Minimal Expected Time in Drawdown through Investment for an Insurance Diffusion Model

Consider an insurance company whose surplus is modelled by an arithmetic Brownian motion of not necessarily positive drift. Additionally, the insurer has the possibility to invest in a stock modelled by a geometric Brownian motion independent of the surplus. Our key variable is the (absolute) drawdown Δ of the surplus X, defined as the distance to its running maximum X¯. Large, long-lasting drawdowns are unfavourable for the insurance company. We consider the stochastic optimisation problem of minimising the expected time that the drawdown is larger than a positive critical value (weighted by a discounting factor) under investment. A fixed-point argument is used to show that the value function is the unique solution to the Hamilton–Jacobi–Bellman equation related to the problem. It turns out that the optimal investment strategy is given by a piecewise monotone and continuously differentiable function of the current drawdown. Several numerical examples illustrate our findings.

Numerical constructions of optimal feedback in models of chemotherapy of a malignant tumor

In this paper, a problem of chemotherapy of a malignant tumor is considered. Dynamics is piecewise monotone and a therapy function has two maxima. The aim of therapy is to minimize the number of tumor cells at the given final instance. The main result of this work is the construction of optimal feedbacks in the chemotherapy task. The construction of optimal feedback is based on the value function in the corresponding problem of optimal control (therapy). The value function is represented as a minimax generalized solution of the Hamilton–Jacobi–Bellman equation. It is proved that optimal feedback is a discontinuous function and the line of discontinuity satisfies the Rankin–Hugoniot conditions. Other results of the work are illustrative numerical examples of the construction of optimal feedbacks and Rankin–Hugoniot lines.

Constant slope maps on the extended real line

For a transitive countably piecewise monotone Markov interval map we consider the question of whether there exists a conjugate map of constant slope. The answer varies depending on whether the map is continuous or only piecewise continuous, whether it is mixing or not, what slope we consider and whether the conjugate map is defined on a bounded interval, half-line or the whole real line (with the infinities included).

Counting preimages

For non-invertible maps, subshifts that are mainly of finite type and piecewise monotone interval maps, we investigate what happens if we follow backward trajectories, which are random in the sense that, at each step, every preimage can be chosen with equal probability. In particular, we ask what happens if we try to compute the entropy this way. It turns out that, instead of the topological entropy, we get the metric entropy of a special measure, which we call the fair measure. In general, this entropy (the fair entropy) is smaller than the topological entropy. In such a way, for the systems that we consider, we get a new natural measure and a new invariant of topological conjugacy.