Stochastic Methods in Finance
A considerable part of the vast development in Mathematical Finance over the last two decades was determined by the application of stochastic methods. These were therefore chosen as the focus of the 2003 School on “Stochastic Methods in Finance”.The growinginterest of the mathematical communityin this ﬁeld was also reﬂected by the extraordinarily high number of applications for the CIME-EMS School. It was attended by 115 scientists and researchers, selected from among over 200 applicants. The attendees came from all continents: 85 were Europeans, among them 35 Italians. The aim of the School was to provide a broad and accurate knowledge of some of the most up-to-date and relevant topics in Mathematical Finance. Particular attention was devoted to the investigation of innovative methods from stochastic analysis that play a fundamental role in mathematical modeling in ﬁnance or insurance: the theory of stochastic processes, optimal and stochastic control, stochastic diﬀerential equations, convex analysis and duality theory. The outstanding and internationally renowned lecturers have themselves contributed in an essential way to the development of the theory and techniques that constituted the subjects of the lectures. The ﬁnancial origin and motivation of the mathematical analysis were presented in a rigorous manner and this facilitated the understanding of the interface between mathematics and ﬁnance. Great emphasis was also placed on the importance and eﬃciency of mathematical instruments for the formalization and resolution of ﬁnancial problems. Moreover, the direct ﬁnancial origin of the development of some theories now of remarkable importance in mathematics emerged with clarity. The selection of the ﬁve topics of the CIME Course was not an easy task because of the wide spectrum of recent developments in Mathematical Finance. Although other topics could have been proposed, we are conﬁdent that the choice made covers some of the areas of greatest current interest. We now propose a brief guided tour through the topics chosen and through themethodologiesthatmodernﬁnancialmathematicshaselaboratedtounveil Risk beneath its diﬀerent masks.