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Affine Markov processes on a general state space
Veerman, E.
Publication date
2011
Link to publication
Citation for published version (APA):
Veerman, E. (2011). Affine Markov processes on a general state space. Uitgeverij BOXPress.
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Contents
Voorwoord vii Notation xi Introduction 1 1 Semimartingales 9 1.1 Definitions . . . 91.2 Itˆo’s formula and characteristics . . . 11
1.3 Girsanov’s Theorem . . . 17
2 The martingale problem 25 2.1 Set-up and notation . . . 26
2.2 Turning X into a semimartingale . . . 28
2.3 The Markov property . . . 33
2.4 The positive maximum principle . . . 37
2.5 Equivalent change of measure . . . 44
3 Markov processes 51 3.1 Definitions . . . 51
3.2 Properties of the state space . . . 54
3.3 The symbol of a regular Markov process . . . 58
3.4 Feller processes . . . 62
4 Affine processes 69 4.1 Definition and characterization . . . 70
4.2 Preliminary results . . . 73
4.3 The infinitesimal generator . . . 79
4.4 Existence under admissibility . . . 87
4.5 Non-vanishing Fourier-Laplace transform . . . 100
5 Admissible parameter sets 105 5.1 Preliminaries . . . 107
5.2 Polyhedral state space . . . 111
5.3 Characterizing all quadratic state spaces . . . 119
5.4 Parabolic state space . . . 123
5.4.1 Admissibility for the diffusion parameters . . . 125
5.4.2 Admissibility for the killing parameters . . . 128
5.4.3 Admissibility for the jump parameters . . . 128
5.4.4 Admissibility for the drift parameters . . . 129
5.5 The Lorentz cone . . . 130
5.5.1 Admissibility for the diffusion parameters . . . 132
5.5.2 Admissibility for the jump and killing parameters . . . 135
5.5.3 Admissibility for the drift parameters . . . 135
6 The affine transform formula 137 6.1 Exponential martingales . . . 138
6.2 Full range of validity . . . 143
6.2.1 Real-valued parameters . . . 144
6.2.2 Complex-valued parameters . . . 150
A Cauchy’s functional equation 153
Bibliography 157
Summary 163
Samenvatting 165
Curriculum Vitae 169
