The book "Probability and Its Applications" is a comprehensive treatise on Malliavin calculus, published in association with the Applied Probability Trust. It is edited by J. Gani, C.C. Heyde, P. Jagers, and T.G. Kurtz, and authored by David Nualart from the University of Kansas. The second edition, published in 2006, includes two additional chapters on fractional Brownian motion and mathematical finance, reflecting new developments and applications since the first edition. The book covers the following main topics: 1. **Analysis on the Wiener Space**: Introduces the derivative and divergence operators in the context of an isonormal Gaussian process and Hilbert space. 2. **Regularity of Probability Laws**: Discusses the smoothness of probability densities, stochastic differential equations, and hypoellipticity. 3. **Anticipating Stochastic Calculus**: Explores the Skorohod integral, substitution formulas, and anticipating stochastic differential equations. 4. **Transformations of the Wiener Measure**: Covers anticipating Girsanov theorems and Markov random fields. 5. **Fractional Brownian Motion**: Provides definitions, properties, and stochastic calculus techniques for this self-similar Gaussian process. 6. **Malliavin Calculus in Finance**: Applications to the Black-Scholes model, computation of "greeks," hedging, and insider trading. The book is designed for graduate students and researchers in probability theory and stochastic processes, assuming familiarity with Itô stochastic calculus and martingale theory. It serves as a valuable resource for both theoretical and applied research in the field.The book "Probability and Its Applications" is a comprehensive treatise on Malliavin calculus, published in association with the Applied Probability Trust. It is edited by J. Gani, C.C. Heyde, P. Jagers, and T.G. Kurtz, and authored by David Nualart from the University of Kansas. The second edition, published in 2006, includes two additional chapters on fractional Brownian motion and mathematical finance, reflecting new developments and applications since the first edition. The book covers the following main topics: 1. **Analysis on the Wiener Space**: Introduces the derivative and divergence operators in the context of an isonormal Gaussian process and Hilbert space. 2. **Regularity of Probability Laws**: Discusses the smoothness of probability densities, stochastic differential equations, and hypoellipticity. 3. **Anticipating Stochastic Calculus**: Explores the Skorohod integral, substitution formulas, and anticipating stochastic differential equations. 4. **Transformations of the Wiener Measure**: Covers anticipating Girsanov theorems and Markov random fields. 5. **Fractional Brownian Motion**: Provides definitions, properties, and stochastic calculus techniques for this self-similar Gaussian process. 6. **Malliavin Calculus in Finance**: Applications to the Black-Scholes model, computation of "greeks," hedging, and insider trading. The book is designed for graduate students and researchers in probability theory and stochastic processes, assuming familiarity with Itô stochastic calculus and martingale theory. It serves as a valuable resource for both theoretical and applied research in the field.