© 2007

Applied Stochastic Control of Jump Diffusions


Part of the Universitext book series (UTX)

About this book


The main purpose of the book is to give a rigorous, yet mostly nontechnical, introduction to the most important and useful solution methods of various types of stochastic control problems for jump diffusions and its applications.

The types of control problems covered include classical stochastic control, optimal stopping, impulse control and singular control. Both the dynamic programming method and the maximum principle method are discussed, as well as the relation between them. Corresponding verification theorems involving the Hamilton-Jacobi Bellman equation and/or (quasi-)variational inequalities are formulated. There are also chapters on the viscosity solution formulation and numerical methods.

The text emphasises applications, mostly to finance. All the main results are illustrated by examples and exercises appear at the end of each chapter with complete solutions. This will help the reader understand the theory and see how to apply it.

The book assumes some basic knowledge of stochastic analysis, measure theory and partial differential equations.

In the 2nd edition there is a new chapter on optimal control of stochastic partial differential equations driven by Lévy processes. There is also a new section on optimal stopping with delayed information. Moreover, corrections and other improvements have been made.


Lévy process Lévy processes Stochastic calculus impulse control jump diffusion jump diffusions linear optimization measure theory stochastic control

Authors and affiliations

  1. 1.Department of MathematicsUniversity of OsloOsloNorway
  2. 2.Domaine de VoluceauINRIA RocquencourtLe Chesnay CXFrance

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From the reviews:

"The main purpose of this excellent monograph is to give a rigorous non-technical introduction to the most important and useful solution methods of various types of optimal stochastic control problems for jump diffusions and their applications. … All the main results are illustrated by examples and exercises … . This really helps the reader to understand the theory and to see how it can be applied. … This book is a very useful text for students, researchers, and practitioners working in stochastic analysis … ." (Pavel Gapeev, Zentralblatt MATH, Vol. 1074, 2005)

"The focus is on the applied aspect of the theory of control diffusion processes with jumps, particularly in finance and economy. … A relatively large number of examples and exercises (with solutions) is provided, mainly typical models in finance, but also examples in biology, physics, or engineering. … Summing up, this book is a very good addition to the stochastic control literature … ." (Jose-Luis Menaldi, SIAM Reviews, Vol. 47 (4), 2005)

"In recent time optimal control in finance is connected with modelling of stock prices by Lévy processes and considering of different transaction costs. In the last ten years the authors and their collaborators obtained a lot of results on this field. The publication of this work in the present book seems to be a good way to attain a big audience. … It is useful for students and practitioners in stochastic analysis." (Hans-Joachim Girlich, OR News, Issue 25, November, 2005)

From the reviews of the second edition:

“The book is a research monograph … . book includes many worked examples (and several more are unsolved exercises) that will serve the dedicated student in good stead. … In summary, this is a good and relatively inexpensive book that should appeal to graduate students and researchers with some prior knowledge of stochastic control who now wish to learn about the jump diffusion case––especially, as applied in the areas of computational finance and economics.” (S. Ramamoorthy, Journal of the Operational Research Society, Vol. 62, 2011)