The Reve’s Puzzle Revisited

The Reve’s puzzle, introduced by the English puzzlist, H.E. Dudeney, is a mathematical puzzle with 10 discs of different sizes and four pegs, designated as S, P1, P2 and D. Initially, the n (  1) discs rest on the source peg, S, in a tower (with the largest disc at the bottom and the smallest disc at the top). The objective is to move the tower from the peg S to the destination peg D, in a minimum number of moves, under the condition that each move can transfer only one disc from one peg to another such that no disc can ever be placed on top of a smaller one. This paper considers the solution of the dynamic programming equation corresponding to the Reve’s puzzle.

Asymptotic Expansions and Strategies in the Online Increasing Subsequence Problem

We study two closely related problems in the online selection of increasing subsequence. In the first problem, introduced by Samuels and Steele (Ann. Probab. 9(6):937–947, 1981), the objective is to maximise the length of a subsequence selected by a nonanticipating strategy from a random sample of given size $n$ n . In the dual problem, recently studied by Arlotto et al. (Random Struct. Algorithms 49:235–252, 2016), the objective is to minimise the expected time needed to choose an increasing subsequence of given length $k$ k from a sequence of infinite length. Developing a method based on the monotonicity of the dynamic programming equation, we derive the two-term asymptotic expansions for the optimal values, with $O(1)$ O ( 1 ) remainder in the first problem and $O(k)$ O ( k ) in the second. Settling a conjecture in Arlotto et al. (Random Struct. Algorithms 52:41–53, 2018), we also design selection strategies to achieve optimality within these bounds, that are, in a sense, best possible.

Discrete Dividend Payments in Continuous Time

We propose a model in which dividend payments occur at regular, deterministic intervals in an otherwise continuous model. This contrasts traditional models where either the payment of continuous dividends is controlled or the dynamics are given by discrete time processes. Moreover, between two dividend payments, the structure allows for other types of control; we consider the possibility of equity issuance at any point in time. The value is characterized as the fixed point of an optimal control problem with periodic initial and terminal conditions. We prove the regularity and uniqueness of the corresponding dynamic programming equation and the convergence of an efficient numerical algorithm that we use to study the problem. The model enables us to find the loss caused by infrequent dividend payments. We show that under realistic parameter values, this loss varies from around 1%–24% depending on the state of the system and that using the optimal policy from the continuous problem further increases the loss.

Optimal Parametric Control of Nonlinear Random Vibrating Systems

Due to the great progresses in the fields of smart structures, especially smart soft materials and structures, the parametric control of nonlinear systems attracts extensive attentions in scientific and industrial communities. This paper devotes to the derivation of the optimal parametric control strategy for nonlinear random vibrating systems, in which the excitations are confined to Gaussian white noises. For a prescribed performance index balancing the control performance and control cost, the stochastic dynamic programming equation with respect to the value function is first derived by the principle of dynamic programming. The optimal feedback control law is established according to the extremum condition. The explicit expression of the value function is determined by approximately expressing as a quadratic function of state variables and by solving the final dynamic programming equation. The application and efficacy of the optimal parametric control are illustrated by a random-excited Duffing oscillator and a dielectric elastomer balloon with random pressure. The numerical results show that the optimal parameter control possesses good effectiveness, high efficiency, and high robustness to excitation intensity, and is superior than the associated optimal bounded parametric control.

Towards Tensor Representation of Controlled Coupled Markov Chains

For a controlled system of coupled Markov chains, which share common control parameters, a tensor description is proposed. A control optimality condition in the form of a dynamic programming equation is derived in tensor form. This condition can be reduced to a system of coupled ordinary differential equations and admits an effective numerical solution. As an application example, the problem of the optimal control for a system of water reservoirs with phase and balance constraints is considered.

Optimal Trading with Online Parameter Revisions

The aim of this paper is to explain how parameters adjustments can be integrated in the design or the control of automates of trading. Typically, we are interested in the online estimation of the market impacts generated by robots or single orders, and how they/the controller should react in an optimal way to the information generated by the observation of the realized impacts. This can be formulated as an optimal impulse control problem with unknown parameters, on which a prior is given. We explain how a mix of the classical Bayesian updating rule and of optimal control techniques allows one to derive the dynamic programming equation satisfied by the corresponding value function, from which the optimal policy can be inferred. We provide an example of convergent finite difference scheme and consider typical examples of applications.