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 Ito formula ..................... 73
4.3. Markov property and invariant measures ................. 91
A. The Bochner Integral ....................................... 105
A.l. 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.l. The main result ....................................... 121
F. Strong, Mild and Weak Solutions ............................ 133
Bibliography .................................................. 137
Index ......................................................... 140
Symbols ....................................................... 143
|
