Class Of Linear-quadratic Problems Research Articles

Stable dividends under linear-quadratic optimisation

The optimisation criterion for dividends from a risky business is most often formalised in terms of the expected present value of future dividends. That criterion disregards a potential, explicit demand for the stability of dividends. In particular, within actuarial risk theory, the maximisation of future dividends has been studied as the so-called de Finetti problem. However, there the optimal strategies typically become so-called barrier strategies. These are far from stable, and suboptimal affine dividend strategies have recently received attention. In contrast, in the class of linear-quadratic problems, the demand for stability is explicitly stressed. These have often been studied in diffusion models different from the actuarial risk models. We bridge the gap between these thinking patterns by deriving optimal affine dividend strategies under a linear-quadratic criterion for an additive process. We characterise the value function by the Hamilton-Jacobi-Bellman equation, solve it, and compare the objective and the optimal controls to the classical objective of maximising the expected present value of future dividends. Thereby we provide a framework within which stability of dividends from a risky business, e.g. in classical risk theory, is explicitly demanded and obtained.

Near-optimal controls of random-switching LQ problems with indefinite control weight costs

In this paper, we consider hybrid controls for a class of linear quadratic problems with white noise perturbation and Markov regime switching, where the regime switching is modeled by a continuous-time Markov chain with a large state space and the control weights are indefinite. The use of large state space enables us to take various factors of uncertain environment into consideration, yet it creates computational overhead and adds difficulties. Aiming at reduction of complexity, we demonstrate how to construct near-optimal controls. First, in the model, we introduce a small parameter to highlight the contrast of the weak and strong interactions and fast and slow motions. This results in a two-time-scale formulation. In view of the recent developments on LQ problems with indefinite control weights and two-time-scale Markov chains, we then establish the convergence of the system of Riccati equations associated with the hybrid LQ problem. Based on the optimal feedback control of the limit system obtained using the system of Riccati equations, we construct controls for the original problem and show that such controls are near-optimal. A numerical demonstration of a simple system is presented here.

On the equivalence between optimal stochastic control and open-loop feedback control

The paper is concerned with the question ‘ when is suboptimal open-loop feedback (OLF) control identical to optimal stochastic control ? ’ A lemma about equivalence between the two controls is formulated and used to obtain sufficient conditions for the equivalence in linear quadratic (LQ) problems. It is then shown that the equivalence exists in a generalized class of LQ gaussian (LQG) problems and also in a class of LQ problems having generalized disturbances and complete state information. An introductory section emphasizes the significance of information structure and provides a notation for its discussion in stochastic control problems.

Minimax design for a class of linear quadratic problems with parameter uncertainty

It is found for a class of linear-quadratic problems that the parameter strategy for the unknown time-varying bounded parameters are found to be independent of the present state. Since the parameter strategy is only a function of time, the feedback gains of the linear-quadratic problem are still easily calculatable functions of time.