Introduction to Stochastic Calculus With Applications

Transcription

1 Introduction to Stochastic Calculus With Applications Fima C Klebaner University of Melbourne \ Imperial College Press

2 Contents Preliminaries From Calculus Continuous and Differentiable Functions Right and Left-Continuous Functions Variation of a Function Riemann Integral Stieltjes Integral Differentials and Integrals Taylor's Formula and other results 16 Concepts of Probability Theory Discrete Probability Model Continuous Probability Model Expectation and Lebesgue Integral Transforms and Convergence Independence and Conditioning Stochastic Processes in Continuous Time 46 Basic Stochastic Processes Brownian Motion Brownian Motion as a Gaussian Process Properties of Brownian Motion Paths Three Martingales of Brownian Motion Markov Property of Brownian Motion Exit Times and Hitting 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 78

3 CONTENTS 3.13 Random Walk Stochastic Integral in Discrete Time Poisson Process Exercises 85 Brownian Motion Calculus Definition of Ito Integral Ito integral process Ito's Formula for Brownian motion Stochastic Differentials and Ito Processes Ito's formula for functions of two variables Stochastic Exponential Ill 4.7 Ito Processes in Higher Dimensions Exercises 114 Stochastic Differential Equations Definition of Stochastic Differential Equations Strong Solutions to SDE's Solutions to Linear SDE's Existence and Uniqueness of Strong Solutions Markov Property of Solutions Weak Solutions to SDE's Existence and Uniqueness of Weak Solutions Backward and Forward Equations Exercises 137 Diffusion Processes Martingales and Dynkin's formula Calculation of Expectations and PDE's Homogeneous Diffusions Exit Times From an Interval...., Representation of Solutions of PDE's Explosion Recurrence and Transience Diffusion on an Interval Stationary Distributions Multidimensional SDE's Exercises 167

4 CONTENTS vii 7 Martingales Definitions Uniform Integrability Martingale Convergence Optional Stopping Localization. Local Martingales Quadratic Variation of Martingales Martingale Inequalities Continuous martingales Change of Time in SDE's Martingale Representations Exercises Calculus For Semimartingales Semimartingales Quadratic Variation and Covariation Predictable Processes Doob-Meyer Decomposition Definition of Stochastic Integral Properties of Stochastic Integrals Ito's Formula: continuous case Local Times Stochastic Exponential Compensators and Sharp Bracket Process Ito's Formula: general case Elements of the General Theory 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 Markov Jump Processes Explosions in Markov Jump Processes Exercises Change of Probability Measure Change of Measure for Random Variables Equivalent Probability Measures Change of Measure for Processes 240

5 viii CONTENTS 10.4 Change of Drift in Diffusion Change of Wiener Measure Change of Measure for Point Processes Likelihood Ratios Exercises Applications in Finance Financial Derivatives and Arbitrage A Finite Market Model Semimartingale Market Model Diffusion and Black-Scholes Model Interest Rates Models Options, Caps, Floors, Swaps and Swaptions Exercises Applications in Biology Branching Diffusion Wright-Fisher Diffusion Birth-Death Processes Exercises Applications in Engineering and Physics Filtering Stratanovich Calculus Random Oscillators Exercises 313 References 315