随机环境中的最优收获问题
On Optimal Harvesting Problems in Random Environments
摘要：
本文研究了随机环境中单个物种的最优收获策略，该环境的生长由状态转换扩散决定。收获作为一个（随机）控制人口的规模。我们的目标是找到一个收获策略，最大限度地提高预期的总折扣收益收获{ \em到灭绝的时间}的物种，允许的收入率是国家和环境的依赖。这是一个消亡时和最优收获策略均依赖于初始条件的奇异随机控制问题。在灭绝的随机时间之前获得付款的一个方面是，当使用相同的收获策略时，初始总体规模的微小变化可能会显著改变灭绝时间消亡时。因此，人们不再获得连续性的价值函数使用标准的参数，无论是定期或奇异控制问题，具有固定的时间范围。本文提出了一个新的充分条件，在此条件下建立了状态转换模型的值函数连续性。进一步证明了该值函数是拟变分不等式耦合系统的黏性解。本文还建立了一个验证定理，并在此基础上，在一定条件下，构造了一个$\varepsilon$ -最优收获策略。并对两个实例进行了详细分析。
英文摘要：
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.
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