Contents
Contents 1
0 Motivation 3
1 Preliminaries 4
1.1 Review of stochastic processes . . . . . . . . . . . . . . . . . . . . . . . 4
1.2 Review of martingales . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.3 Poisson process and Brownian motion . . . . . . . . . . . . . . . . . . 9
1.4 Lévy processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.5 Lévy measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.6 Localization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.7 Integration with respect to processes of finite variation . . . . . . . . 21
1.8 Naïve stochastic integration is impossible . . . . . . . . . . . . . . . . 23
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12 Semimartingales and stochastic integration 24
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.2 Stability properties of semimartingales . . . . . . . . . . . . . . . . . . 25
2.3 Elementary examples of semimartingales . . . . . . . . . . . . . . . . 26
2.4 The stochastic integral as a process . . . . . . . . . . . . . . . . . . . . 27
2.5 Properties of the stochastic integral . . . . . . . . . . . . . . . . . . . . 29
2.6 The quadratic variation of a semimartingale . . . . . . . . . . . . . . 31
2.7 Itô’s formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.8 Applications of Itô’s formula . . . . . . . . . . . . . . . . . . . . . . . . 40
3 The Bichteler-Dellacherie Theorem and its connexions to arbitrage 43
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.2 Proofs of Theorems 3.1.7 and 3.1.8 . . . . . . . . . . . . . . . . . . . . 45
3.3 A short proof of the Doob-Meyer theorem . . . . . . . . . . . . . . . . 52
3.4 Fundamental theorem of local martingales . . . . . . . . . . . . . . . 54
3.5 Quasimartingales, compensators, and the fundamental theorem of
local martingales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.6 Special semimartingales and another decomposition theorem for
local martingales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.7 Girsanov’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4 General stochastic integration 65
4.1 Stochastic integrals with respect to predictable processes . . . . . . 65
Index 72