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Full Description
The book constitutes an introduction to stochastic calculus, stochastic differential equations and related topics such as Malliavin calculus. On the other hand it focuses on the techniques of stochastic integration and calculus via regularization initiated by the authors. The definitions relies on a smoothing procedure of the integrator process, they generalize the usual Itô and Stratonovich integrals for Brownian motion but the integrator could also not be a semimartingale and the integrand is allowed to be anticipating. The resulting calculus requires a simple formalism: nevertheless it entails pathwise techniques even though it takes into account randomness. It allows connecting different types of pathwise and non pathwise integrals such as Young, fractional, Skorohod integrals, enlargement of filtration and rough paths. The covariation, but also high order variations, play a fundamental role in the calculus via regularization, which can also be applied for irregularintegrators. A large class of Gaussian processes, various generalizations of semimartingales such that Dirichlet and weak Dirichlet processes are revisited. Stochastic calculus via regularization has been successfully used in applications, for instance in robust finance and on modeling vortex filaments in turbulence. The book is addressed to PhD students and researchers in stochastic analysis and applications to various fields.
Contents
- 1. Review on Basic Probability Theory. - 2. Processes, Brownian Motion and Martingales. - 3. Fractional Brownian Motion and Related Processes. - 4. Stochastic Integration via Regularization. - 5. Itô Integrals. - 6. Stability of the Covariation and Itô's Formula. - 7. Change of probability and martingale representation. - 8. About finite quadratic variation: examples. - 9. Hermite Polynomials and Wiener Chaos. - 10. Elements of Wiener Analysis. - 11. Elements of Non-causal Calculus. - 12. Itô Classical Stochastic Differential Equations. - 13. Itô SDEs with Non-Lipschitz Coefficients. - 14. Föllmer-Dirichlet Processes. - 15. Weak Dirichlet Processes. - Stochastic Calculus with n-Covariations. - Calculus via Regularization and Rough Paths.
