商品描述
This book is an essential reference for researchers and advanced students working in stochastic control, applied probability, mathematical finance, engineering systems, and the growing field of mean-field modeling.
Optimal Control in Random Environments offers a modern and comprehensive treatment of stochastic optimal control in systems driven simultaneously by Brownian noise and marked Poisson jumps with random intensity. A central contribution of this work is its rigorous integration of random environments--probability-measure-valued processes that shape both the coefficients of the governing SDEs and the jump intensities themselves. These environments may arise exogenously, representing external or contextual uncertainty, or endogenously, emerging from the collective behavior of large interacting systems. Originally motivated by mean-field control, where particle dynamics generate their own evolving environment, this framework proves equally powerful in settings where the environment acts independently of the system's internal state.
By unifying these viewpoints, this book develops a broad and flexible class of models capable of capturing realistic sources of randomness across applications. Through the use of forward-backward stochastic differential equations, generalized intensity kernels, and an extended Pontryagin Maximum Principle, the text provides both the theoretical foundation and the analytical tools needed to study optimal decisions in complex, jump-driven stochastic systems.
商品描述(中文翻譯)
本書是從事隨機控制、應用概率、數學金融、工程系統以及日益增長的均場建模領域的研究人員和高級學生的重要參考資料。《隨機環境中的最佳控制》提供了對同時受到布朗噪聲和隨機強度的標記泊松跳躍驅動的系統中隨機最佳控制的現代且全面的處理。本書的一個核心貢獻是其對隨機環境的嚴謹整合——這些環境是概率測度值過程，塑造了控制隨機微分方程（SDEs）的係數以及跳躍強度本身。
這些環境可能是外生產生的，代表外部或情境不確定性，或是內生產生的，源自大型互動系統的集體行為。最初受到均場控制的啟發，其中粒子動力學生成自身不斷演變的環境，這一框架在環境獨立於系統內部狀態的情況下同樣強大。
通過統一這些觀點，本書發展出一類廣泛且靈活的模型，能夠捕捉各種應用中的現實隨機性來源。通過使用前向-後向隨機微分方程、廣義強度核以及擴展的龐特里亞金最大原則，文本提供了研究複雜的、由跳躍驅動的隨機系統中最佳決策所需的理論基礎和分析工具。
作者簡介
Daniel Hernández-Hernández has been a professor in the Department of Probability and Statistics at the Research Center for Mathematics (CIMAT) in Guanajuato, Mexico, since 1999, and is an internationally recognized researcher in the field of optimal control of stochastic systems. His research interests include HJB equations, stochastic dynamic games, stochastic optimization, and financial modeling. His academic background began with a bachelor's degree in Applied Mathematics from the Universidad Juárez del Estado de Durango (1988), followed by Master's and Doctoral degrees in Mathematical Sciences (1991 and 1993, respectively) from the Center for Research and Advanced Studies (CINVESTAV) of the National Polytechnic Institute. He subsequently completed postdoctoral fellowships at Brown University (1994) and the University of Maryland (1995).
He is a member of the National System of Researchers in the area of Physics, Mathematics, and Earth Sciences, holding Level III status since 2011, and a member of the Mexican Academy of Sciences. Joshué Helí Ricalde-Guerrero is a mathematician working in probability theory and stochastic analysis, with a focus on interacting stochastic systems and their applications to economics, finance, and large-scale decision models. He obtained his Ph.D. in Mathematics from the Center for Research in Mathematics (CIMAT), Mexico, in December 2023, under the supervision of Prof. Daniel Hernández-Hernández. He is currently a postdoctoral researcher in the Department of Mathematics at ETH Zürich, working with Prof. Dylan Possamaï. His doctoral research focused on Mean-Field Games. In particular, he developed a version of Pontryagin's Maximum Principle for Mean-Field Games conditioned on a random Poisson measure, allowing for the analysis of models with jump-driven dynamics and heterogeneous sources of randomness. At ETH Zürich, his research has expanded toward the study of heterogeneous interacting systems beyond the classical mean-field framework.
作者簡介(中文翻譯)
丹尼爾·埃爾南德斯-埃爾南德斯自1999年以來一直擔任墨西哥瓜納華托數學研究中心（CIMAT）概率與統計系的教授，是隨機系統最佳控制領域的國際知名研究者。他的研究興趣包括HJB方程、隨機動態博弈、隨機優化和金融建模。他的學術背景始於1988年在杜蘭戈州哈雷斯大學（Universidad Juárez del Estado de Durango）獲得的應用數學學士學位，隨後在國立理工學院的高級研究中心（CINVESTAV）獲得數學科學碩士和博士學位（分別為1991年和1993年）。之後，他在布朗大學（1994年）和馬里蘭大學（1995年）完成了博士後研究。
他是國家研究系統（National System of Researchers）物理、數學和地球科學領域的成員，自2011年以來擁有三級資格，並且是墨西哥科學院的成員。
喬舒亞·赫利·里卡爾德-格雷羅是一位專注於概率論和隨機分析的數學家，研究重點為互動隨機系統及其在經濟學、金融學和大規模決策模型中的應用。他於2023年12月在墨西哥數學研究中心（CIMAT）獲得數學博士學位，指導教授為丹尼爾·埃爾南德斯-埃爾南德斯。目前，他是蘇黎世聯邦理工學院（ETH Zürich）數學系的博士後研究員，與迪倫·波薩邁教授合作。他的博士研究集中於均場博弈，特別是他為隨機泊松測度條件下的均場博弈開發了一個龐特里亞金最大原則的版本，這使得能夠分析具有跳躍驅動動力學和異質隨機源的模型。在ETH Zürich，他的研究擴展到超越傳統均場框架的異質互動系統的研究。
