On Optimal Harvesting Problems in Random Environments
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It is shown that the value function is a viscosity solution of a coupled system of quasi-variational inequalities and an $\varepsilon$-optimal harvesting strategy is constructed under certain conditions on the regime-switching model.Abstract:
This paper investigates the optimal harvesting strategy for a single species living in random environments whose growth is given by a regime-switching diffusion. Harvesting acts as a (stochastic) control on the size of the population. The objective is to find a harvesting strategy which maximizes the expected total discounted income from harvesting {\em up to the time of extinction} of the species; the income rate is allowed to be state- and environment-dependent. This is a singular stochastic control problem with both the extinction time and the optimal harvesting policy depending on the initial condition. One aspect of receiving payments up to the random time of extinction is that small changes in the initial population size may significantly alter the extinction time when using the same harvesting policy. Consequently, one no longer obtains continuity of the value function using standard arguments for either regular or singular control problems having a fixed time horizon. This paper introduces a new sufficient condition under which the continuity of the value function for the regime-switching model is established. Further, it is shown that the value function is a viscosity solution of a coupled system of quasi-variational inequalities. The paper also establishes a verification theorem and, based on this theorem, an $\varepsilon$-optimal harvesting strategy is constructed under certain conditions on the model. Two examples are analyzed in detail.read more
Citations
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Stochastic Differential Equations
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Continuous-time stochastic control and optimization with financial applications / Huyen Pham
TL;DR: This volume provides a systematic treatment of stochastic optimization problems applied to finance by presenting the different existing methods: dynamic programming, viscosity solutions, backward stochastically differential equations, and martingale duality methods.
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Numerical methods for stochastic control problems in continuous time
TL;DR: K Kushner and P.H. Dupuis as discussed by the authors have published a book called "Kushner and Duyguluis, 1992: A History of the World Wide Web".
Journal ArticleDOI
Analysis of a stochastic tri-trophic food-chain model with harvesting
Meng Liu,Chuanzhi Bai +1 more
TL;DR: This work establishes critical values between persistence in mean and extinction for each species and provides a necessary and sufficient condition for existence of optimal harvesting strategy and gives the optimal harvesting effort and maximum of sustainable yield.
References
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