Stochastic Calculus: An Introduction with
                 Applications
                Gregory F. Lawler
               ©2014Gregory F. Lawler
                All rights reserved
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                    Contents
                    1 Martingales in discrete time                                                      3
                        1.1   Conditional expectation      . . . . . . . . . . . . . . . . . . . . .    3
                        1.2   Martingales . . . . . . . . . . . . . . . . . . . . . . . . . . . .     10
                        1.3   Optional sampling theorem . . . . . . . . . . . . . . . . . . . .       14
                        1.4   Martingale convergence theorem . . . . . . . . . . . . . . . . .        19
                        1.5   Square integrable martingales . . . . . . . . . . . . . . . . . .       24
                        1.6   Integrals with respect to random walk        . . . . . . . . . . . . .  26
                        1.7   Amaximal inequality . . . . . . . . . . . . . . . . . . . . . . .       27
                        1.8   Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .   28
                    2 Brownian motion                                                                 35
                        2.1   Limits of sums of independent variables . . . . . . . . . . . . .       35
                        2.2   Multivariate normal distribution . . . . . . . . . . . . . . . . .      38
                        2.3   Limits of random walks . . . . . . . . . . . . . . . . . . . . . .      42
                        2.4   Brownian motion . . . . . . . . . . . . . . . . . . . . . . . . .       43
                        2.5   Construction of Brownian motion . . . . . . . . . . . . . . . .         46
                        2.6   Understanding Brownian motion . . . . . . . . . . . . . . . . .         51
                              2.6.1   Brownian motion as a continuous martingale . . . . . .          54
                              2.6.2   Brownian motion as a Markov process . . . . . . . . . 56
                              2.6.3   Brownian motion as a Gaussian process . . . . . . . . .         57
                              2.6.4   Brownian motion as a self-similar process . . . . . . . .       58
                        2.7   Computations for Brownian motion . . . . . . . . . . . . . . .          58
                        2.8   Quadratic variation . . . . . . . . . . . . . . . . . . . . . . . .     63
                        2.9   Multidimensional Brownian motion . . . . . . . . . . . . . . .          66
                        2.10 Heat equation and generator . . . . . . . . . . . . . . . . . . .        68
                              2.10.1 One dimension . . . . . . . . . . . . . . . . . . . . . .        68
                              2.10.2 Expected value at a future time . . . . . . . . . . . . .        74
                        2.11 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .    78
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                              iv                                                                    CONTENTS
                              3 Stochastic integration                                                          83
                                  3.1   What is stochastic calculus? . . . . . . . . . . . . . . . . . . .      83
                                  3.2   Stochastic integral    . . . . . . . . . . . . . . . . . . . . . . . .  85
                                        3.2.1   Review of Riemann integration . . . . . . . . . . . . .         85
                                        3.2.2   Integration of simple processes . . . . . . . . . . . . . .     86
                                        3.2.3   Integration of continuous processes . . . . . . . . . . .       89
                                  3.3   Itˆo’s formula . . . . . . . . . . . . . . . . . . . . . . . . . . . .  99
                                  3.4   More versions of Itˆo’s formula . . . . . . . . . . . . . . . . . . 105
                                  3.5   Diffusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
                                  3.6   Covariation and the product rule       . . . . . . . . . . . . . . . . 116
                                  3.7   Several Brownian motions . . . . . . . . . . . . . . . . . . . . 117
                                  3.8   Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
                              4 More stochastic calculus                                                      125
                                  4.1   Martingales and local martingales . . . . . . . . . . . . . . . . 125
                                  4.2   An example: the Bessel process . . . . . . . . . . . . . . . . . 131
                                  4.3   Feynman-Kac formula . . . . . . . . . . . . . . . . . . . . . . 133
                                  4.4   Binomial approximations . . . . . . . . . . . . . . . . . . . . . 137
                                  4.5   Continuous martingales . . . . . . . . . . . . . . . . . . . . . . 141
                                  4.6   Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
                              5 Change of measure and Girsanov theorem                                        145
                                  5.1   Absolutely continuous measures . . . . . . . . . . . . . . . . . 145
                                  5.2   Giving drift to a Brownian motion        . . . . . . . . . . . . . . . 150
                                  5.3   Girsanov theorem . . . . . . . . . . . . . . . . . . . . . . . . . 153
                                  5.4   Black-Scholes formula . . . . . . . . . . . . . . . . . . . . . . . 162
                                  5.5   Martingale approach to Black-Scholes equation . . . . . . . . . 166
                                  5.6   Martingale approach to pricing       . . . . . . . . . . . . . . . . . 169
                                  5.7   Martingale representation theorem        . . . . . . . . . . . . . . . 178
                                  5.8   Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
                              6 Jump processes                                                                185
                                  6.1   L´evy processes . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
                                  6.2   Poisson process . . . . . . . . . . . . . . . . . . . . . . . . . . 188
                                  6.3   Compound Poisson process . . . . . . . . . . . . . . . . . . . . 192
                                  6.4   Integration with respect to compound Poisson processes . . . . 200
                                  6.5   Change of measure . . . . . . . . . . . . . . . . . . . . . . . . 205
                                  6.6   Generalized Poisson processes I . . . . . . . . . . . . . . . . . 206