【书名】 A Concise Course on Stochastic Partial Differential Equations
【作者】Claudia Pr´evˆot ¢ Michael R¨ockner
【出版社】Springer Berlin / Heidelberg
【版本】
【出版日期】2007
【文件格式】PDF
【文件大小】2.48 MB
【页数】109 pages
【ISBN出版号】ISSN 0075-8434 ，ISBN 978-3-540-70780-6
【资料类别】计量经济学，数学
【市面定价】31.80 Dollars (Amazon Paperback Price)
【扫描版还是影印版】影印版
【是否缺页】完整
【关键词】Stochastic Partial Differential Equations
【内容简介】
These lectures concentrate on (nonlinear) stochastic partial differential equations (SPDE) of evolutionary type. All kinds of dynamics with stochastic influence in nature or man-made complex systems can be modelled by such equations. To keep the technicalities minimal we confine ourselves to the case where the noise term is given by a stochastic integral w.r.t. a cylindrical Wiener process.But all results can be easily generalized to SPDE with more general noises such as, for instance, stochastic integral w.r.t. a continuous local martingale.
There are basically three approaches to analyze SPDE: the "martingale measure approach", the "mild solution approach and the "variational approach". The purpose of these notes is to give a concise and as self-contained as possible an introduction to the "variational approach. A large part of necessary background material, such as definitions and results from the theory of Hilbert spaces, are included in appendices.
【目录】
Contents
1. Motivation, Aims and Examples 1
2. Stochastic Integral in Hilbert Spaces 5
2.1. Infinite-dimensional Wiener processes . . . . . . . . . . . . . . 5
2.2. Martingales in general Banach spaces . . . . . . . . . . . . . . . 17
2.3. The definition of the stochastic integral . . . . . . . . . . . . . 21
2.3.1. Scheme of the construction of the stochastic integral . . 22
2.3.2. The construction of the stochastic integral
in detail . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.4. Properties of the stochastic integral . . . . . . . . . . . . . . . . 35
2.5. The stochastic integral for cylindrical Wiener processes . . . . 39
2.5.1. Cylindrical Wiener processes . . . . . . . . . . . . . . . 39
2.5.2. The definition of the stochastic integral . . . . . . . . . 41
3. Stochastic Differential Equations in Finite Dimensions 43
3.1. Main result and a localization lemma . . . . . . . . . . . . . . . 43
3.2. Proof of existence and uniqueness . . . . . . . . . . . . . . . . . 49
4. A Class of Stochastic Differential Equations 55
4.1. Gelfand triples, conditions on the coefficients and examples . . 55
4.2. The main result and an Itˆo formula . . . . . . . . . . . . . . . . 73
4.3. Markov property and invariant measures . . . . . . . . . . . . . 91
A. The Bochner Integral 105
A.1. Definition of the Bochner integral . . . . . . . . . . . . . . . . . 105
A.2. Properties of the Bochner integral . . . . . . . . . . . . . . . . 107
B. Nuclear and Hilbert–Schmidt Operators 109
C. Pseudo Inverse of Linear Operators 115
D. Some Tools from Real Martingale Theory 119
E. Weak and Strong Solutions: Yamada–Watanabe Theorem 121
E.1. The main result . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
F. Strong, Mild and Weak Solutions 133
V
VI Contents
Bibliography 137
Index 140
Symbols 143
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