| |
| |
| Preface | |
| |
| |
| |
| Preliminaries From Calculus | |
| |
| |
| |
| Functions in Calculus | |
| |
| |
| |
| Variation of a Function | |
| |
| |
| |
| Riemann Integral and Stieltjes Integral | |
| |
| |
| |
| Lebesgue's Method of Integration | |
| |
| |
| |
| Differentials and Integrals | |
| |
| |
| |
| Taylor's Formula and Other Results | |
| |
| |
| |
| Concepts of Probability Theory | |
| |
| |
| |
| Discrete Probability Model | |
| |
| |
| |
| Continuous Probability Model | |
| |
| |
| |
| Expectation and Lebesgue Integral | |
| |
| |
| |
| Transforms and Convergence | |
| |
| |
| |
| Independence and Covariance | |
| |
| |
| |
| Normal (Gaussian) Distributions | |
| |
| |
| |
| Conditional Expectation | |
| |
| |
| |
| Stochastic Processes in Continuous Time | |
| |
| |
| |
| Basic Stochastic Processes | |
| |
| |
| |
| Brownian Motion | |
| |
| |
| |
| Properties of Brownian Motion Paths | |
| |
| |
| |
| Three Martingales of Brownian Motion | |
| |
| |
| |
| Markov Property of Brownian Motion | |
| |
| |
| |
| Hitting Times and Exit Times | |
| |
| |
| |
| Maximum and Minimum of Brownian Motion | |
| |
| |
| |
| Distribution of Hitting Times | |
| |
| |
| |
| Reflection Principle and Joint Distributions | |
| |
| |
| |
| Zeros of Brownian Motion - Arcsine Law | |
| |
| |
| |
| Size of Increments of Brownian Motion | |
| |
| |
| |
| Brownian Motion in Higher Dimensions | |
| |
| |
| |
| Random Walk | |
| |
| |
| |
| Stochastic Integral in Discrete Time | |
| |
| |
| |
| Poisson Process | |
| |
| |
| |
| Exercises | |
| |
| |
| |
| Brownian Motion Calculus | |
| |
| |
| |
| Definition of Ito Integral | |
| |
| |
| |
| Ito Integral Process | |
| |
| |
| |
| Ito Integral and Gaussian Processes | |
| |
| |
| |
| Ito's Formula for Brownian Motion | |
| |
| |
| |
| Ito Processes and Stochastic Differentials | |
| |
| |
| |
| Ito's Formula for Ito Processes | |
| |
| |
| |
| Ito Processes in Higher Dimensions | |
| |
| |
| |
| Exercises | |
| |
| |
| |
| Stochastic Differential Equations | |
| |
| |
| |
| Definition of Stochastic Differential Equations (SDEs) | |
| |
| |
| |
| Stochastic Exponential and Logarithm | |
| |
| |
| |
| Solutions to Linear SDEs | |
| |
| |
| |
| Existence and Uniqueness of Strong Solutions | |
| |
| |
| |
| Markov Property of Solutions | |
| |
| |
| |
| Weak Solutions to SDEs | |
| |
| |
| |
| Construction of Weak Solutions | |
| |
| |
| |
| Backward and Forward Equations | |
| |
| |
| |
| Stratonovich Stochastic Calculus | |
| |
| |
| |
| Exercises | |
| |
| |
| |
| Diffusion Processes | |
| |
| |
| |
| Martingales and Dynkin's Formula | |
| |
| |
| |
| Calculation of Expectations and PDEs | |
| |
| |
| |
| Time-Homogeneous Diffusions | |
| |
| |
| |
| Exit Times from an Interval | |
| |
| |
| |
| Representation of Solutions of ODES | |
| |
| |
| |
| Explosion | |
| |
| |
| |
| Recurrence and Transience | |
| |
| |
| |
| Diffusion on an Interval | |
| |
| |
| |
| Stationary Distributions | |
| |
| |
| |
| Multi-dimensional SDEs | |
| |
| |
| |
| Exercises | |
| |
| |
| |
| Martingales | |
| |
| |
| |
| Definitions | |
| |
| |
| |
| Uniform Integrability | |
| |
| |
| |
| Martingale Convergence | |
| |
| |
| |
| Optional Stopping | |
| |
| |
| |
| Localization and Local Martingales | |
| |
| |
| |
| Quadratic Variation of Martingales | |
| |
| |
| |
| Martingale Inequalities | |
| |
| |
| |
| Continuous Martingales - Change of Time | |
| |
| |
| |
| Exercises | |
| |
| |
| |
| Calculus For Semimartingales | |
| |
| |
| |
| Semimartingales | |
| |
| |
| |
| Predictable Processes | |
| |
| |
| |
| Doob-Meyer Decomposition | |
| |
| |
| |
| Integrals with Respect to Semimartingales | |
| |
| |
| |
| Quadratic Variation and Covariation | |
| |
| |
| |
| Ito's Formula for Continuous Semimartingales | |
| |
| |
| |
| Local Times | |
| |
| |
| |
| Stochastic Exponential | |
| |
| |
| |
| Compensators and Sharp Bracket Process | |
| |
| |
| |
| It�'s Formula for Semimartingales | |
| |
| |
| |
| Stochastic Exponential and Logarithm | |
| |
| |
| |
| Martingale (Predictable) Representations | |
| |
| |
| |
| Elements of the General Theory | |
| |
| |
| |
| Random Measures and Canonical Decomposition | |
| |
| |
| |
| Exercises | |
| |
| |
| |
| Pure Jump Processes | |
| |
| |
| |
| Definitions | |
| |
| |
| |
| Pure Jump Process Filtration | |
| |
| |
| |
| Ito's Formula for Processes of Finite Variation | |
| |
| |
| |
| Counting Processes | |
| |
| |
| |
| Markov Jump Processes | |
| |
| |
| |
| Stochastic Equation for Jump Processes | |
| |
| |
| |
| Generators and Dynkin's Formula | |
| |
| |
| |
| Explosions in Markov Jump Processes | |
| |
| |
| |
| Exercises | |
| |
| |
| |
| Change of Probability Measure | |
| |
| |
| |
| Change of Measure for Random Variables | |
| |
| |
| |
| Change of Measure on a General Space | |
| |
| |
| |
| Change of Measure for Processes | |
| |
| |
| |
| Change of Wiener Measure | |
| |
| |
| |
| Change of Measure for Point Processes | |
| |
| |
| |
| Likelihood Functions | |
| |
| |
| |
| Exercises | |
| |
| |
| |
| Applications in Finance: Stock and FX Options | |
| |
| |
| |
| Financial Derivatives and Arbitrage | |
| |
| |
| |
| A Finite Market Model | |
| |
| |
| |
| Semimartingale Market Model | |
| |
| |
| |
| Diffusion and the Black-Scholes Model | |
| |
| |
| |
| Change of Numeraire | |
| |
| |
| |
| Currency (FX) Options | |
| |
| |
| |
| Asian, Lookback, and Barrier Options | |
| |
| |
| |
| Exercises | |
| |
| |
| |
| Applications in Finance: Bonds, Rates, and Options | |
| |
| |
| |
| Bonds and the Yield Curve | |
| |
| |
| |
| Models Adapted to Brownian Motion | |
| |
| |
| |
| Models Based on the Spot Rate | |
| |
| |
| |
| Merton's Model and Vasicek's Model | |
| |
| |
| |
| Heath-Jarrow-Morton (HJM) Model | |
| |
| |
| |
| Forward Measures - Bond as a Numeraire | |
| |
| |
| |
| Options, Caps, and Floors | |
| |
| |
| |
| Brace-Gatarek-Musiela (BGM) Model | |
| |
| |
| |
| Swaps and Swaptions | |
| |
| |
| |
| Exercises | |
| |
| |
| |
| Applications in Biology | |
| |
| |
| |
| Feller's Branching Diffusion | |
| |
| |
| |
| Wright-Fisher Diffusion | |
| |
| |
| |
| Birth-Death Processes | |
| |
| |
| |
| Growth of Birth-Death Processes | |
| |
| |
| |
| Extinction, Probability, and Time to Exit | |
| |
| |
| |
| Processes in Genetics | |
| |
| |
| |
| Birth-Death Processes in Many Dimensions | |
| |
| |
| |
| Cancer Models | |
| |
| |
| |
| Branching Processes | |
| |
| |
| |
| Stochastic Lotka-Volterra Model | |
| |
| |
| |
| Exercises | |
| |
| |
| |
| Applications in Engineering and Physics | |
| |
| |
| |
| Filtering | |
| |
| |
| |
| Random Oscillators | |
| |
| |
| |
| Exercises | |
| |
| |
| Solutions to Selected Exercises | |
| |
| |
| References | |
| |
| |
| Index | |