Optimal Control In Random Environments: Pontryagin Maximum Principle With Environment-Dependent Jumps And Mean-Field Games With Common Poissonian Noise

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.

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.