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Applications of Variational Inequalities in Stochastic Control
"The general aim of this book is to establish and study the relations that exist, via dynamic programming, between, on the one hand, stochastic control, and on the other hand variational and quasi-variational inequalities, with the intention of obtaining constructive methods of solution by numerical methods. It begins with numerous examples which occur in applications and goes on to study, from an analytical viewpoint, both elliptic and parabolic quasi-variational inequalities. Finally the authors reconstruct an optimal control starting from the solution of the quasi-variational inequality."--Amazon.
Here is a rigorous introduction to the most important and useful solution methods of various types of stochastic control problems for jump diffusions and its applications. Discussion includes the dynamic programming method and the maximum principle method, and their relationship. The text emphasises real-world applications, primarily in finance. Results are illustrated by examples, with end-of-chapter exercises including complete solutions. The 2nd edition adds a chapter on optimal control of stochastic partial differential equations driven by Lévy processes, and a new section on optimal stopping with delayed information. Basic knowledge of stochastic analysis, measure theory and partial differential equations is assumed.
Stochastic Control by Functional Analysis Methods
Inverse problems of identifying parameters and initial/boundary conditions in deterministic and stochastic partial differential equations constitute a vibrant and emerging research area that has found numerous applications. A related problem of paramount importance is the optimal control problem for stochastic differential equations. This edited volume comprises invited contributions from world-renowned researchers in the subject of control and inverse problems. There are several contributions on optimal control and inverse problems covering different aspects of the theory, numerical methods, and applications. Besides a unified presentation of the most recent and relevant developments, this volume also presents some survey articles to make the material self-contained. To maintain the highest level of scientific quality, all manuscripts have been thoroughly reviewed.
Here is a rigorous introduction to the most important and useful solution methods of various types of stochastic control problems for jump diffusions and its applications. Discussion includes the dynamic programming method and the maximum principle method, and their relationship. The text emphasises real-world applications, primarily in finance. Results are illustrated by examples, with end-of-chapter exercises including complete solutions. The 2nd edition adds a chapter on optimal control of stochastic partial differential equations driven by Lévy processes, and a new section on optimal stopping with delayed information. Basic knowledge of stochastic analysis, measure theory and partial differential equations is assumed.
Optimal control problems and differential Nash games have been employed by many scholars in the study of dynamic pricing, supply chain management and transportation network flow problems. This dissertation emphasizes the extensionof frequently employed deterministic, open-loop modeling paradigms into feedback and stochastic cases respectively with a focus on the computational perspective.For the feedback differential Nash games, this dissertation briefly reviews the classical theory of Hamilton-Jacobi-Bellman equation and the general technique to synthesis feedback optimal control from its solution. Such techniques are then applied to the investigation of a dynamic competitive pricing problem of perishable products with fixed initial inventories (DPFI). Other qualitative analysis and numerical extensions of the DPFI model are also provided.In the study of differential Nash games with Ito-type of stochastic dynamics, this dissertation starts from reviewing the stochastic maximum principle. It then proposes stochastic differential variational inequality (S-DVI) as the necessary condition for stochastic differential Nash games. As an application, this dissertation provides formulation, qualitative analysis and algorithm for a stochastic differential oligopsony problem where multiple agents compete in the procurement of key rawmaterial which follows Ito-type of stochastic price dynamics.