Heat Kernel Method and its Applications

Description

The heart of the book is the development of a short-time asymptotic expansion for the heat kernel. This is explained in detail and explicit examples of some advanced calculations are given. In addition some advanced methods and extensions, including path integrals, jump diffusion and others are presented. The book consists of four parts: Analysis, Geometry, Perturbations and Applications. The first part shortly reviews of some background material and gives an introduction to PDEs. The second part is devoted to a short introduction to various aspects of differential geometry that will be needed later. The third part and heart of the book presents a systematic development of effective methods for various approximation schemes for parabolic differential equations. The last part is devoted to applications in financial mathematics, in particular, stochastic differential equations. Although this book is intended for advanced undergraduate or beginning graduate students in, it should also provide a useful reference for professional physicists, applied mathematicians as well as quantitative analysts with an interest in PDEs.    Front Matter....Pages i-xix Front Matter....Pages 1-1 Background in Analysis....Pages 3-41 Introduction to Partial Differential Equations....Pages 43-84 Front Matter....Pages 85-85 Introduction to Differential Geometry....Pages 87-166 Front Matter....Pages 167-167 Singular Perturbations....Pages 169-196 Heat Kernel Asymptotics....Pages 197-238 Advanced Topics....Pages 239-304 Front Matter....Pages 305-305 Stochastic Processes....Pages 307-328 Applications in Mathematical Finance....Pages 329-377 Back Matter....Pages 379-390