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.
This book deals with the partial differential equations and some of theirapplications. The book consists of four parts. The first part presents somebackground material and an introduction to partial differential equations. Thesecond part is a short introduction to various aspects of differential geometry.The third part deals with a systematic treatment of singularly perturbed parabolicpartial differential equations. The main theme in this part is the development of ashort-time asymptotic expansion for the heat kernel. Some advanced methodsand extensions are discussed as well. The forth part is devoted to applications. Itstarts with a short introduction to financial mathematics including stochasticdifferential equations and the description of some basic models. The advancedperturbational techniques developed earlier are applied to some models ofmathematical finance.The book is addressed primarily to non-mathematicians interested in partialdifferential equations. The presentation is rather informal and is as self-containedas possible from some elementary introductory concepts to rather advancedtechnical methods. The style of the presentation makes it accessible for a wideraudience. Although this book is intended for advanced undergraduate orbeginning graduate students in, it should also provide a useful reference forprofessional physicists, applied mathematicians as well as quantitative analystswith an interest in partial differential equations, mathematical physics, differentialgeometry, singular perturbations and mathematical finance."
