In this second edition, we have added a chapter on optimal control of random jump ?elds (solutions of stochastic partial di?erential equations) and partial information control (Chap.10). We have also added a section on optimal st- ping with delayed information (Sect.2.3). It has always been our intention to give a contemporary presentation of applied stochastic control, and we hope thattheadditionoftheserecentdevelopmentswillcontributeinthisdirection. Wehavealsomadeanumberofcorrectionsandotherimprovements,many of them based on helpful comments from our readers. In particular, we would like to thank Andreas Kyprianou for his valuable communications. We are also grateful to (in alphabetical order) Knut Aase, Jean-Philippe Chancelier, Inga Eide, Emil Framnes, Arne-Christian Lund, Jose-Luis Menaldi, Tam´ as K. Papp, Atle Seierstad, and Jens Arne Sukkestad for pointing out errors and suggesting improvements. Our special thanks go to Martine Verneuille for her skillful typing. Oslo and Paris, November 2006 Bernt Øksendal and Agn` es Sulem Preface of the First Edition Jump di?usions are solutions of stochastic di?erential equations driven by L´ evy processes. Since a L´ evy process ?(t) can be written as a linear com- nation of t, a Brownian motion B(t) and a pure jump process, jump di- sions represent a natural and useful generalization of Itˆ o di?usions. They have received a lot of attention in the last years because of their many applications, particularly in economics.