英文标题：
《An Optimal Extraction Problem with Price Impact》
---
作者：
Giorgio Ferrari, Torben Koch
---
最新提交年份：
2018
---
英文摘要：
  A price-maker company extracts an exhaustible commodity from a reservoir, and sells it instantaneously in the spot market. In absence of any actions of the company, the commodity\'s spot price evolves either as a drifted Brownian motion or as an Ornstein-Uhlenbeck process. While extracting, the company affects the market price of the commodity, and its actions have an impact on the dynamics of the commodity\'s spot price. The company aims at maximizing the total expected profits from selling the commodity, net of the total expected proportional costs of extraction. We model this problem as a two-dimensional degenerate singular stochastic control problem with finite fuel. To determine its solution, we construct an explicit solution to the associated Hamilton-Jacobi-Bellman equation, and then verify its actual optimality through a verification theorem. On the one hand, when the (uncontrolled) price is a drifted Brownian motion, it is optimal to extract whenever the current price level is larger or equal than an endogenously determined constant threshold. On the other hand, when the (uncontrolled) price evolves as an Ornstein-Uhlenbeck process, we show that the optimal extraction rule is triggered by a curve depending on the current level of the reservoir. Such a curve is a strictly decreasing $C^{\\infty}$-function for which we are able to provide an explicit expression. Finally, our study is complemented by a theoretical and numerical analysis of the dependency of the optimal extraction strategy and value function on the model\'s parameters.
---
中文摘要：
定价公司从水库中提取出一种可耗尽的商品，并在现货市场上即时出售。在公司没有采取任何行动的情况下，商品的现货价格要么是漂移布朗运动，要么是奥恩斯坦-乌伦贝克过程。提取时，公司会影响商品的市场价格，其行为会影响商品现货价格的动态。该公司的目标是最大化销售商品的总预期利润，扣除总预期比例提取成本。我们将该问题建模为具有有限燃料的二维退化奇异随机控制问题。为了确定其解，我们构造了一个关联的Hamilton-Jacobi-Bellman方程的显式解，然后通过一个验证定理来验证其实际最优性。一方面，当（非受控）价格是漂移布朗运动时，当当前价格水平大于或等于内生确定的常数阈值时，提取最优价格。另一方面，当（非受控）价格演变为Ornstein-Uhlenbeck过程时，我们表明，最优开采规则由一条取决于水库当前水位的曲线触发。这样的曲线是一个严格递减的$C ^{\\infty}$-函数，我们可以为它提供一个显式表达式。最后，我们通过理论和数值分析对最优提取策略和值函数对模型参数的依赖性进行了补充。
---
分类信息：

一级分类：Mathematics        数学
二级分类：Optimization and Control        优化与控制
分类描述：Operations research, linear programming, control theory, systems theory, optimal control, game theory
运筹学，线性规划，控制论，系统论，最优控制，博弈论
--
一级分类：Quantitative Finance        数量金融学
二级分类：Mathematical Finance        数学金融学
分类描述：Mathematical and analytical methods of finance, including stochastic, probabilistic and functional analysis, algebraic, geometric and other methods
金融的数学和分析方法，包括随机、概率和泛函分析、代数、几何和其他方法
--

---
PDF下载：
-->