An impulse control of a geometric Brownian motion with quadratic costs
An impulse control of a geometric Brownian motion with quadratic costs
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DOI：
10.1016/j.ejor.2004.07.006
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发表时间：
2006-01
期刊：
影响因子：
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通讯作者：
M. Ohnishi;M. Tsujimura
中科院分区：
文献类型：
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作者：
M. Ohnishi;M. Tsujimura
研究了状态服从几何布朗运动的随机系统的最优脉冲控制问题。我们假设，当代理干预系统时，它需要由系统状态的二次形式组成的成本。除了干预成本外，运行成本对系统来说也是持续不断的，而且它们也是二次形式的。我们的目标是找到一个最优脉冲控制，最小化在无限时间范围内发生的干预成本和运行成本的预期总折扣和。为了解决这一问题，我们将其描述为一个随机脉冲控制问题，通过拟变分不等式(QVI)来逼近。对于给定的问题参数，在一组适当的充分条件下，我们证明了最优脉冲控制的存在性，使得当系统状态达到某一水平时，代理介入系统。因此，它瞬间降低到另一个水平。
We examine an optimal impulse control problem of a stochastic system whose state follows a geometric Brownian motion. We suppose that, when an agent intervenes in the system, it requires costs consisting of a quadratic form of the system state. Besides the intervention costs, running costs are continuously incurred to the system, and they are also of a quadratic form. Our objective is to find an optimal impulse control of minimizing the expected total discounted sum of the intervention costs and running costs incurred over the infinite time horizon. In order to solve this problem, we formulate it as a stochastic impulse control problem, which is approached via quasi-variational inequalities (QVI). Under a suitable set of sufficient conditions on the given problem parameters, we prove the existence of an optimal impulse control such that, whenever the system state reaches a certain level, the agent intervenes in the system. Consequently it instantaneously reduces to another level.