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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.

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Part I Analysis.- 1 Background in Analysis.- 2 Introduction to Partial Differential Equations.- Part II Geometry.- 3 Introduction to Differential Geometry.- Part III Perturbations.- 4 Singular Perturbations.- 5 Heat Kernel Asymptotics.- 6 Advanced Topics.- Part IV Applications.- 7 Stochastic Processes.- 8 Applications in Mathematical Finance.- Summary.- References.- Index.

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"Every chapter in this book contains a lot of information. Suggested further reading and precise references can be found at the end of each one. ... It is aimed towards non-specialists, allowing them to quickly enter into the heart of the matters described above. This volume will be very useful to those interested in asymptotic expansions of heat kernels and stochastic models from financial mathematics." (Pablo Raúl Stinga, Mathematical Reviews, July, 2017)
"The main theme of the monograph is the development of a short-time expansion for the heat kernel. ... The monograph is well organized; each chapter starts with an abstract and ends with a section of notes, that can be effectively used to navigate through the contents. Thanks to an extensive presentation of background material, the book is well suited for undergraduate and graduate students, but can be of interest also for applied mathematicians and physicists." (Paolo Musolino, zbMATH 1342.35001, 2016)

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