Stochastic Differential Equations: Theory and Applications

Title Page......Page 1
Copyright Page......Page 2
Preface......Page 3
Contents......Page 5
Introduction......Page 9
Notation and Abbreviations......Page 13
1.1 Events and random variables......Page 17
1.2 Probability and distribution functions......Page 19
1.3 Integration theory, expectation......Page 23
1.4 Convergence concepts......Page 28
1.5 Products of probability spaces, independence......Page 30
1.6 Limit theorems......Page 33
1.7 Conditional expectations and conditional probabilities......Page 34
1.8 Stochastic processes......Page 37
1.9 Martingales......Page 41
2.1 The Markov property......Page 43
2.2 Transition probabilities, the Chapman-Kolmogorov equation......Page 45
2.3 Examples......Page 50
2.4 The infinitesimal operator......Page 52
2.5 Diffusion processes......Page 55
2.6 Backward and forward equations......Page 57
3.1 Wiener process......Page 61
3.2 White noise......Page 66
4.1 Introduction......Page 73
4.2 An example......Page 74
4.3 Nonanticipating functions......Page 77
4.4 Definition of the stochastic integral......Page 80
4.5 Examples and remarks......Page 91
5.1 The stochastic integral as a function of the upper limit......Page 95
5.2 Examples and remarks......Page 100
5.3 Stochastic differentials. Ito's theorem......Page 104
5.4 Examples and remarks in connection with It6's theorem......Page 108
5.5 Proof of Ito's theorem......Page 112
6.1 Definition and examples......Page 116
6.2 Existence and uniqueness of a solution......Page 121
6.3 Supplements to the existence-and-uniqueness theorem       I I I......Page 0
7.1 The moments of the solutions......Page 132
7.2 Analytical properties of the solutions......Page 136
7.3 Dependence of the solutions on parameters and initial values......Page 138
8.1 Introduction......Page 141
8.2 Linear equations in the narrow sense......Page 144
8.3 The Ornstein-Uhlenbeck-process......Page 150
8.4 The general scalar linear equation......Page 152
8.5 The general vector linear equation......Page 157
9.1 Introduction......Page 161
9.2 The solutions as Markov processes......Page 162
9.3 The solutions as diffusion processes......Page 168
9.4 Transition probabilities......Page 172
10.1 The shift from a real to a Markov process......Page 179
10.2 Stratonovich's stochastic integral......Page 183
10.3 Approximation of stochastic differential equations......Page 188
11.1 Stability of deterministic systems......Page 192
11.2 The basic ideas of stochastic stability theory......Page 195
11.3 Stability of the moments......Page 204
11.4 Linear equations......Page 206
11.5 The disturbed n th-order linear equation......Page 212
11.6 Proof of stability by linearization......Page 214
11.7 An example from satellite dynamics......Page 215
12.1 Description of the problem......Page 218
12.2 The conditional expectation as optimal estimate......Page 221
12.3 The Kalman-Bucy filter......Page 222
12.4 Optimal filters for linear systems......Page 224
13.1 Bellman's equation......Page 227
13.2 Linear systems......Page 229
13.3 Control on the basis of filtered observations......Page 231
Bibliography......Page 233
Name and Subject Index......Page 239