Chapter I Martingale Theory
Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1. Convergence of Random Variables . . . . . . . . . . . . . . . . . . 2
1.a Forms of convergence . . . . . . . . . . . . . . . . . . . . . . 2
1.b Norm convergence and uniform integrability . . . . . . . . . . . 3
2. Conditioning . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.a Sigma fields, information and conditional expectation . . . . . . . 8
2.b Conditional expectation . . . . . . . . . . . . . . . . . . . . 10
3. Submartingales . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.a Adapted stochastic processes . . . . . . . . . . . . . . . . . . 19
3.b Sampling at optional times . . . . . . . . . . . . . . . . . . . 22
3.c Application to the gambler’s ruin problem . . . . . . . . . . . . 25
4. Convergence Theorems . . . . . . . . . . . . . . . . . . . . . . . 29
4.a Upcrossings . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4.b Reversed submartingales . . . . . . . . . . . . . . . . . . . . 34
4.c Levi’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . 36
4.d Strong Law of Large Numbers . . . . . . . . . . . . . . . . . . 38
5. Optional Sampling of Closed Submartingale Sequences . . . . . . . . 42
5.a Uniform integrability, last elements, closure . . . . . . . . . . . . 42
5.b Sampling of closed submartingale sequences . . . . . . . . . . . . 44
6. Maximal Inequalities for Submartingale Sequences . . . . . . . . . . 47
6.a Expectations as Lebesgue integrals . . . . . . . . . . . . . . . . 47
6.b Maximal inequalities for submartingale sequences . . . . . . . . . 47
7. Continuous Time Martingales . . . . . . . . . . . . . . . . . . . . 50
7.a Filtration, optional times, sampling . . . . . . . . . . . . . . . 50
7.b Pathwise continuity . . . . . . . . . . . . . . . . . . . . . . 56
7.c Convergence theorems . . . . . . . . . . . . . . . . . . . . . 59
7.d Optional sampling theorem . . . . . . . . . . . . . . . . . . . 62
7.e Continuous time Lp-inequalities . . . . . . . . . . . . . . . . . 64
8. Local Martingales . . . . . . . . . . . . . . . . . . . . . . . . . 65
8.a Localization . . . . . . . . . . . . . . . . . . . . . . . . . . 65
8.b Bayes Theorem . . . . . . . . . . . . . . . . . . . . . . . . 71

9. Quadratic Variation . . . . . . . . . . . . . . . . . . . . . . . . 73
9.a Square integrable martingales . . . . . . . . . . . . . . . . . . 73
9.b Quadratic variation . . . . . . . . . . . . . . . . . . . . . . 74
9.c Quadratic variation and L2-bounded martingales . . . . . . . . . 86
9.d Quadratic variation and L1-bounded martingales . . . . . . . . . 88
10. The Covariation Process . . . . . . . . . . . . . . . . . . . . . . 90
10.a Definition and elementary properties . . . . . . . . . . . . . . 90
10.b Integration with respect to continuous bounded variation processes . 91
10.c Kunita-Watanabe inequality . . . . . . . . . . . . . . . . . . 94
11. Semimartingales . . . . . . . . . . . . . . . . . . . . . . . . . 98
11.a Definition and basic properties . . . . . . . . . . . . . . . . . 98
11.b Quadratic variation and covariation . . . . . . . . . . . . . . . 99
Chapter II Brownian Motion
1. Gaussian Processes . . . . . . . . . . . . . . . . . . . . . . . . 103
1.a Gaussian random variables in Rk . . . . . . . . . . . . . . . 103
1.b Gaussian processes . . . . . . . . . . . . . . . . . . . . . . 109
1.c Isonormal processes . . . . . . . . . . . . . . . . . . . . . 111
2. One Dimensional Brownian Motion . . . . . . . . . . . . . . . . 112
2.a One dimensional Brownian motion starting at zero . . . . . . . . 112
2.b Pathspace and Wiener measure . . . . . . . . . . . . . . . . 116
2.c The measures Px . . . . . . . . . . . . . . . . . . . . . . 118
2.d Brownian motion in higher dimensions . . . . . . . . . . . . . 118
2.e Markovprop erty . . . . . . . . . . . . . . . . . . . . . . . 120
2.f The augmented filtration (Ft) . . . . . . . . . . . . . . . . . 127
2.g Miscellaneous properties . . . . . . . . . . . . . . . . . . . 128
Chapter III Stochastic Integration
1. Measurability Properties of Stochastic Processes . . . . . . . . . . 131
1.a The progressive and predictable σ-fields on Π . . . . . . . . . . 131
1.b Stochastic intervals and the optional σ-field . . . . . . . . . . . 134
2. Stochastic Integration with Respect to Continuous Semimartingales . . 135
2.a Integration with respect to continuous local martingales . . . . . 135
2.b M-integrable processes . . . . . . . . . . . . . . . . . . . . 140
2.c Properties of stochastic integrals with respect to continuous
local martingales . . . . . . . . . . . . . . . . . . . . . . 142
2.d Integration with respect to continuous semimartingales . . . . . . 147
2.e The stochastic integral as a limit of certain Riemann type sums . . 150
2.f Integration with respect to vector valued continuous semimartingales 153

3. Ito’s Formula . . . . . . . . . . . . . . . . . . . . . . . . . . 157
3.a Ito’s formula . . . . . . . . . . . . . . . . . . . . . . . . 157
3.b Differential notation . . . . . . . . . . . . . . . . . . . . . 160
3.c Consequences of Ito’s formula . . . . . . . . . . . . . . . . . 161
3.d Stock prices . . . . . . . . . . . . . . . . . . . . . . . . . 165
3.e Levi’s characterization of Brownian motion . . . . . . . . . . . 166
3.f The multiplicative compensator UX . . . . . . . . . . . . . . 168
3.g Harmonic functions of Brownian motion . . . . . . . . . . . . 169
4. Change of Measure . . . . . . . . . . . . . . . . . . . . . . . 170
4.a Locally equivalent change of probability . . . . . . . . . . . . 170
4.b The exponential local martingale . . . . . . . . . . . . . . . 173
4.c Girsanov’s theorem . . . . . . . . . . . . . . . . . . . . . 175
4.d The Novikov condition . . . . . . . . . . . . . . . . . . . . 180
5. Representation of Continuous Local Martingales . . . . . . . . . . 183
5.a Time change for continuous local martingales . . . . . . . . . . 183
5.b Brownian functionals as stochastic integrals . . . . . . . . . . . 187
5.c Integral representation of square integrable Brownian martingales . 192
5.d Integral representation of Brownian local martingales . . . . . . 195
5.e Representation of positive Brownian martingales . . . . . . . . . 196
5.f Kunita-Watanabe decomposition . . . . . . . . . . . . . . . . 196
6. Miscellaneous . . . . . . . . . . . . . . . . . . . . . . . . . . 200
6.a Ito processes . . . . . . . . . . . . . . . . . . . . . . . . 200
6.b Volatilities . . . . . . . . . . . . . . . . . . . . . . . . . 203
6.c Call option lemmas . . . . . . . . . . . . . . . . . . . . . 205
6.d Log-Gaussian processes . . . . . . . . . . . . . . . . . . . . 208
6.e Processes with finite time horizon . . . . . . . . . . . . . . . 209
Chapter IV Application to Finance
1. The Simple Black Scholes Market . . . . . . . . . . . . . . . . . 211
1.aThemodel . . . . . . . . . . . . . . . . . . . . . . . . . 211
1.b Equivalent martingale measure . . . . . . . . . . . . . . . . 212
1.c Trading strategies and absence of arbitrage . . . . . . . . . . . 213
2. Pricing of Contingent Claims . . . . . . . . . . . . . . . . . . . 218
2.a Replication of contingent claims . . . . . . . . . . . . . . . . 218
2.b Derivatives of the form h = f(ST) . . . . . . . . . . . . . . . 221
2.c Derivatives of securities paying dividends . . . . . . . . . . . . 225
3. The General Market Model . . . . . . . . . . . . . . . . . . . . 228
3.a Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . 228
3.b Markets and trading strategies . . . . . . . . . . . . . . . . 229

3.c Deflators . . . . . . . . . . . . . . . . . . . . . . . . . . 232
3.d Numeraires and associated equivalent probabilities . . . . . . . . 235
3.e Absence of arbitrage and existence of a local spot
martingale measure . . . . . . . . . . . . . . . . . . . . . 238
3.f Zero coupon bonds and interest rates . . . . . . . . . . . . . . 243
3.g General Black Scholes model and market price of risk . . . . . . 246
4. Pricing of Random Payoffs at Fixed Future Dates . . . . . . . . . . 251
4.a European options . . . . . . . . . . . . . . . . . . . . . . 251
4.b Forward contracts and forward prices . . . . . . . . . . . . . 254
4.c Option to exchange assets . . . . . . . . . . . . . . . . . . . 254
4.d Valuation of non-path-dependent options in Gaussian models . . . 259
4.e Delta hedging . . . . . . . . . . . . . . . . . . . . . . . . 265
4.f Connection with partial differential equations . . . . . . . . . . 267
5. Interest Rate Derivatives . . . . . . . . . . . . . . . . . . . . . 276
5.a Floating and fixed rate bonds . . . . . . . . . . . . . . . . . 276
5.b Interest rate swaps . . . . . . . . . . . . . . . . . . . . . . 277
5.c Swaptions . . . . . . . . . . . . . . . . . . . . . . . . . . 278
5.d Interest rate caps and floors . . . . . . . . . . . . . . . . . . 280
5.e Dynamics of the Libor process . . . . . . . . . . . . . . . . . 281
5.f Libor models with prescribed volatilities . . . . . . . . . . . . 282
5.g Cap valuation in the log-Gaussian Libor model . . . . . . . . . 285
5.h Dynamics of forward swap rates . . . . . . . . . . . . . . . . 286
5.i Swap rate models with prescribed volatilities . . . . . . . . . . 288
5.j Valuation of swaptions in the log-Gaussian swap rate model . . . . 291
5.k Replication of claims . . . . . . . . . . . . . . . . . . . . . 292
Appendix
A. Separation of convex sets . . . . . . . . . . . . . . . . . . . 297
B. The basic extension procedure . . . . . . . . . . . . . . . . . 299
C. Positive semidefinite matrices . . . . . . . . . . . . . . . . . 305
D. Kolmogoroff existence theorem . . . . . . . . . . . . . . . . . 306