Stochastic Calculus A Practical Introduction [Hardcover]

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, It?'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, L?vy's Theorem
Burkholder Davis Gundy Inequalities
Martingales Adapted to Brownian Filtrations
CHAPTER 4. PARTIAL DIFFERENTIAL EQUATIONS
A. Parabolic El#<