Introduction to Stochastic Calculus With Applications Fima C Klebaner University of Melbourne \ Imperial College Press
Contents Preliminaries From Calculus 1 1.1 Continuous and Differentiable Functions. 1 1.2 Right and Left-Continuous Functions 2 1.3 Variation of a Function 5 1.4 Riemann Integral 10 1.5 Stieltjes Integral 11 1.6 Differentials and Integrals 15 1.7 Taylor's Formula and other results 16 Concepts of Probability Theory 23 2.1 Discrete Probability Model 23 2.2 Continuous Probability Model 30 2.3 Expectation and Lebesgue Integral 35 2.4 Transforms and Convergence 39 2.5 Independence and Conditioning 40 2.6 Stochastic Processes in Continuous Time 46 Basic Stochastic Processes 55 3.1 Brownian Motion 56 3.2 Brownian Motion as a Gaussian Process 58 3.3 Properties of Brownian Motion Paths 61 3.4 - Three Martingales of Brownian Motion 63 3.5 Markov Property of Brownian Motion 65 3.6 Exit Times and Hitting Times 68 3.7 Maximum and Minimum of Brownian Motion 70 3.8 Distribution of Hitting Times 72 3.9 Reflection Principle and Joint Distributions 73 3.10 Zeros of Brownian Motion. Arcsine Law 74 3.11 Size of Increments of Brownian Motion 77 3.12 Brownian Motion in Higher Dimensions 78
CONTENTS 3.13 Random Walk 80 3.14 Stochastic Integral in Discrete Time 81 3.15 Poisson Process 82 3.16 Exercises 85 Brownian Motion Calculus 87 4.1 Definition of Ito Integral 87 4.2 Ito integral process 95 4.3 Ito's Formula for Brownian motion 99 4.4 Stochastic Differentials and Ito Processes 102 4.5 Ito's formula for functions of two variables 109 4.6 Stochastic Exponential Ill 4.7 Ito Processes in Higher Dimensions 112 4.8 Exercises 114 Stochastic Differential Equations 117 5.1 Definition of Stochastic Differential Equations 117 5.2 Strong Solutions to SDE's 120 5.3 Solutions to Linear SDE's 121 5.4 Existence and Uniqueness of Strong Solutions 125 5.5 Markov Property of Solutions 126 5.6 Weak Solutions to SDE's 128 5.7 Existence and Uniqueness of Weak Solutions 130 5.8 Backward and Forward Equations 134 5.9 Exercises 137 Diffusion Processes 139 6.1 Martingales and Dynkin's formula 139 6.2 Calculation of Expectations and PDE's 143 6.3 Homogeneous Diffusions 145 6.4. Exit Times From an Interval...., 148 6.5 Representation of Solutions of PDE's 152 6.6 Explosion 153 6.7 Recurrence and Transience. 155 6.8 Diffusion on an Interval 156 6.9 Stationary Distributions 157 6.10 Multidimensional SDE's 160 6.11 Exercises 167
CONTENTS vii 7 Martingales 169 7.1 Definitions 169 7.2 Uniform Integrability 171 7.3 Martingale Convergence 173 7.4 Optional Stopping 175 7.5 Localization. Local Martingales 177 7.6 Quadratic Variation of Martingales 180 7.7 Martingale Inequalities 182 7.8 Continuous martingales 184 7.9 Change of Time in SDE's 185 7.10 Martingale Representations 187 7.11 Exercises 189 8 Calculus For Semimartingales 191 8.1 Semimartingales 191 8.2 Quadratic Variation and Covariation 192 8.3 Predictable Processes 194 8.4 Doob-Meyer Decomposition 195 8.5 Definition of Stochastic Integral 196 8.6 Properties of Stochastic Integrals 199 8.7 Ito's Formula: continuous case 200 8.8 Local Times 202 8.9 Stochastic Exponential 203 8.10 Compensators and Sharp Bracket Process 207 8.11 Ito's Formula: general case 212 8.12 Elements of the General Theory 214 8.13 Exercises 217 9 Pure Jump Processes ' 219 9.1 Definitions 219 9.2 Pure Jump Process Filtration 220 9.3 Ito's Formula for Processes of Finite Variation 221 9.4 Counting Processes 222 9.5 Markov Jump Processes 229 9.6 Stochastic equation for Markov Jump Processes 231 9.7 Explosions in Markov Jump Processes 233 9.8 Exercises 234 10 Change of Probability Measure 237 10.1 Change of Measure for Random Variables 237 10.2 Equivalent Probability Measures 238 10.3 Change of Measure for Processes 240
viii CONTENTS 10.4 Change of Drift in Diffusion 243 10.5 Change of Wiener Measure 244 10.6 Change of Measure for Point Processes 245 10.7 Likelihood Ratios 247 10.8 Exercises 250 11 Applications in Finance 253 11.1 Financial Derivatives and Arbitrage 253 11.2 A Finite Market Model 256 11.3 Semimartingale Market Model 260 11.4 Diffusion and Black-Scholes Model 265 11.5 Interest Rates Models 273 11.6 Options, Caps, Floors, Swaps and Swaptions 281 11.7 Exercises 283 12 Applications in Biology 289 12.1 Branching Diffusion 289 12.2 Wright-Fisher Diffusion 292 12.3 Birth-Death Processes 293 12.4 Exercises 297 13 Applications in Engineering and Physics 299 13.1 Filtering 299 13.2 Stratanovich Calculus... 304 13.3 Random Oscillators 305 13.4 Exercises 313 References 315