Basics of Stochastic Analysis
Timo Sepp¨al¨ainen
Department of Mathematics, University of Wisconsin–Madison,
Madison, Wisconsin 53706
E-mail address: seppalai@math.wisc.edu

Contents
Chapter 1. Measures, Integrals, and Foundations of Probability
Theory 1
§1.1. Measure theory and integration 1
§1.2. Basic concepts of probability theory 19
Exercises 30
Chapter 2. Stochastic Processes 35
§2.1. Filtrations and stopping times 35
§2.2. Quadratic variation 44
§2.3. Path spaces, Markov and martingale property 50
§2.4. Brownian motion 52
§2.5. Poisson processes 64
Exercises 67
Chapter 3. Martingales 69
§3.1. Optional Stopping 71
§3.2. Inequalities 75
§3.3. Local martingales and semimartingales 78
§3.4. Quadratic variation for local martingales 79
§3.5. Doob-Meyer decomposition 84
§3.6. Spaces of martingales 86
Exercises 90
Chapter 4. Stochastic Integral with respect to Brownian Motion 91
iv Contents
Exercises 105
Chapter 5. Stochastic Integration of Predictable Processes 107
§5.1. Square-integrable martingale integrator 108
§5.2. Local square-integrable martingale integrator 135
§5.3. Semimartingale integrator 145
§5.4. Further properties of stochastic integrals 150
§5.5. Integrator with absolutely continuous Dol′eans measure 162
Exercises 167
Chapter 6. Itˆo’s formula 171
§6.1. Quadratic variation 171
§6.2. Itˆo’s formula 179
§6.3. Applications of Itˆo’s formula 187
Exercises 193
Chapter 7. Stochastic Differential Equations 195
§7.1. Examples of stochastic equations and solutions 196
§7.2. Existence and uniqueness for a semimartingale equation 203
§7.3. Proof of the existence and uniqueness theorem 208
Exercises 234
Appendix A. Analysis 235
§A.1. Continuous, cadlag and BV functions 236
§A.2. Differentiation and integration 247
Exercises 250
Appendix B. Probability 251
§B.1. General matters 251
§B.2. Construction of Brownian motion 257
Appendix. Bibliography 269
Appendix. Index 271