## Stochastic Calculus: A Practical IntroductionThis 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. |

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### Contents

Brownian Motion | 1 |

Stochastic Integration | 33 |

Brownian Motion II | 95 |

Partial Differential Equations | 125 |

Stochastic Differential Equations | 177 |

One Dimensional Diffusions | 211 |

Diffusions as Markov Processes | 245 |

Weak Convergence | 271 |

Solutions to Exercises | 311 |

335 | |

### Common terms and phrases

apply assume assumption begin boundary bounded Brownian motion called Chapter Combining complete the proof compute conclusion condition consider constant construct continuous local martingale convergence theorem define definition derivative desired result differential diffusion dimensional distribution Durrett easy equality equation Example Exercise exists expected fact finite follows function given gives hence holds implies increasing independent inequality inf{t integral integrands Lemma limit local martingale mean measure o-field observe optional stopping theorem paths pick predictable probability problem Proof Proof Let prove prove the result reader recall Remark respect result right-hand side satisfies says semimartingale sequence shows solution solve space starting step stochastic suffices Suppose term trivial uniqueness write