Stochastic Calculus A Practical Introduction

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 MOTIONDefinition and ConstructionMarkov Property, Blumenthal's 0-1 LawStopping Times, Strong Markov PropertyFirst FormulasCHAPTER 2. STOCHASTIC INTEGRATIONIntegrands: Predictable ProcessesIntegrators: Continuous Local MartingalesVariance and Covariance ProcessesIntegration w.r.t. Bounded MartingalesThe Kunita-Watanabe InequalityIntegration w.r.t. Local MartingalesChange of Variables, Itô's FormulaIntegration w.r.t. SemimartingalesAssociative LawFunctions of Several SemimartingalesChapter SummaryMeyer-Tanaka Formula, Local TimeGirsanov's FormulaCHAPTER 3. BROWNIAN MOTION, IIRecurrence and TransienceOccupation TimesExit TimesChange of Time, Lévy's TheoremBurkholder Davis Gundy InequalitiesMartingales Adapted to Brownian FiltrationsCHAPTER 4. PARTIAL DIFFERENTIAL EQUATIONSA. Parabolic EquationsThe Heat EquationThe Inhomogeneous EquationThe Feynman-Kac FormulaB. Elliptic EquationsThe Dirichlet ProblemPoisson's EquationThe Schrödinger EquationC. Applications to Brownian MotionExit Distributions for the BallOccupation Times for the BallLaplace Transforms, Arcsine LawCHAPTER 5. STOCHASTIC DIFFERENTIAL EQUATIONSExamplesItô's ApproachExtensionWeak SolutionsChange of MeasureChange of TimeCHAPTER 6. ONE DIMENSIONAL DIFFUSIONSConstructionFeller's TestRecurrence and TransienceGreen's FunctionsBoundary BehaviorApplications to Higher DimensionsCHAPTER 7. DIFFUSIONS AS MARKOV PROCESSESSemigroups and GeneratorsExamplesTransition ProbabilitiesHarris ChainsConvergence TheoremsCHAPTER 8. WEAK CONVERGENCEIn Metric SpacesProkhorov's TheoremsThe Space CSkorohod's Existence Theorem for SDEDonsker's TheoremThe Space DConvergence to DiffusionsExamplesSolutions to ExercisesReferencesIndex