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This work is concerned with optimal harvesting problems in random environments. In contrast to the existing literature, the Markov chain is hidden and can only be observed in a Gaussian white noise in our work. We first use the Wonham filter to estimate the Markov chain from the observable evolution of the given process so as to convert the original problem to a completely observable one. Then we treat the resulting optimal control problem. Because the problem is virtually impossible to solve in closed form, our main effort is devoted to developing numerical approximation algorithms. To approximate the value function and optimal strategies, Markov chain approximation methods are used to construct a discrete-time controlled Markov chain. Convergence of the algorithm is proved by weak convergence method and suitable scaling. A numerical example is provided to demonstrate the results. [ABSTRACT FROM AUTHOR]

Abstract: This paper develops numerical methods for finding optimal dividend pay-out and reinsurance policies. A generalized singular control formulation of surplus and discounted payoff function is introduced, where the surplus is modeled by a regime-switching process subject to both regular and singular controls. To approximate the value function and optimal controls, Markov chain approximation techniques are used to construct a discrete-time controlled Markov chain. The proofs of the convergence of the approximation sequence to the surplus process and the value function are given. Examples of proportional and excess-of-loss reinsurance are presented to illustrate the applicability of numerical methods. [Copyright &y& Elsevier]

Abstract: This paper is concerned with asymptotic properties of consensus-type algorithms for networked systems whose topologies switch randomly. The regime-switching process is modeled as a discrete-time Markov chain with a finite state space. The consensus control is achieved by using stochastic approximation methods. In the setup, the regime-switching process (the Markov chain) contains a rate parameter in the transition probability matrix that characterizes how frequently the topology switches. On the other hand, the consensus control algorithm uses a stepsize that defines how fast the network states are updated. Depending on their relative values, three distinct scenarios emerge. Under suitable conditions, we show that when , a continuous-time interpolation of the iterates converges weakly to a system of randomly switching ordinary differential equations modulated by a continuous-time Markov chain. In this case a scaled sequence of tracking errors converges to a system of switching diffusion. When , the network topology is almost non-switching during consensus control transient intervals, and hence the limit dynamic system is simply an autonomous differential equation. When , the Markov chain acts as a fast varying noise, and only its averaged network matrices are relevant, resulting in a limit differential equation that is an average with respect to the stationary measure of the Markov chain. Simulation results are presented to demonstrate these findings. [Copyright &y& Elsevier]