Enhanced descriptions from Syndetics:

This book presents a concise and rigorous treatment of stochastic calculus. It also gives its main applications in finance, biology and engineering. In finance, the stochastic calculus is applied to pricing options by no arbitrage. In biology, it is applied to populations' models, and in engineering it is applied to filter signal from noise. Not everything is proved, but enough proofs are given to make it a mathematically rigorous exposition.This book aims to present the theory of stochastic calculus and its applications to an audience which possesses only a basic knowledge of calculus and probability. It may be used as a textbook by graduate and advanced undergraduate students in stochastic processes, financial mathematics and engineering. It is also suitable for researchers to gain working knowledge of the subject. It contains many solved examples and exercises making it suitable for self study.In the book many of the concepts are introduced through worked-out examples, eventually leading to a complete, rigorous statement of the general result, and either a complete proof, a partial proof or a reference. Using such structure, the text will provide a mathematically literate reader with rapid introduction to the subject and its advanced applications. The book covers models in mathematical finance, biology and engineering. For mathematicians, this book can be used as a first text on stochastic calculus or as a companion to more rigorous texts by a way of examples and exercises.

Table of contents provided by Syndetics

• Preface (p. xi)
• 1 Preliminaries From Calculus (p. 1)
• 1.1 Functions in Calculus (p. 1)
• 1.2 Variation of a Function (p. 4)
• 1.3 Riemann Integral and Stieltjes Integral (p. 9)
• 1.4 Lebesgue's Method of Integration (p. 14)
• 1.5 Differentials and Integrals (p. 14)
• 1.6 Taylor's Formula and Other Results (p. 15)
• 2 Concepts of Probability Theory (p. 21)
• 2.1 Discrete Probability Model (p. 21)
• 2.2 Continuous Probability Model (p. 28)
• 2.3 Expectation and Lebesgue Integral (p. 33)
• 2.4 Transforms and Convergence (p. 37)
• 2.5 Independence and Covariance (p. 39)
• 2.6 Normal (Gaussian) Distributions (p. 41)
• 2.7 Conditional Expectation (p. 43)
• 2.8 Stochastic Processes in Continuous Time (p. 47)
• 3 Basic Stochastic Processes (p. 55)
• 3.1 Brownian Motion (p. 56)
• 3.2 Properties of Brownian Motion Paths (p. 63)
• 3.3 Three Martingales of Brownian Motion (p. 65)
• 3.4 Markov Property of Brownian Motion (p. 67)
• 3.5 Hitting Times and Exit Times (p. 69)
• 3.6 Maximum and Minimum of Brownian Motion (p. 71)
• 3.7 Distribution of Hitting Times (p. 73)
• 3.8 Reflection Principle and Joint Distributions (p. 74)
• 3.9 Zeros of Brownian Motion - Arcsine Law (p. 75)
• 3.10 Size of Increments of Brownian Motion (p. 78)
• 3.11 Brownian Motion in Higher Dimensions (p. 81)
• 3.12 Random Walk (p. 81)
• 3.13 Stochastic Integral in Discrete Time (p. 83)
• 3.14 Poisson Process (p. 86)
• 3.15 Exercises (p. 88)
• 4 Brownian Motion Calculus (p. 91)
• 4.1 Definition of Ito Integral (p. 91)
• 4.2 Ito Integral Process (p. 100)
• 4.3 Ito Integral and Gaussian Processes (p. 103)
• 4.4 Ito's Formula for Brownian Motion (p. 106)
• 4.5 Ito Processes and Stochastic Differentials (p. 108)
• 4.6 Ito's Formula for Ito Processes (p. 112)
• 4.7 Ito Processes in Higher Dimensions (p. 118)
• 4.8 Exercises (p. 121)
• 5 Stochastic Differential Equations (p. 123)
• 5.1 Definition of Stochastic Differential Equations (SDEs) (p. 123)
• 5.2 Stochastic Exponential and Logarithm (p. 129)
• 5.3 Solutions to Linear SDEs (p. 131)
• 5.4 Existence and Uniqueness of Strong Solutions (p. 134)
• 5.5 Markov Property of Solutions (p. 136)
• 5.6 Weak Solutions to SDEs (p. 137)
• 5.7 Construction of Weak Solutions (p. 139)
• 5.8 Backward and Forward Equations (p. 144)
• 5.9 Stratonovich Stochastic Calculus (p. 146)
• 5.10 Exercises (p. 148)
• 6 Diffusion Processes (p. 151)
• 6.1 Martingales and Dynkin's Formula (p. 151)
• 6.2 Calculation of Expectations and PDEs (p. 155)
• 6.3 Time-Homogeneous Diffusions (p. 159)
• 6.4 Exit Times from an Interval (p. 163)
• 6.5 Representation of Solutions of ODES (p. 167)
• 6.6 Explosion (p. 168)
• 6.7 Recurrence and Transience (p. 170)
• 6.8 Diffusion on an Interval (p. 171)
• 6.9 Stationary Distributions (p. 172)
• 6.10 Multi-dimensional SDEs (p. 175)
• 6.11 Exercises (p. 183)
• 7 Martingales (p. 185)
• 7.1 Definitions (p. 185)
• 7.2 Uniform Integrability (p. 187)
• 7.3 Martingale Convergence (p. 189)
• 7.4 Optional Stopping (p. 191)
• 7.5 Localization and Local Martingales (p. 197)
• 7.6 Quadratic Variation of Martingales (p. 200)
• 7.7 Martingale Inequalities (p. 203)
• 7.8 Continuous Martingales - Change of Time (p. 205)
• 7.9 Exercises (p. 211)
• 8 Calculus For Semimartingales (p. 213)
• 8.1 Semimartingales (p. 213)
• 8.2 Predictable Processes (p. 214)
• 8.3 Doob-Meyer Decomposition (p. 215)
• 8.4 Integrals with Respect to Semimartingales (p. 217)
• 8.5 Quadratic Variation and Covariation (p. 220)
• 8.6 Ito's Formula for Continuous Semimartingales (p. 222)
• 8.7 Local Times (p. 224)
• 8.8 Stochastic Exponential (p. 226)
• 8.9 Compensators and Sharp Bracket Process (p. 230)
• 8.10 Itô's Formula for Semimartingales (p. 236)
• 8.11 Stochastic Exponential and Logarithm (p. 238)
• 8.12 Martingale (Predictable) Representations (p. 239)
• 8.13 Elements of the General Theory (p. 242)
• 8.14 Random Measures and Canonical Decomposition (p. 246)
• 8.15 Exercises (p. 249)
• 9 Pure Jump Processes (p. 251)
• 9.1 Definitions (p. 251)
• 9.2 Pure Jump Process Filtration (p. 252)
• 9.3 Ito's Formula for Processes of Finite Variation (p. 253)
• 9.4 Counting Processes (p. 254)
• 9.5 Markov Jump Processes (p. 261)
• 9.6 Stochastic Equation for Jump Processes (p. 264)
• 9.7 Generators and Dynkin's Formula (p. 265)
• 9.8 Explosions in Markov Jump Processes (p. 267)
• 9.9 Exercises (p. 268)
• 10 Change of Probability Measure (p. 269)
• 10.1 Change of Measure for Random Variables (p. 269)
• 10.2 Change of Measure on a General Space (p. 273)
• 10.3 Change of Measure for Processes (p. 276)
• 10.4 Change of Wiener Measure (p. 281)
• 10.5 Change of Measure for Point Processes (p. 283)
• 10.6 Likelihood Functions (p. 284)
• 10.7 Exercises (p. 287)
• 11 Applications in Finance: Stock and FX Options (p. 289)
• 11.1 Financial Derivatives and Arbitrage (p. 289)
• 11.2 A Finite Market Model (p. 295)
• 11.3 Semimartingale Market Model (p. 299)
• 11.4 Diffusion and the Black-Scholes Model (p. 304)
• 11.5 Change of Numeraire (p. 312)
• 11.6 Currency (FX) Options (p. 315)
• 11.7 Asian, Lookback, and Barrier Options (p. 318)
• 11.8 Exercises (p. 321)
• 12 Applications in Finance: Bonds, Rates, and Options (p. 325)
• 12.1 Bonds and the Yield Curve (p. 325)
• 12.2 Models Adapted to Brownian Motion (p. 327)
• 12.3 Models Based on the Spot Rate (p. 328)
• 12.4 Merton's Model and Vasicek's Model (p. 329)
• 12.5 Heath-Jarrow-Morton (HJM) Model (p. 333)
• 12.6 Forward Measures - Bond as a Numeraire (p. 338)
• 12.7 Options, Caps, and Floors (p. 341)
• 12.8 Brace-Gatarek-Musiela (BGM) Model (p. 343)
• 12.9 Swaps and Swaptions (p. 347)
• 12.10 Exercises (p. 349)
• 13 Applications in Biology (p. 353)
• 13.1 Feller's Branching Diffusion (p. 353)
• 13.2 Wright-Fisher Diffusion (p. 357)
• 13.3 Birth-Death Processes (p. 359)
• 13.4 Growth of Birth-Death Processes (p. 363)
• 13.5 Extinction, Probability, and Time to Exit (p. 366)
• 13.6 Processes in Genetics (p. 369)
• 13.7 Birth-Death Processes in Many Dimensions (p. 375)
• 13.8 Cancer Models (p. 377)
• 13.9 Branching Processes (p. 379)
• 13.10 Stochastic Lotka-Volterra Model (p. 386)
• 13.11 Exercises (p. 393)
• 14 Applications in Engineering and Physics (p. 395)
• 14.1 Filtering (p. 395)
• 14.2 Random Oscillators (p. 402)
• 14.3 Exercises (p. 408)
• Solutions to Selected Exercises (p. 411)
• References (p. 429)
• Index (p. 435)