obtained by solving

Let $K$ be an allocation vector in the feasible set $\mathcal{K}$, let $D(K)$ denote the maximum drawdown, and let $P(D(K)\leq\varepsilon)$ denote the probability that the maximum…

Consider the objective … with $\gamma(x)=\gamma/x$, where the $\gamma$ on the right-hand side is a constant. Let $\hat u$ denote the equilibrium control. Linear-feedback conjecture…

Let $\rho_{OCE}^{s}$ be the true optimized certainty equivalent risk measure, let $\rho_{Spec}$ be the reference spectral risk measure, and let $\rho_{IC}^{s}$ be the imputed risk…

Let $v(t,x;d)$ be the value function for the stochastic control problem … where $t\in[0,T]$, $x\in\mathbb{R}$, and $d\in\mathbb{R}$. Convexity conjecture. The function $v(t,x;d)$ i…

Let $m^*<m<c/r$, and let $\pi^*(w,m)$ denote the optimal amount invested in the risky asset when wealth is $w$ and the drawdown level is $m$. Monotonicity conjecture. The first par…

The finite-horizon objective is scale-invariant under simultaneous doubling of the initial risky and safe holdings and of the shares held over time. Let $t$ denote calendar time, l…

No-trade interval conjecture. The no-trade region is given by an interval $[y_-,y_+]$ for some endpoints $y_-$ and $y_+$. This conjecture extends the familiar interval-shaped no-tr…

Long-run exponential-growth ansatz. The long-run value function is conjectured to have the form

Let $\gamma$ denote the manager's coefficient of relative risk aversion, and consider the optimal-policy and welfare result established in the paper for $\gamma\leq 1$. Extension c…

Let $V^T$ and $\vartheta^T$ be the portfolios and strategies in Theorem, and let $V^*$ and $\vartheta^*$ be the corresponding robust growth-optimal quantities. Assume Assumption. T…