This compact yet thorough text zeros in on the parts of the theory that are useful for applications to mathematical finance, queuing theory, biology, and physics. The book 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 by treating semigroups and generators, applying the theory of Harris chains to diffusions, and presenting a quick course in weak convergence of Markov chains to diffusions.

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, Ito'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, Levy's Theorem Burkholder Davis Gundy Inequalities Martingales Adapted to Brownian Filtrations 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 Schrodinger 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 Ito'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