This book is an introduction to Malliavin calculus as a generalization of the classical non-anticipating Ito calculus to an anticipating setting. It presents the development of the theory and its use in new fields of application.

While the original works on Malliavin calculus aimed to study the smoothness of densities of solutions to stochastic differential equations, this book has another goal. It portrays the most important and innovative applications in stochastic control and finance, such as hedging in complete and incomplete markets, optimisation in the presence of asymmetric information and also pricing and sensitivity analysis. In a self-contained fashion, both the Malliavin calculus with respect to Brownian motion and general Lévy type of noise are treated. Besides, forward integration is included and indeed extended to general Lévy processes. The forward integration is a recent development within anticipative stochastic calculus that, together with the Malliavin calculus, provides new methods for the study of insider trading problems. To allow more flexibility in the treatment of the mathematical tools, the generalization of Malliavin calculus to the white noise framework is also discussed. This book is a valuable resource for graduate students, lecturers in stochastic analysis and applied researchers.

IntroductionINTEGRATION BY PARTS AND ABSOLUTE CONTINUITY OF PROBABILITY LAWSFINITE DIMENSIONAL MALLIAVIN CALCULUSThe Ornstein-Uhlenbeck OperatorThe Adjoint of the differentialAn Interration by Parts Fromula: Existence of a DensityTHE BASIC OPERATORS OF MALLIAVIN CALCULUSThe Ornstein-Uhlenbeck OperatorThe Derivative OperatorThe Integral or Divergence OperatorDifferential CalculusCalculus with Multiple Wiener IntergralsLocal Property of the OperatorsREPRESENTATION OF WIENER FUNCTIONALThe Ito Integral and the Divergence OperatorThe Cark-Ocone FormulaGeneralized Clark-Ocone FormulaApplication to Option PricingCRITERIA FOR ABSOLUTE CONTINUITY AND SMOOTHNESS OF PROBABILITY LAWSExistence of a DensitySmoothness of the DensitySTOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS DRIVEN BY SPATIALLY HOMOGENEOUS GAUSSIAN NOISEStochastic Integration with Respect to Coloured NoiseStochastic Partial Differential Equations Driven by a Coloured NoiseMALLIAVIN REGULARITY OF SOLUTIONS OF SPDEsANALYSIS OF THE MALLIAVIN MATRIC OF SOLUTIONS OF SPDEsOne Dimensional CaseExamplesMultidimensional CaseDEFINITION OF SPACES USED THROUGHOUT THE COURSE.

After functional, measure and stochastic analysis prerequisites, the author covers chaos decomposition, Skorohod integral processes, Malliavin derivative and Girsanov transformations.

This book introduces the study of stochastic volatility (SV) models via Malliavin Calculus. Malliavin calculus has had a profound impact on stochastic analysis. It shows that Malliavin calculus is an easy-to-apply tool that allows us to recover, unify, and generalize several previous results in the literature on SV modeling.

The Malliavin calculus was developed to provide a probabilistic proof of Hormander's hypoellipticity theorem. The theory has expanded to encompass other significant applications. The main application of the Malliavin calculus is to establish the regularity of the probability distribution of functionals of an underlying Gaussian process. In this way, one can prove the existence and smoothness of the density for solutions of various stochastic differential equations. More recently, applications of the Malliavin calculus in areas such as stochastic calculus for fractional Brownian motion, central limit theorems for multiple stochastic integrals, and mathematical finance have emerged. The first part of the book covers the basic results of the Malliavin calculus. The middle part establishes the existence and smoothness results that then lead to the proof of Hormander's hypoellipticity theorem. The last part discusses the recent developments for Brownian motion, central limit theorems, and mathematical finance.