## Introduction to Stochastic Calculus with ApplicationsThis book presents a concise and rigorous treatment of stochastic calculus and its applications. It gives a simple but rigorous treatment of the subject including a range of advanced topics, it is useful for practitioners who use advanced theoretical results. It covers advanced applications, such as models in mathematical finance, biology and engineering. Self-contained and unified in presentation, the book contains many solved examples and exercises. It may be used as a textbook by advanced undergraduates and graduate students in stochastic calculus and financial mathematics. It is also suitable for practitioners who wish to gain an understanding or working knowledge of the subject. For mathematicians, this book could be a first text on stochastic calculus; it is good companion to more advanced texts by a way of examples and exercises. For people from other fields, it provides a way to gain a working knowledge of stochastic calculus. It shows all readers the applications of stochastic calculus methods and takes readers to the technical level required in research and sophisticated modelling. This second edition contains a new chapter on bonds, interest rates and their options. New materials include more worked out examples in all chapters, best estimators, more results on change of time, change of measure, random measures, new results on exotic options, FX options, stochastic and implied volatility, models of the age-dependent branching process and the stochastic Lotka--Volterra model in biology, non-linear filtering in engineering and five new figures. |

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Its a very good introductory book about stochastic calculus esp. for continous part.

Brownian Motion Part was explained very in detail.

however, there are tons of typo in the book. You must read it very carefully and better discuss what you thought with your classmates.

### Contents

1 Preliminaries From Calculus | 1 |

2 Concepts of Probability Theory | 21 |

3 Basic Stochastic Processes | 55 |

4 Brownian Motion Calculus | 91 |

5 Stochastic Differential Equations | 123 |

6 Diffusion Processes | 149 |

7 Martingales | 183 |

8 Calculus For Semimartingales | 211 |

10 Change of Probability Measure | 267 |

Stock and FX Options | 287 |

Bonds Rates and Options | 323 |

13 Applications in Biology | 351 |

14 Applications in Engineering and Physics | 375 |

Solutions to Selected Exercises | 391 |

407 | |

413 | |

### Common terms and phrases

adapted process arbitrage Black-Scholes bond bounded Brownian motion called conditional distribution conditional expectation constant continuous function continuous martingale converges Corollary covariation dB(t deﬁned Deﬁnition denotes density derivative diﬀerent differential diﬀusion dX(t Example Exercise exists ﬁeld ﬁltration ﬁnd ﬁnite variation ﬁrst given implies increments independent inﬁnite integral with respect interval Itˆo Itˆo’s Ito integral Ito's formula jump process Lebesgue Lebesgue measure Let X(t limit linear Lipschitz Markov property martingale martingale property measure Q non-random Normal numeraire obtain option partition Poisson process portfolio predictable process probability measure process X(t Proof Q-Brownian Q-martingale quadratic variation random variable representation result follows right-continuous satisﬁes semimartingale Show square integrable square integrable martingale Stochastic Calculus stochastic exponential stochastic integral stochastic process stopping submartingale taking Theorem twice continuously uniformly integrable unique variation process vector weak solution