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
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