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Introduction to Stochastic Analysis: Integrals and Differential Equations

Book Description

This is an introduction to stochastic integration and stochastic differential equations written in an understandable way for a wide audience, from students of mathematics to practitioners in biology, chemistry, physics, and finances. The presentation is based on the naïve stochastic integration, rather than on abstract theories of measure and stochastic processes. The proofs are rather simple for practitioners and, at the same time, rather rigorous for mathematicians. Detailed application examples in natural sciences and finance are presented. Much attention is paid to simulation diffusion processes.

The topics covered include Brownian motion; motivation of stochastic models with Brownian motion; Itô and Stratonovich stochastic integrals, Itô's formula; stochastic differential equations (SDEs); solutions of SDEs as Markov processes; application examples in physical sciences and finance; simulation of solutions of SDEs (strong and weak approximations). Exercises with hints and/or solutions are also provided.

Table of Contents

  1. Cover
  2. Title Page
  3. Copyright
  4. Preface
  5. Notation
  6. Chapter 1: Introduction: Basic Notions of Probability Theory
    1. 1.1. Probability space
    2. 1.2. Random variables
    3. 1.3. Characteristics of a random variable
    4. 1.4. Types of random variables
    5. 1.5. Conditional probabilities and distributions
    6. 1.6. Conditional expectations as random variables
    7. 1.7. Independent events and random variables
    8. 1.8. Convergence of random variables
    9. 1.9. Cauchy criterion
    10. 1.10. Series of random variables
    11. 1.11. Lebesgue theorem
    12. 1.12. Fubini theorem
    13. 1.13. Random processes
    14. 1.14. Kolmogorov theorem
  7. Chapter 2: Brownian Motion
    1. 2.1. Definition and properties
    2. 2.2. White noise and Brownian motion
    3. 2.3. Exercises
  8. Chapter 3: Stochastic Models with Brownian Motion and White Noise
    1. 3.1. Discrete time
    2. 3.2. Continuous time
  9. Chapter 4: Stochastic Integral with Respect to Brownian Motion
    1. 4.1. Preliminaries. Stochastic integral with respect to a step process
    2. 4.2. Definition and properties
    3. 4.3. Extensions
    4. 4.4. Exercises
  10. Chapter 5: Itô's Formula
    1. 5.1. Exercises
  11. Chapter 6: Stochastic Differential Equations
    1. 6.1. Exercises
  12. Chapter 7: Itô Processes
    1. 7.1. Exercises
  13. Chapter 8: Stratonovich Integral and Equations
    1. 8.1. Exercises
  14. Chapter 9: Linear Stochastic Differential Equations
    1. 9.1. Explicit solution of a linear SDE
    2. 9.2. Expectation and variance of a solution of an LSDE
    3. 9.3. Other explicitly solvable equations
    4. 9.4. Stochastic exponential equation
    5. 9.5. Exercises
  15. Chapter 10: Solutions of SDEs as Markov Diffusion Processes
    1. 10.1. Introduction
    2. 10.2. Backward and forward Kolmogorov equations
    3. 10.3. Stationary density of a diffusion process
    4. 10.4. Exercises
  16. Chapter 11: Examples
    1. 11.1. Additive noise: Langevin equation
    2. 11.2. Additive noise: general case
    3. 11.3. Multiplicative noise: general remarks
    4. 11.4. Multiplicative noise: Verhulst equation
    5. 11.5. Multiplicative noise: genetic model
  17. Chapter 12: Example in Finance: Black–Scholes Model
    1. 12.1. Introduction: what is an option?
    2. 12.2. Self-financing strategies
      1. 12.2.1. Portfolio and its trading strategy
      2. 12.2.2. Self-financing strategies
      3. 12.2.3. Stock discount
    3. 12.3. Option pricing problem: the Black–Scholes model
    4. 12.4. Black–Scholes formula
    5. 12.5.  Risk-neutral probabilities: alternative derivation of Black–Scholes formula
    6. 12.6. Exercises
  18. Chapter 13: Numerical Solution of Stochastic Differential Equations
    1. 13.1. Memories of approximations of ordinary differential equations
    2. 13.2. Euler approximation
    3. 13.3. Higher-order strong approximations
    4. 13.4. First-order weak approximations
    5. 13.5. Higher-order weak approximations
    6. 13.6. Example: Milstein-type approximations
    7. 13.7. Example: Runge–Kutta approximations
    8. 13.8. Exercises
  19. Chapter 14: Elements of Multidimensional Stochastic Analysis
    1. 14.1. Multidimensional Brownian motion
    2. 14.2. Itô's formula for a multidimensional Brownian motion
    3. 14.3. Stochastic differential equations
    4. 14.4. Itô processes
    5. 14.5. Itô's formula for multidimensional Itô processes
    6. 14.6. Linear stochastic differential equations
    7. 14.7. Diffusion processes
    8. 14.8. Approximations of stochastic differential equations
  20. Solutions, Hints, and Answers
  21. Bibliography
  22. Index