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Minimal Kullback–Leibler Divergence for Constrained Lévy–Itô Processes
SIAM Journal on Control and Optimization ( IF 2.2 ) Pub Date : 2024-03-18 , DOI: 10.1137/23m1555697 Sebastian Jaimungal 1 , Silvana M. Pesenti 1 , Leandro Sánchez-Betancourt 2
SIAM Journal on Control and Optimization ( IF 2.2 ) Pub Date : 2024-03-18 , DOI: 10.1137/23m1555697 Sebastian Jaimungal 1 , Silvana M. Pesenti 1 , Leandro Sánchez-Betancourt 2
Affiliation
SIAM Journal on Control and Optimization, Volume 62, Issue 2, Page 982-1005, April 2024.
Abstract. Given an [math]-dimensional stochastic process [math] driven by [math]-Brownian motions and Poisson random measures, we search for a probability measure [math], with minimal relative entropy to [math], such that the [math]-expectations of some terminal and running costs are constrained. We prove existence and uniqueness of the optimal probability measure, derive the explicit form of the measure change, and characterize the optimal drift and compensator adjustments under the optimal measure. We provide an analytical solution for Value-at-Risk (quantile) constraints, discuss how to perturb a Brownian motion to have arbitrary variance, and show that pinned measures arise as a limiting case of optimal measures. The results are illustrated in a risk management setting—including an algorithm to simulate under the optimal measure—and explore an example where an agent seeks to answer the question what dynamics are induced by a perturbation of the Value-at-Risk and the average time spent below a barrier on the reference process?
中文翻译：
约束 Lévy-Itô 过程的最小 Kullback-Leibler 散度
SIAM 控制与优化杂志，第 62 卷，第 2 期，第 982-1005 页，2024 年 4 月。
摘要。给定由[数学]-布朗运动和泊松随机测量驱动的[数学]维随机过程[数学]，我们搜索概率测量[数学]，与[数学]的相对熵最小，使得[数学] -对某些终端和运营成本的预期受到限制。我们证明了最优概率测度的存在性和唯一性，推导了测度变化的显式形式，并描述了最优测度下的最优漂移和补偿器调整。我们提供了风险值（分位数）约束的分析解决方案，讨论了如何扰动布朗运动以具有任意方差，并表明固定测量是作为最佳测量的极限情况而出现的。结果在风险管理设置中进行了说明（包括在最佳测量下进行模拟的算法），并探索了一个示例，其中代理试图回答风险值和平均时间的扰动会引发哪些动态的问题参考流程的花费低于障碍？
更新日期：2024-03-19
Abstract. Given an [math]-dimensional stochastic process [math] driven by [math]-Brownian motions and Poisson random measures, we search for a probability measure [math], with minimal relative entropy to [math], such that the [math]-expectations of some terminal and running costs are constrained. We prove existence and uniqueness of the optimal probability measure, derive the explicit form of the measure change, and characterize the optimal drift and compensator adjustments under the optimal measure. We provide an analytical solution for Value-at-Risk (quantile) constraints, discuss how to perturb a Brownian motion to have arbitrary variance, and show that pinned measures arise as a limiting case of optimal measures. The results are illustrated in a risk management setting—including an algorithm to simulate under the optimal measure—and explore an example where an agent seeks to answer the question what dynamics are induced by a perturbation of the Value-at-Risk and the average time spent below a barrier on the reference process?
中文翻译：
约束 Lévy-Itô 过程的最小 Kullback-Leibler 散度
SIAM 控制与优化杂志，第 62 卷，第 2 期，第 982-1005 页，2024 年 4 月。
摘要。给定由[数学]-布朗运动和泊松随机测量驱动的[数学]维随机过程[数学]，我们搜索概率测量[数学]，与[数学]的相对熵最小，使得[数学] -对某些终端和运营成本的预期受到限制。我们证明了最优概率测度的存在性和唯一性，推导了测度变化的显式形式，并描述了最优测度下的最优漂移和补偿器调整。我们提供了风险值（分位数）约束的分析解决方案，讨论了如何扰动布朗运动以具有任意方差，并表明固定测量是作为最佳测量的极限情况而出现的。结果在风险管理设置中进行了说明（包括在最佳测量下进行模拟的算法），并探索了一个示例，其中代理试图回答风险值和平均时间的扰动会引发哪些动态的问题参考流程的花费低于障碍？
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