Stochastic Calculus and Applications to Finance (SCAF)
                         ´  ´
                      Pierre ETORE
                 Master 2 MSIAM, track Data Science
                      Year 2019-2020
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                   Contents
                   1 Stochastic processes and Brownian motion                                                                          5
                       1.1   Stochastic processes: general definitions and properties . . . . . . . . . . . . . . . . . . . .            5
                       1.2   Markov processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .         7
                       1.3   Continuous time martingales: first definitions . . . . . . . . . . . . . . . . . . . . . . . . .             8
                       1.4   Afundamental stochastic process: the Brownian motion . . . . . . . . . . . . . . . . . . .                 9
                   2 Processes of finite variation and quadratic variation of martingales                                              17
                       2.1   Functions of finite variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .       17
                       2.2   Processes of finite variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .       18
                       2.3   Quadratic variation of martingales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .         20
                   3 Stochastic integration and Itˆo formula                                                                          23
                       3.1   Stochastic integration     . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .   23
                       3.2   Itˆo formula   . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .   27
                   4 L´evy and Girsanov theorems                                                                                      31
                       4.1   Exponential martingale and theorem of L´evy . . . . . . . . . . . . . . . . . . . . . . . . .            31
                       4.2   Girsanov theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .       32
                   5 Applications to Finance, Stochastic Differential Equations and link with Partial Dif-
                       ferential Equations                                                                                            37
                       5.1   Introduction and motivations, one-dimensional Black and Scholes model . . . . . . . . . .                37
                       5.2   Adigression on SDE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .         38
                       5.3   Self-financing portfolio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .      39
                       5.4   Risk -neutral probability measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .        40
                       5.5   Construction of the self financing replicating portfolio, pricing and hedging formulae, link
                             with PDE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .       41
                   6 Appendix                                                                                                         43
                       6.1   Functions of finite variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .       43
                   Bibliography                                                                                                       45
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