Minimal Kullback–Leibler Divergence for Constrained Lévy–Itô Processes

IF 2.2 2区数学 Q2 AUTOMATION & CONTROL SYSTEMS SIAM Journal on Control and Optimization Pub Date : 2024-03-18 DOI:10.1137/23m1555697

Sebastian Jaimungal, Silvana M. Pesenti, Leandro Sánchez-Betancourt

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Abstract

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?

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受约束莱维-伊托过程的最小库尔巴克-莱伯勒发散性

SIAM 控制与优化期刊》第 62 卷第 2 期第 982-1005 页，2024 年 4 月。摘要。给定一个由布朗运动和泊松随机度量驱动的[math]维随机过程[math]，我们寻找一个与[math]相对熵最小的概率度量[math]，使得某些终端和运行成本的[math]期望受到约束。我们证明了最优概率度量的存在性和唯一性，导出了度量变化的显式，并描述了最优度量下的最优漂移和补偿器调整。我们提供了风险价值（量化）约束的解析解，讨论了如何扰动布朗运动使其具有任意方差，并证明了针状度量是最优度量的极限情况。这些结果在一个风险管理环境中进行了说明--包括一种在最优度量下进行模拟的算法--并探讨了一个例子，在这个例子中，一个代理试图回答这样一个问题：风险价值的扰动和参考过程中低于障碍的平均时间会诱发怎样的动态变化？

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来源期刊

SIAM Journal on Control and Optimization 数学-应用数学

CiteScore

4.00

自引率

4.50%

发文量

143

审稿时长

12 months

期刊介绍： SIAM Journal on Control and Optimization (SICON) publishes original research articles on the mathematics and applications of control theory and certain parts of optimization theory. Papers considered for publication must be significant at both the mathematical level and the level of applications or potential applications. Papers containing mostly routine mathematics or those with no discernible connection to control and systems theory or optimization will not be considered for publication. From time to time, the journal will also publish authoritative surveys of important subject areas in control theory and optimization whose level of maturity permits a clear and unified exposition. The broad areas mentioned above are intended to encompass a wide range of mathematical techniques and scientific, engineering, economic, and industrial applications. These include stochastic and deterministic methods in control, estimation, and identification of systems; modeling and realization of complex control systems; the numerical analysis and related computational methodology of control processes and allied issues; and the development of mathematical theories and techniques that give new insights into old problems or provide the basis for further progress in control theory and optimization. Within the field of optimization, the journal focuses on the parts that are relevant to dynamic and control systems. Contributions to numerical methodology are also welcome in accordance with these aims, especially as related to large-scale problems and decomposition as well as to fundamental questions of convergence and approximation.