1st Edition

# Stochastic Calculus A Practical Introduction

352 Pages
by CRC Press

This compact yet thorough text zeros in on the parts of the theory that are particularly relevant to applications . It begins with a description of Brownian motion and the associated stochastic calculus, including their relationship to partial differential equations. It solves stochastic differential equations by a variety of methods and studies in detail the one-dimensional case. The book concludes with a treatment of semigroups and generators, applying the theory of Harris chains to diffusions, and presenting a quick course in weak convergence of Markov chains to diffusions.

The presentation is unparalleled in its clarity and simplicity. Whether your students are interested in probability, analysis, differential geometry or applications in operations research, physics, finance, or the many other areas to which the subject applies, you'll find that this text brings together the material you need to effectively and efficiently impart the practical background they need.

CHAPTER 1. BROWNIAN MOTION
Definition and Construction
Markov Property, Blumenthal's 0-1 Law
Stopping Times, Strong Markov Property
First Formulas
CHAPTER 2. STOCHASTIC INTEGRATION
Integrands: Predictable Processes
Integrators: Continuous Local Martingales
Variance and Covariance Processes
Integration w.r.t. Bounded Martingales
The Kunita-Watanabe Inequality
Integration w.r.t. Local Martingales
Change of Variables, Itô's Formula
Integration w.r.t. Semimartingales
Associative Law
Functions of Several Semimartingales
Chapter Summary
Meyer-Tanaka Formula, Local Time
Girsanov's Formula
CHAPTER 3. BROWNIAN MOTION, II
Recurrence and Transience
Occupation Times
Exit Times
Change of Time, Lévy's Theorem
Burkholder Davis Gundy Inequalities
CHAPTER 4. PARTIAL DIFFERENTIAL EQUATIONS
A. Parabolic Equations
The Heat Equation
The Inhomogeneous Equation
The Feynman-Kac Formula
B. Elliptic Equations
The Dirichlet Problem
Poisson's Equation
The Schrödinger Equation
C. Applications to Brownian Motion
Exit Distributions for the Ball
Occupation Times for the Ball
Laplace Transforms, Arcsine Law
CHAPTER 5. STOCHASTIC DIFFERENTIAL EQUATIONS
Examples
Itô's Approach
Extension
Weak Solutions
Change of Measure
Change of Time
CHAPTER 6. ONE DIMENSIONAL DIFFUSIONS
Construction
Feller's Test
Recurrence and Transience
Green's Functions
Boundary Behavior
Applications to Higher Dimensions
CHAPTER 7. DIFFUSIONS AS MARKOV PROCESSES
Semigroups and Generators
Examples
Transition Probabilities
Harris Chains
Convergence Theorems
CHAPTER 8. WEAK CONVERGENCE
In Metric Spaces
Prokhorov's Theorems
The Space C
Skorohod's Existence Theorem for SDE
Donsker's Theorem
The Space D
Convergence to Diffusions
Examples
Solutions to Exercises
References
Index

### Biography

Richard Durrett