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On the Modeling of Impulse Control with Random Effects for Continuous Markov Processes
SIAM Journal on Control and Optimization ( IF 2.2 ) Pub Date : 2024-02-15 , DOI: 10.1137/19m1286967 Kurt L. Helmes 1 , Richard H. Stockbridge 2 , Chao Zhu 2
SIAM Journal on Control and Optimization ( IF 2.2 ) Pub Date : 2024-02-15 , DOI: 10.1137/19m1286967 Kurt L. Helmes 1 , Richard H. Stockbridge 2 , Chao Zhu 2
Affiliation
SIAM Journal on Control and Optimization, Volume 62, Issue 1, Page 699-723, February 2024.
Abstract. The use of coordinate processes for the modeling of impulse control for general Markov processes typically involves the construction of a probability measure on a countable product of copies of the path space. In addition, admissibility of an impulse control policy requires that the random times of the interventions be stopping times with respect to different filtrations arising from the different component coordinate processes. When the underlying Markov process has continuous paths, however, a simpler model can be developed which takes the single path space as its probability space and uses the natural filtration with respect to which the intervention times must be stopping times. Moreover, this model construction allows for impulse control with random effects whereby the decision maker selects a distribution of the new state. This paper gives the construction of the probability measure on the path space for an admissible intervention policy subject to a randomized impulse mechanism. In addition, a class of polices is defined for which the paths between interventions are independent and a further subclass for which the cycles following the initial cycle are identically distributed. A benefit of this smaller subclass of policies is that one is allowed to use classical renewal arguments to analyze long-term average control problems. Further, the paper defines a class of stationary impulse policies for which the family of models gives a Markov family. The decision to use an [math] ordering policy in inventory management provides an example of an impulse policy for which the process has independent and identically distributed cycles and the family of models forms a Markov family.
中文翻译：
连续马尔可夫过程的随机效应脉冲控制建模
SIAM 控制与优化杂志，第 62 卷，第 1 期，第 699-723 页，2024 年 2 月。
摘要。使用坐标过程对一般马尔可夫过程的脉冲控制进行建模通常涉及对路径空间副本的可数乘积构建概率度量。此外，脉冲控制策略的可接受性要求干预的随机时间是相对于不同组件协调过程产生的不同过滤的停止时间。然而，当底层马尔可夫过程具有连续路径时，可以开发一个更简单的模型，该模型将单路径空间作为其概率空间，并使用自然过滤，相对于此，干预时间必须是停止时间。此外，这种模型构造允许具有随机效应的脉冲控制，决策者可以选择新状态的分布。本文给出了受随机脉冲机制影响的可接受干预政策的路径空间概率测度的构造。此外，还定义了一类策略，其中干预之间的路径是独立的，并定义了另一个子类，其中初始周期之后的周期是相同分布的。这一较小的政策子类的一个好处是，人们可以使用经典的更新论据来分析长期平均控制问题。此外，本文定义了一类固定脉冲策略，模型族给出了马尔可夫族。在库存管理中使用[数学]订购策略的决定提供了一个脉冲策略的示例，该过程具有独立且相同分布的周期，并且模型族形成马尔可夫族。
更新日期：2024-02-15
Abstract. The use of coordinate processes for the modeling of impulse control for general Markov processes typically involves the construction of a probability measure on a countable product of copies of the path space. In addition, admissibility of an impulse control policy requires that the random times of the interventions be stopping times with respect to different filtrations arising from the different component coordinate processes. When the underlying Markov process has continuous paths, however, a simpler model can be developed which takes the single path space as its probability space and uses the natural filtration with respect to which the intervention times must be stopping times. Moreover, this model construction allows for impulse control with random effects whereby the decision maker selects a distribution of the new state. This paper gives the construction of the probability measure on the path space for an admissible intervention policy subject to a randomized impulse mechanism. In addition, a class of polices is defined for which the paths between interventions are independent and a further subclass for which the cycles following the initial cycle are identically distributed. A benefit of this smaller subclass of policies is that one is allowed to use classical renewal arguments to analyze long-term average control problems. Further, the paper defines a class of stationary impulse policies for which the family of models gives a Markov family. The decision to use an [math] ordering policy in inventory management provides an example of an impulse policy for which the process has independent and identically distributed cycles and the family of models forms a Markov family.
中文翻译：
连续马尔可夫过程的随机效应脉冲控制建模
SIAM 控制与优化杂志，第 62 卷，第 1 期，第 699-723 页，2024 年 2 月。
摘要。使用坐标过程对一般马尔可夫过程的脉冲控制进行建模通常涉及对路径空间副本的可数乘积构建概率度量。此外，脉冲控制策略的可接受性要求干预的随机时间是相对于不同组件协调过程产生的不同过滤的停止时间。然而，当底层马尔可夫过程具有连续路径时，可以开发一个更简单的模型，该模型将单路径空间作为其概率空间，并使用自然过滤，相对于此，干预时间必须是停止时间。此外，这种模型构造允许具有随机效应的脉冲控制，决策者可以选择新状态的分布。本文给出了受随机脉冲机制影响的可接受干预政策的路径空间概率测度的构造。此外，还定义了一类策略，其中干预之间的路径是独立的，并定义了另一个子类，其中初始周期之后的周期是相同分布的。这一较小的政策子类的一个好处是，人们可以使用经典的更新论据来分析长期平均控制问题。此外，本文定义了一类固定脉冲策略，模型族给出了马尔可夫族。在库存管理中使用[数学]订购策略的决定提供了一个脉冲策略的示例，该过程具有独立且相同分布的周期，并且模型族形成马尔可夫族。
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