16 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…

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 $u_1\leq u_2<w_0=\infty$ be the thresholds in the collaborating business-lines model, let $M_1<\infty$ be the upper bound on the reinsurance control, and let $\pi^*_1(x)$ denot…

Let $mu_1$ and $mu_2$ denote the drift parameters in regimes 1 and 2, respectively. The company starts with any surplus and may choose a dividend strategy, including liquidation by…

Let $\overline{D}_n$ denote the principal determinant of the system matrix in (2x2). The system matrix arises in the linear equations for the finite-horizon ruin probabilities of t…

Let … be the principal determinant associated with the system defining the survival probabilities $\varphi(n)$. Monotonicity and sign conjecture. For every $n\in\mathbb{N}_0$, … Th…

Consider the Brownian diffusion risk model with a dividend-rate control that cannot be decreased over time. A curve strategy is a strategy whose curve divides the state space into…

Let $\overline{\zeta}$ be the solution of the differential equation referred to as (Ecuacion diferencial de z0), with the boundary condition referred to as (Condicion de Optimo en…

Conjecture on optimal periodic strategies. An optimal periodic strategy will be of the form $(b_u,b_l)$. This conjecture is motivated by the form of optimal strategies in earlier w…

When the discounting factor is a geometric Brownian motion and claims have continuous distribution function $G(y)$, the surplus process is considered under a stochastic interest ra…

Ruin-between-observations conjecture. This difference is due to the chance of jumping to ruin between Poisson observation times, which can be made negligible in the absence of down…

Let $p$ be a general premium function, and suppose that the candidate optimal barrier level … is well-defined. The conjecture. The barrier strategy need not always be optimal for t…

In model (2.1), let $\{p_n,n\geq 1\}$ be DFR and let $P$ be DFR. Let $b^*$ denote the barrier level and (2.5) the stochastic control problem. Conjecture 2. The barrier strategy at…

The conjectured characterization. On $(0,\infty)$, the function $V$ solves

Let $\rho=\frac{2a}{\sigma^2}$, where $a>0$ is the drift and $\sigma>0$ is the volatility of the geometric Brownian motion governing the invested risk asset. Let $\psi(u)$ denote t…