Stochastic Calculus: An Introduction Through Theory and Exercises (Universitext)

Description

This book provides a comprehensive introduction to the theory of stochastic calculus and some of its applications. It is the only textbook on the subject to include more than two hundred exercises with complete solutions. After explaining the basic elements of probability, the author introduces more advanced topics such as Brownian motion, martingales and Markov processes. The core of the book covers stochastic calculus, including stochastic differential equations, the relationship to partial differential equations, numerical methods and simulation, as well as applications of stochastic processes to finance. The final chapter provides detailed solutions to all exercises, in some cases presenting various solution techniques together with a discussion of advantages and drawbacks of the methods used. Stochastic Calculus will be particularly useful to advanced undergraduate and graduate students wishing to acquire a solid understanding of the subject through the theory and exercises. Including full mathematical statements and rigorous proofs, this book is completely self-contained and suitable for lecture courses as well as self-study. Preface Contents Common Notations Real, complex numbers, Rm Derivatives Functional spaces 1 Elements of Probability 1.1 Probability spaces, random variables 1.2 Variance, covariance, law of a r.v. 1.3 Independence, product measure 1.4 Probabilities on Rm 1.5 Convergence of probabilities and random variables 1.6 Characteristic functions 1.7 Gaussian laws 1.8 Simulation 1.9 Measure-theoretic arguments Exercises 2 Stochastic Processes 2.1 General facts 2.2 Kolmogorov's continuity theorem 2.3 Construction of stochastic processes 2.4 Next… Exercises 3 Brownian Motion 3.1 Definition and general facts 3.2 The law of a continuous process, Wiener measure 3.3 Regularity of the paths 3.4 Asymptotics 3.5 Stopping times 3.6 The stopping theorem 3.7 The simulation of Brownian motion Exercises 4 Conditional Probability 4.1 Conditioning 4.2 Conditional expectations 4.3 Conditional laws 4.4 Conditional laws of Gaussian vectors 4.5 The augmented Brownian filtration Exercises 5 Martingales 5.1 Definitions and general facts 5.2 Discrete time martingales 5.3 Discrete time martingales: a.s. convergence 5.4 Doob's inequality; Lp convergence, the p>1 case 5.5 Uniform integrability and convergence in L1 5.6 Continuous time martingales 5.7 Complements: the Laplace transform Exercises 6 Markov Processes 6.1 Definitions and general facts 6.2 The Feller and strong Markov properties 6.3 Semigroups, generators, diffusions Exercises 7 The Stochastic Integral 7.1 Introduction 7.2 Elementary processes 7.3 The stochastic integral 7.4 The martingale property 7.5 The stochastic integral in M2loc 7.6 Local martingales Exercises 8 Stochastic Calculus 8.1 Ito's formula 8.2 Application: exponential martingales 8.3 Application: Lp estimates 8.4 The multidimensional stochastic integral 8.5 *A case study: recurrence of multidimensional Brownian motion Exercises 9 Stochastic Differential Equations 9.1 Definitions 9.2 Examples 9.3 An a priori estimate 9.4 Existence for Lipschitz continuous coefficients 9.5 Localization and existence for locally Lipschitz coefficients 9.6 Uniqueness in law 9.7 The Markov property 9.8 Lp bounds and dependence on the initial data 9.9 The square root of a matrix field and the problemof diffusions 9.10 Further reading Exercises 10 PDE Problems and Diffusions 10.1 Representation of the solutions of a PDE problem 10.2 The Dirichlet problem 10.3 Parabolic equations 10.4 The Feynman–Kac formula 10.5 The density of the transition function, the backwardequation 10.6 Construction of the solutions of the Dirichlet problem Exercises 11 *Simulation 11.1 Numerical approximations of an SDE 11.2 Strong approximation 11.3 Weak approximation 11.4 Simulation and PDEs 11.5 Other schemes 11.6 Practical remarks Exercises 12 Back to Stochastic Calculus 12.1 Girsanov's theorem 12.2 The Cameron–Martin formula 12.3 The martingales of the Brownian filtration 12.4 Equivalent probability measures Exercises 13 An Application: Finance 13.1 Stocks and options 13.2 Trading strategies and arbitrage 13.3 Equivalent martingale measures 13.4 Replicating strategies and market completeness 13.5 The generalized Black–Scholes models 13.6 Pricing and hedging in the generalized Black–Scholes model 13.7 The standard Black–Scholes model Exercises Solutions of the Exercises References Index